On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      2. Multiplicity of solutions for a fourth order equation with power-type nonlinearity      Juan Dávila  Isabel Flores  Ignacio Guerra 《Mathematische Annalen》2010,348(1):143-193 Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbRN{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      3. Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems      Mikil Foss  Antonia Passarelli di Napoli  Anna Verde 《Annali di Matematica Pura ed Applicata》2010,189(1):127-162 We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rn{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      4. Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity      Jiabao Su 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,21(2):51-62 We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation {ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ￥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.      5. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      6. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      7. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      8. Two-dimensional variational problems with a wide range of anisotropy      M. Fuchs 《Journal of Mathematical Sciences》2011,175(3):375-389 We consider local minimizers u:\mathbbR2 é W? \mathbbRM u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral òW H( ?u )dx \int\limits_\Omega {H\left( {\nabla u} \right)dx}      9. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      12. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      13. Some estimates of Schrödinger type operators on the Heisenberg group      Yu Liu  Jizheng Huang  Dongmei Xie 《Archiv der Mathematik》2010,94(3):255-264 In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D\mathbb Hn)2 +V 2{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class Bq1 for q1 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D\mathbb Hn{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hn{\mathbb {H}^n}. An L p estimate and a weak type L 1 estimate for the operator ?4\mathbb Hn H-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? Bq1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq12{1 < p \leq \frac{q_{1}}{2}} are obtained.      14. Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems      M. F. Betta  F. Chiacchio  A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):37-52 We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( guxi )xi = lgu \textin W ì \mathbbRn - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRn \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.      15. Uniqueness of a solution to the problem of atmosphere dynamics      A. V. Gorshkov 《Journal of Mathematical Sciences》2010,167(3):340-357 We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm || f ||V(Q)2 = \textvrai  supt ? [ 0,T ] || f ||L2( W)2 + || ?3f ||L2(Q)2. \left\| f \right\|_{V(Q)}^2 = \mathop {{\text{vrai}}\,{ \sup }}\limits_{t \in \left[ {0,T} \right]} \left\| f \right\|_{{L_2}\left( \Omega \right)}^2 + \left\| {{\nabla_3}f} \right\|_{{L_2}(Q)}^2.      16. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      17. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      18. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      19. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      20. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      3. Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems      Mikil Foss  Antonia Passarelli di Napoli  Anna Verde 《Annali di Matematica Pura ed Applicata》2010,189(1):127-162 We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rn{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      4. Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity      Jiabao Su 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,21(2):51-62 We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation {ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ￥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.      5. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      6. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      7. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      8. Two-dimensional variational problems with a wide range of anisotropy      M. Fuchs 《Journal of Mathematical Sciences》2011,175(3):375-389 We consider local minimizers u:\mathbbR2 é W? \mathbbRM u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral òW H( ?u )dx \int\limits_\Omega {H\left( {\nabla u} \right)dx}      9. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      12. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      13. Some estimates of Schrödinger type operators on the Heisenberg group      Yu Liu  Jizheng Huang  Dongmei Xie 《Archiv der Mathematik》2010,94(3):255-264 In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D\mathbb Hn)2 +V 2{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class Bq1 for q1 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D\mathbb Hn{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hn{\mathbb {H}^n}. An L p estimate and a weak type L 1 estimate for the operator ?4\mathbb Hn H-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? Bq1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq12{1 < p \leq \frac{q_{1}}{2}} are obtained.      14. Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems      M. F. Betta  F. Chiacchio  A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):37-52 We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( guxi )xi = lgu \textin W ì \mathbbRn - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRn \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.      15. Uniqueness of a solution to the problem of atmosphere dynamics      A. V. Gorshkov 《Journal of Mathematical Sciences》2010,167(3):340-357 We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm || f ||V(Q)2 = \textvrai  supt ? [ 0,T ] || f ||L2( W)2 + || ?3f ||L2(Q)2. \left\| f \right\|_{V(Q)}^2 = \mathop {{\text{vrai}}\,{ \sup }}\limits_{t \in \left[ {0,T} \right]} \left\| f \right\|_{{L_2}\left( \Omega \right)}^2 + \left\| {{\nabla_3}f} \right\|_{{L_2}(Q)}^2.      16. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      17. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      18. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      19. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      20. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Global Lipschitz regularity for almost minimizers of asymptotically convex variational problems

We prove global, up to the boundary of a domain ${{\it \Omega}\subset\mathbb {R}^n}We prove global, up to the boundary of a domain W ì \mathbb Rⁿ{{\it \Omega}\subset\mathbb {R}^n}, Lipschitz regularity results for almost minimizers of functionals of the form

u ? òW g(x, u(x), ?u(x)) dx.u \mapsto \int \limits_{\Omega} g(x, u(x), \nabla u(x))\,dx.      4. Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity      Jiabao Su 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2012,21(2):51-62 We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation {ll-div(|?u|p-2?u)+V(|x|)|u|p-2u=Q(|x|)f(u),    x ? \mathbbRN,u(x) ? 0,     |x|? ￥\left\{\begin{array}{ll}-{div}(|\nabla u|^{p-2}\nabla u)+V(|x|)|u|^{p-2}u=Q(|x|)f(u),\quad x\in \mathbb{R}^N,\\u(x) \rightarrow 0, \quad |x|\rightarrow \infty \end{array}\right.      5. Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential      Mónica Clapp  Andrzej Szulkin 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):229-248 We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      6. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      7. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      8. Two-dimensional variational problems with a wide range of anisotropy      M. Fuchs 《Journal of Mathematical Sciences》2011,175(3):375-389 We consider local minimizers u:\mathbbR2 é W? \mathbbRM u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral òW H( ?u )dx \int\limits_\Omega {H\left( {\nabla u} \right)dx}      9. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      12. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      13. Some estimates of Schrödinger type operators on the Heisenberg group      Yu Liu  Jizheng Huang  Dongmei Xie 《Archiv der Mathematik》2010,94(3):255-264 In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D\mathbb Hn)2 +V 2{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class Bq1 for q1 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D\mathbb Hn{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hn{\mathbb {H}^n}. An L p estimate and a weak type L 1 estimate for the operator ?4\mathbb Hn H-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? Bq1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq12{1 < p \leq \frac{q_{1}}{2}} are obtained.      14. Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems      M. F. Betta  F. Chiacchio  A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):37-52 We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( guxi )xi = lgu \textin W ì \mathbbRn - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRn \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.      15. Uniqueness of a solution to the problem of atmosphere dynamics      A. V. Gorshkov 《Journal of Mathematical Sciences》2010,167(3):340-357 We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm || f ||V(Q)2 = \textvrai  supt ? [ 0,T ] || f ||L2( W)2 + || ?3f ||L2(Q)2. \left\| f \right\|_{V(Q)}^2 = \mathop {{\text{vrai}}\,{ \sup }}\limits_{t \in \left[ {0,T} \right]} \left\| f \right\|_{{L_2}\left( \Omega \right)}^2 + \left\| {{\nabla_3}f} \right\|_{{L_2}(Q)}^2.      16. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      17. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      18. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      19. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      20. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Quasilinear elliptic equations on \mathbbRN{\mathbb{R}^{N}} with singular potentials and bounded nonlinearity

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equation

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

We consider the magnetic nonlinear Schrödinger equations $\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}$ in ${\Omega=\mathcal{O}\times \mathbb{R}}We consider the magnetic nonlinear Schr?dinger equations

ll(-i?+ sA)2 u + u   =  |u|p-2 u,     p ? (2, 6),         (-i?+ sA) 2u   =  |u|4 u,\begin{array}{ll}{\left(-i\nabla + sA\right)^{2} u + u \, = \, |u|^{p-2}\, u, \quad p \in (2, 6),} \\ \quad \quad {\left(-i\nabla + sA\right) ^{2}u \, = \, |u|^{4}\, u,}\end{array}      6. Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres      Ali Taheri 《NoDEA : Nonlinear Differential Equations and Applications》2012,19(1):79-95 Let X ì \mathbb Rn{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional \mathbb E[u; X]:=\frac12 ò\nolimitsX |?u (x)|2  dx, {\mathbb E}[u; {\bf X}]:=\frac{1}{2} \int\nolimits_{\bf X} |\nabla u (x)|^2 \, dx,      7. Global existence and uniqueness for the magnetic Hartree equation      Pei Cao 《Journal of Evolution Equations》2011,11(4):811-825 In this paper, we consider lliut=Hu+\frac1|x|*|u|2u,    (x,t) ? \mathbbRN×\mathbbR.\begin{array}{ll}iu_{t}=Hu+\frac{1}{|x|}*|u|^{2}u,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}.\end{array}      8. Two-dimensional variational problems with a wide range of anisotropy      M. Fuchs 《Journal of Mathematical Sciences》2011,175(3):375-389 We consider local minimizers u:\mathbbR2 é W? \mathbbRM u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral òW H( ?u )dx \int\limits_\Omega {H\left( {\nabla u} \right)dx}      9. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      12. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      13. Some estimates of Schrödinger type operators on the Heisenberg group      Yu Liu  Jizheng Huang  Dongmei Xie 《Archiv der Mathematik》2010,94(3):255-264 In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D\mathbb Hn)2 +V 2{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class Bq1 for q1 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D\mathbb Hn{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hn{\mathbb {H}^n}. An L p estimate and a weak type L 1 estimate for the operator ?4\mathbb Hn H-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? Bq1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq12{1 < p \leq \frac{q_{1}}{2}} are obtained.      14. Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems      M. F. Betta  F. Chiacchio  A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):37-52 We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( guxi )xi = lgu \textin W ì \mathbbRn - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRn \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.      15. Uniqueness of a solution to the problem of atmosphere dynamics      A. V. Gorshkov 《Journal of Mathematical Sciences》2010,167(3):340-357 We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm || f ||V(Q)2 = \textvrai  supt ? [ 0,T ] || f ||L2( W)2 + || ?3f ||L2(Q)2. \left\| f \right\|_{V(Q)}^2 = \mathop {{\text{vrai}}\,{ \sup }}\limits_{t \in \left[ {0,T} \right]} \left\| f \right\|_{{L_2}\left( \Omega \right)}^2 + \left\| {{\nabla_3}f} \right\|_{{L_2}(Q)}^2.      16. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      17. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      18. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      19. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      20. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Spherical twists, stationary paths and harmonic maps from generalised annuli into spheres

Let X ì \mathbb Rⁿ{{\bf X} \subset {\mathbb R}^n} be a generalised annulus and consider the Dirichlet energy functional

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider

Two-dimensional variational problems with a wide range of anisotropy

We consider local minimizers u:\mathbbR² é W? \mathbbR^M u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      10. On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}      Manass��s de Souza 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(2):199-215 In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbRN{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form -DN u+ V(x)|u|N-2u=\fracf(x,u)|x|a   in  \mathbbRN,    N 3 2,-\Delta_N\,u+ V(x)|u|^{N-2}u=\frac{f(x,u)}{|x|^a}\quad{\rm in} \, \mathbb{R}^N,\quad N\geq 2,      11. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      12. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      13. Some estimates of Schrödinger type operators on the Heisenberg group      Yu Liu  Jizheng Huang  Dongmei Xie 《Archiv der Mathematik》2010,94(3):255-264 In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D\mathbb Hn)2 +V 2{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class Bq1 for q1 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D\mathbb Hn{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hn{\mathbb {H}^n}. An L p estimate and a weak type L 1 estimate for the operator ?4\mathbb Hn H-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? Bq1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq12{1 < p \leq \frac{q_{1}}{2}} are obtained.      14. Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems      M. F. Betta  F. Chiacchio  A. Ferone 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,58(2):37-52 We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( guxi )xi = lgu \textin W ì \mathbbRn - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRn \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.      15. Uniqueness of a solution to the problem of atmosphere dynamics      A. V. Gorshkov 《Journal of Mathematical Sciences》2010,167(3):340-357 We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm || f ||V(Q)2 = \textvrai  supt ? [ 0,T ] || f ||L2( W)2 + || ?3f ||L2(Q)2. \left\| f \right\|_{V(Q)}^2 = \mathop {{\text{vrai}}\,{ \sup }}\limits_{t \in \left[ {0,T} \right]} \left\| f \right\|_{{L_2}\left( \Omega \right)}^2 + \left\| {{\nabla_3}f} \right\|_{{L_2}(Q)}^2.      16. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      17. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      18. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      19. Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient      M. Fuchs 《Journal of Mathematical Sciences》2010,167(3):418-434 We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR2 é W? \mathbbR2 u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form òW h( | eD(u) | )dx \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx}      20. Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators      A. A. Albanese  D. Jornet  A. Oliaro 《Integral Equations and Operator Theory》2010,66(2):153-181 In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF *, of classical distributions. In particular, we have the following inclusion WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On a singular elliptic problem involving critical growth in {\mathbb{R}^{\bf N}}

In this paper, we prove a suitable Trudinger–Moser inequality with a singular weight in \mathbbR^N{\mathbb{R}^N} and as an application of this result, using the mountain-pass theorem we establish sufficient conditions for the existence of nontrivial solutions to quasilinear elliptic partial differential equations of the form

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Some estimates of Schrödinger type operators on the Heisenberg group

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D_{\mathbb H}ⁿ)² +V ²{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class B_q1 for q₁ 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D_{\mathbb Hⁿ}{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hⁿ{\mathbb {H}^n}. An L ^p estimate and a weak type L ¹ estimate for the operator ?⁴_{\mathbb Hⁿ} H^-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? B_q1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq₁2{1 < p \leq \frac{q_{1}}{2}} are obtained.     

Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems

We find some optimal estimates for the first eigenfunction of a class of elliptic equations whose prototype is - ( gu_xi )_xi = lgu \textin W ì \mathbbRⁿ - {\left( {\gamma u_{{x_{i} }} } \right)}_{{x_{i} }} = \lambda \gamma u\,{\text{in}}\,\Omega \subset \mathbb{R}^{n} with Dirichlet boundary condition, where γ is the normalized Gaussian function in \mathbbRⁿ \mathbb{R}^{n} . To this aim we make use of the Gaussian symmetrization which transforms a domain into an half-space with the same Gaussian measure. The main tools we use are the properties of the weighted rearrangements and in particular the isoperimetric inequality with respect to Gaussian measure.     

Uniqueness of a solution to the problem of atmosphere dynamics

We consider the model of atmosphere dynamics and prove the uniqueness of a solution in a bounded domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} in the space V(Q) of weak solutions equipped with the finite norm

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbF_m,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbF_m,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbF_m,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbF_m,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?_j=1^m X_j²{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d ^2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient

We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields u:\mathbbR² é W? \mathbbR² u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form

Quasianalytic Wave Front Sets for Solutions of Linear Partial Differential Operators

In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion $WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),$ where Ω is an open subset of ${\mathbb {R}^n}In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, WF _*, of classical distributions. In particular, we have the following inclusion

WF*(u) ì WF*(Pu) èS,     u ? D￠(W),WF_{*}(u) \subset WF_{*}(Pu) \cup \Sigma, \quad u \in \mathcal {D}^{\prime}(\Omega),