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.      2. 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}}.      3. Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $      Saïd Benachour  Philippe Laurenccedil;ot 《Journal of Evolution Equations》2003,3(1):27-37 The large time behaviour of the Lq L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ ut - Du + ?i=1m |uxi|pi = 0      in   \mathbbR+×\mathbbRN,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=￥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and pi ? [1,+￥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L1 L^1 -norm is identified, and temporal decay estimates for the L￥ L^\infty -norm are obtained, according to the values of the pi p_i 's. The main tool in our approach is the derivation of L￥ L^\infty -decay estimates for ?(ua ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.      4. Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces      K. Kh. Boimatov 《Journal of Mathematical Sciences》2002,108(4):543-573 Let W ì \mathbbRn \Omega \subset \mathbb{R}^n be an open set and l(x) | u |p,l = ( òW lp (x)| u(x) |p dx )1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}      5. On the Euler–Lagrange equation for a variational problem: the general case II      Stefano Bianchini  Matteo Gloyer 《Mathematische Zeitschrift》2010,265(4):889-923 In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L￥loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      6. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      7. Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition      Mikhail Perepelitsa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,114(1):267-276 We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR3+{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ 0 is bounded and the magnitude of the initial velocity u 0 is suitably restricted in the norm ||?{r0(·)}u0(·)||L2(W) + ||?u0(·)||L2(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.      8. Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces      V. S. Guliyev  J. J. Hasanov  S. G. Samko 《Journal of Mathematical Sciences》2010,170(4):423-443 We consider generalized Morrey type spaces Mp( ·),q( ·),w( ·)( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRn \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem Mp( ·),q1( ·),w1( ·)( W) ? Mq( ·),q2( ·),w2( ·)( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.      9. On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations      N. M. Ivochkina 《Journal of Mathematical Sciences》2010,170(4):496-509 We construct an a priori estimate of the seminorm á uxx ña, [`(W)] {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem Fm[ u ] = f;    u |?W = F {F_m}\left[ u \right] = f;\quad \left. u \right|{_{\partial \Omega }} = \Phi      10. 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.      11. 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}      12. Embeddings between weighted Copson and Cesàro function spaces      Amiran Gogatishvili  Rza Mustafayev  Tuğçe Ünver 《Czechoslovak Mathematical Journal》2017,67(4):1105-1132 In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality $${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$ where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.       13. A regularity criterion for axially symmetric solutions to the Navier–Stokes equations      W. M. Zaja̧czkowski 《Journal of Mathematical Sciences》2011,178(3):265-273 The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v r ) of the velocity belongs to L ∞(0, T;L 3(Ω0)) or v r /r belongs to L ∞(0, T;L 3/2(Ω0)), where Ω0 is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω k , k = 1, . . . , N, such that W0 ì èk = 1N Wk {\Omega_0} \subset \bigcup\limits_{k = 1}^N {{\Omega_k}} , and assume that there exist constants α 1, α 2 such that either || vr ||L￥ ( 0,T;L3( Wk ) ) ￡ a1 or  || \fracvrr ||L￥ ( 0,T;L3/2( Wk ) ) ￡ a2 {\left\| {{v_r}} \right\|_{{L_\infty }\left( {0,T;{L_3}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_1}\,or\;{\left\| {\frac{{{v_r}}}{r}} \right\|_{{L_\infty }\left( {0,T;{L_{3/2}}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W22,1( W×( 0,T ) ),?p ? L2( W×( 0,T ) ) ) \left( {v \in W_2^{2,1}\left( {\Omega \times \left( {0,T} \right)} \right),\nabla p \in {L_2}\left( {\Omega \times \left( {0,T} \right)} \right)} \right) . Bibliography: 28 titles.      14. Ultraproducts of real interpolation spaces between L p -spaces      J. A. López Molina  M. E. Puerta  M. J. Rivera 《Bulletin of the Brazilian Mathematical Society》2006,37(2):191-216 Let , be a family of compatible couples of Lp-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω3) defined on some purely non atomic measure space (Ω3, ν3) in such a way that Ω2 ∪ Ω3 ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.      15. Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient      M. Fuchs  S. Repin 《Journal of Mathematical Sciences》2011,178(3):367-372 It is proved that if Ω ⊂ Rn {R^n}  is a bounded Lipschitz domain, then the inequality || u ||1 \leqslant c(n)\textdiam( W)òW | eD(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here eD(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and òW | eD(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure eD(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.      16. 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}      17. Interior Differentiability Results for Nonlinear Variational Parabolic Systems with Nonlinearity 1 <?q < 2      Luisa Fattorusso  Mario Marino 《Mediterranean Journal of Mathematics》2011,8(1):97-112 In this work we prove that the solutions u ? Lq(-T, 0,H1,q(W,\mathbbRN)) ?C0,l([`(Q)],\mathbbRN)u\in L^{q}(-T, 0,H^{1,q}(\Omega,\mathbb{R}^{N})) \cap C^{0,\lambda}(\overline{Q},\mathbb{R}^{N})      18. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      19. Triple Hilbert Transforms Along Polynomial Surfaces      Yong-Kum Cho  Sunggeum Hong  Joonil Kim  Chan Woo Yang 《Integral Equations and Operator Theory》2009,65(4):485-528 Given W ì \mathbbZ+3\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in Lp(\mathbbR4)L^p({\mathbb{R}}^4).      20. 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.      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

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}}.     

Decay estimates for "anisotropic" viscous Hamilton-Jacobi equations in $ \mathbb{R}^{N} $

The large time behaviour of the L^q L^q -norm of nonnegative solutions to the "anisotropic" viscous Hamilton-Jacobi equation¶¶ u_t - Du + ?_i=1^m |u_xi|^p_i = 0      in   \mathbbR₊×\mathbbR^N,u_t - \Delta u + \sum_{i=1}^m \vert u_{x_i}\vert^{p_i} = 0 \;\;\mbox{ in }\; {\mathbb{R}}_+\times{\mathbb{R}}^N,¶¶is studied for q=1 q=1 and q=￥ q=\infty , where m ? {1,?,N} m\in\{1,\ldots,N\} and p_i ? [1,+￥) p_i\in [1,+\infty) for i ? {1,?,m} i\in\{1,\ldots,m\} . The limit of the L¹ L^1 -norm is identified, and temporal decay estimates for the L^￥ L^\infty -norm are obtained, according to the values of the p_i p_i 's. The main tool in our approach is the derivation of L^￥ L^\infty -decay estimates for ?(u^a ), a ? (0,1] \nabla\left(u^\alpha \right), \alpha\in (0,1] , by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.     

Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Let W ì \mathbbRⁿ \Omega \subset \mathbb{R}^n be an open set and l(x) | u |_p,l = ( ò_W l^p (x)| u(x) |^p dx )^1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      6. Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential      Y. Belaud  A. Shishkov 《Journal of Evolution Equations》2010,10(4):857-882 We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò01 s-1 (meas\nolimits {x ? W: |a(x)| ￡ s })q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.      7. Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition      Mikhail Perepelitsa 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,114(1):267-276 We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR3+{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ 0 is bounded and the magnitude of the initial velocity u 0 is suitably restricted in the norm ||?{r0(·)}u0(·)||L2(W) + ||?u0(·)||L2(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.      8. Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces      V. S. Guliyev  J. J. Hasanov  S. G. Samko 《Journal of Mathematical Sciences》2010,170(4):423-443 We consider generalized Morrey type spaces Mp( ·),q( ·),w( ·)( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRn \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem Mp( ·),q1( ·),w1( ·)( W) ? Mq( ·),q2( ·),w2( ·)( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.      9. On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations      N. M. Ivochkina 《Journal of Mathematical Sciences》2010,170(4):496-509 We construct an a priori estimate of the seminorm á uxx ña, [`(W)] {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem Fm[ u ] = f;    u |?W = F {F_m}\left[ u \right] = f;\quad \left. u \right|{_{\partial \Omega }} = \Phi      10. 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.      11. 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}      12. Embeddings between weighted Copson and Cesàro function spaces      Amiran Gogatishvili  Rza Mustafayev  Tuğçe Ünver 《Czechoslovak Mathematical Journal》2017,67(4):1105-1132 In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality $${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$ where p1, p2, q1, q2 ∈ (0,∞), p2 ≤ q2 and u1, u2, v1, v2 are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p1 and p2 and possibly different inner weights v1 and v2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.       13. A regularity criterion for axially symmetric solutions to the Navier–Stokes equations      W. M. Zaja̧czkowski 《Journal of Mathematical Sciences》2011,178(3):265-273 The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v r ) of the velocity belongs to L ∞(0, T;L 3(Ω0)) or v r /r belongs to L ∞(0, T;L 3/2(Ω0)), where Ω0 is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω k , k = 1, . . . , N, such that W0 ì èk = 1N Wk {\Omega_0} \subset \bigcup\limits_{k = 1}^N {{\Omega_k}} , and assume that there exist constants α 1, α 2 such that either || vr ||L￥ ( 0,T;L3( Wk ) ) ￡ a1 or  || \fracvrr ||L￥ ( 0,T;L3/2( Wk ) ) ￡ a2 {\left\| {{v_r}} \right\|_{{L_\infty }\left( {0,T;{L_3}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_1}\,or\;{\left\| {\frac{{{v_r}}}{r}} \right\|_{{L_\infty }\left( {0,T;{L_{3/2}}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W22,1( W×( 0,T ) ),?p ? L2( W×( 0,T ) ) ) \left( {v \in W_2^{2,1}\left( {\Omega \times \left( {0,T} \right)} \right),\nabla p \in {L_2}\left( {\Omega \times \left( {0,T} \right)} \right)} \right) . Bibliography: 28 titles.      14. Ultraproducts of real interpolation spaces between L p -spaces      J. A. López Molina  M. E. Puerta  M. J. Rivera 《Bulletin of the Brazilian Mathematical Society》2006,37(2):191-216 Let , be a family of compatible couples of Lp-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω3) defined on some purely non atomic measure space (Ω3, ν3) in such a way that Ω2 ∪ Ω3 ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.      15. Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient      M. Fuchs  S. Repin 《Journal of Mathematical Sciences》2011,178(3):367-372 It is proved that if Ω ⊂ Rn {R^n}  is a bounded Lipschitz domain, then the inequality || u ||1 \leqslant c(n)\textdiam( W)òW | eD(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here eD(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and òW | eD(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure eD(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.      16. 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}      17. Interior Differentiability Results for Nonlinear Variational Parabolic Systems with Nonlinearity 1 <?q < 2      Luisa Fattorusso  Mario Marino 《Mediterranean Journal of Mathematics》2011,8(1):97-112 In this work we prove that the solutions u ? Lq(-T, 0,H1,q(W,\mathbbRN)) ?C0,l([`(Q)],\mathbbRN)u\in L^{q}(-T, 0,H^{1,q}(\Omega,\mathbb{R}^{N})) \cap C^{0,\lambda}(\overline{Q},\mathbb{R}^{N})      18. Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces      Markus Haase 《Integral Equations and Operator Theory》2006,56(2):197-228 We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( Aa ) ?R( Aa ) )q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as { x  ?  X|t - q\textRea y1 ( tA )x, t - q\textRea y2 ( tA )x ? L*p ( ( 0,￥ );X ) } {\left\{ {x\, \in \,X|t^{{ - \theta {\text{Re}}\alpha }} \psi _{1} {\left( {tA} \right)}x,\,t^{{ - \theta {\text{Re}}\alpha }} \psi _{2} {\left( {tA} \right)}x \in L_{*}^{p} {\left( {{\left( {0,\infty } \right)};X} \right)}} \right\}}      19. Triple Hilbert Transforms Along Polynomial Surfaces      Yong-Kum Cho  Sunggeum Hong  Joonil Kim  Chan Woo Yang 《Integral Equations and Operator Theory》2009,65(4):485-528 Given W ì \mathbbZ+3\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in Lp(\mathbbR4)L^p({\mathbb{R}}^4).      20. 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.      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

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

On the Hölder constant for the second order derivatives of admissible solutions to m-Hessian equations

We construct an a priori estimate of the seminorm á u_xx ñ_{a, [`(W)]} {\left\langle {{u_{xx}}} \right\rangle_{\alpha, \bar{\Omega }}} for solutions to the problem

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.     

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

Embeddings between weighted Copson and Cesàro function spaces

In this paper, characterizations of the embeddings between weighted Copson function spaces \(Co{p_{{p_1},{q_1}}}\left( {{u_1},{v_1}} \right)\) and weighted Cesàro function spaces \(Ce{s_{{p_2},{q_2}}}\left( {{u_2},{v_2}} \right)\) are given. In particular, two-sided estimates of the optimal constant c in the inequality

$${\left( {\int_0^\infty {{{\left( {\int_0^t {f{{\left( \tau \right)}^{{p_2}}}{v_2}\left( \tau \right)d\tau } } \right)}^{{q_2}/{p_2}}}{u_2}\left( t \right)dt} } \right)^{1/{q_2}}} \leqslant c{\left( {\int_0^\infty {{{\left( {\int_t^\infty {f{{\left( \tau \right)}^{{p_1}}}{v_1}\left( \tau \right)d\tau } } \right)}^{{q_1}/{p_1}}}{u_1}\left( t \right)dt} } \right)^{1/{q_1}}},$$

where p₁, p₂, q₁, q₂ ∈ (0,∞), p₂ ≤ q₂ and u₁, u₂, v₁, v₂ are weights on (0,∞), are obtained. The most innovative part consists of the fact that possibly different parameters p₁ and p₂ and possibly different inner weights v₁ and v₂ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

     

A regularity criterion for axially symmetric solutions to the Navier–Stokes equations

The axially symmetric solutions to the Navier–Stokes equations are studied. Assume that either the radial component (v _r ) of the velocity belongs to L _∞(0, T;L ₃(Ω₀)) or v _r /r belongs to L _∞(0, T;L _3/2(Ω₀)), where Ω₀ is a neighborhood of the axis of symmetry. Assume additionally that there exist subdomains Ω _k , k = 1, . . . , N, such that W₀ ì è_{k = 1}^N W_k {\Omega_0} \subset \bigcup\limits_{k = 1}^N {{\Omega_k}} , and assume that there exist constants α ₁, α ₂ such that either || v_r ||_{L￥ ( 0,T;L3( Wk ) )} ￡ a₁ or  || \fracv_rr ||_{L￥ ( 0,T;L3/2( Wk ) )} ￡ a₂ {\left\| {{v_r}} \right\|_{{L_\infty }\left( {0,T;{L_3}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_1}\,or\;{\left\| {\frac{{{v_r}}}{r}} \right\|_{{L_\infty }\left( {0,T;{L_{3/2}}\left( {{\Omega_k}} \right)} \right)}} \leq {\alpha_2} for k = 1, . . . , N. Then the weak solution becomes strong ( v ? W₂^2,1( W×( 0,T ) ),?p ? L₂( W×( 0,T ) ) ) \left( {v \in W_2^{2,1}\left( {\Omega \times \left( {0,T} \right)} \right),\nabla p \in {L_2}\left( {\Omega \times \left( {0,T} \right)} \right)} \right) . Bibliography: 28 titles.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

Some Poincaré-type inequalities for functions of bounded deformation involving the deviatoric part of the symmetric gradient

It is proved that if Ω ⊂ Rⁿ {R^n}  is a bounded Lipschitz domain, then the inequality || u ||₁ \leqslant c(n)\textdiam( W)ò_W | e^D(u) | {\left\| u \right\|_1} \leqslant c(n){\text{diam}}\left( \Omega \right)\int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} is valid for functions of bounded deformation vanishing on ∂Ω. Here e^D(u) {\varepsilon^D}(u) denotes the deviatoric part of the symmetric gradient and ò_W | e^D(u) | \int\limits_\Omega {\left| {{\varepsilon^D}(u)} \right|} stands for the total variation of the tensor-valued measure e^D(u) {\varepsilon^D}(u) . Further results concern possible extensions of this Poincaré-type inequality. Bibliography: 27 titles.     

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

Interior Differentiability Results for Nonlinear Variational Parabolic Systems with Nonlinearity 1 <?q < 2

In this work we prove that the solutions

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

Triple Hilbert Transforms Along Polynomial Surfaces

Given W ì \mathbbZ₊³\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in L^p(\mathbbR⁴)L^p({\mathbb{R}}^4).     

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.