On Quasi-static Non=Newtonian Fluids with Power-law

We discuss certain classes of quasi-static non-Newtonian fluids for which a power-law of the form σ^D=∇ϕ(ℰv) holds. Here σ^D is the stress deviator, v the velocity field, ℰv its symmetric derivative and ϕ is the function \[ \phi ({\cal E}v)=\frac 12\mu _\infty \left| {\cal E}v\right| ⁁2+\frac 1p\mu _0\left\{ \begin{array}{c} \left( 1+\left| {\cal E}v\right| ⁁2\right) ⁁{p/2} \\ \text{or} \\ \left| {\cal E}v\right| ⁁p \end{array} \right\}, \] ϕ(ℰv)=1 2 μ_∞∣ℰv∣²+1 p μ₀ (1+∣ℰv∣²)^p/2 or ∣ℰv∣^p, μ_∞⩾0, μ₀⩾0, μ_∞+μ₀>0, 1<p<∞. We then prove various regularity results for the velocity field v, for example differentiability almost everywhere and local boundedness of the tensor ℰv.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

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}      4. Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid      S. Poborchi 《Journal of Mathematical Sciences》2011,175(3):363-374 We consider the Dirichlet problem for the equation - \textdiv( | ?u |p - 2?u ) + a| u |p - 2u = 0, - {\text{div}}\left( {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \right) + a{\left| u \right|^{p - 2}}u = 0,      5. Berger-Shaw's theorem forp-hyponormal operators      Atsushi Uchiyama 《Integral Equations and Operator Theory》1999,33(2):221-230 For an-multicyclicp-hyponormal operatorT, we shall show that |T|2p –|T *|2p belongs to the Schatten and that tr Area ((T)).      6. Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality      W. B. Liu  John W. Barrett 《Advances in Computational Mathematics》1993,1(2):223-239 In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu 0; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW 0 1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.      7. 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{),}}      8. 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.      9. Regularity results for a class of obstacle problems in Heisenberg groups      Francesco Bigolin 《Applications of Mathematics》2013,58(5):531-554 We study regularity results for solutions u ∈ HW 1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? n . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.      10. Asymptotics for the minimization of a Ginzburg-Landau functional      Fabrice Bethuel  Haim Brezis  Frédéric Hélein 《Calculus of Variations and Partial Differential Equations》1993,1(2):123-148 LetΩ ? ?2 be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH g 1 ={u εH 1(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu ε be a minimizer for Eε onH g 1 . We prove that uε → u0 in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u0 is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .      11. The Two-Dimensional Exterior Stokes Problem,Existence, Regularity and Decay Properties      Maria Specovius Neugebauer 《Mathematical Methods in the Applied Sciences》1996,19(7):507-528 The Stokes problem −Δu+∇p = f, div u = g in Ω, u∣∂Ω = h is investigated for two-dimensional exterior domains Ω. By means of potential theory, existence, uniqueness and regularity results for weak solutions are proved in weighted Sobolev spaces with weights proportional to ∣x∣δ as ∣x∣→∞. For f = 0,g = 0, explicit decay formulas are obtained for the solutions u and p. Finally, the results are compared with the theory of r-generalized solutions, i.e. ∇u∈Lr.      12. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 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}      13. 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.      14. Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities      De-xiang Ma Wei-gao Ge Xue-gang Chen 《应用数学学报(英文版)》2005,21(4):661-670 In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0      15. On a nonlocal problem modelling Ohmic heating in planar domains      Fei Liang  Qi Lin Liu  Yu Xiang Li 《数学学报(英文版)》2013,29(3):523-534 In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.      16. Homogenization of quasi-linear equations with natural growth terms      Valeria Chiadò Piat  Anneliese Defranceschi 《manuscripta mathematica》1990,68(1):229-247 In this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|p. Under these assumptions on a and H we prove that there exists a function H0=H0(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (uε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|p-2u = H0(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.      17. Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity      Da Cheng Cui  Ji Hui Zhang  Ming Wen Fei 《数学学报(英文版)》2012,28(11):2243-2274 In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α 0 > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H 1(?2)-solution u ? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u ? is also established as ? tends to zero.      18. On the Range of the Aluthge Transform   总被引：1，自引：0，他引：1   Guoxing Ji  Yongfeng Pang  Ze Li 《Integral Equations and Operator Theory》2007,57(2):209-215 Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .      19. Removable singularities of some strongly nonlinear elliptic equations      Juan-Luis Vàzquez  Laurent Véron 《manuscripta mathematica》1980,33(2):129-144 Let Ω be some open subset of ?N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L loc ∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?+ and above on ??, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L loc ∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .      20. A Littlewood-Paley type inequality   总被引：2，自引：0，他引：2   Stevo?Stevi?Email author 《Bulletin of the Brazilian Mathematical Society》2003,34(2):211-217 In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that .      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

Existence and uniqueness of a solution to the Dirichlet problem for a quasilinear equation inside and outside a paraboloid

We consider the Dirichlet problem for the equation

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? ^d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu ₀; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW ₀ ^1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.     

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{),}}     

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.      9. Regularity results for a class of obstacle problems in Heisenberg groups      Francesco Bigolin 《Applications of Mathematics》2013,58(5):531-554 We study regularity results for solutions u ∈ HW 1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? n . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.      10. Asymptotics for the minimization of a Ginzburg-Landau functional      Fabrice Bethuel  Haim Brezis  Frédéric Hélein 《Calculus of Variations and Partial Differential Equations》1993,1(2):123-148 LetΩ ? ?2 be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH g 1 ={u εH 1(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu ε be a minimizer for Eε onH g 1 . We prove that uε → u0 in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u0 is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .      11. The Two-Dimensional Exterior Stokes Problem,Existence, Regularity and Decay Properties      Maria Specovius Neugebauer 《Mathematical Methods in the Applied Sciences》1996,19(7):507-528 The Stokes problem −Δu+∇p = f, div u = g in Ω, u∣∂Ω = h is investigated for two-dimensional exterior domains Ω. By means of potential theory, existence, uniqueness and regularity results for weak solutions are proved in weighted Sobolev spaces with weights proportional to ∣x∣δ as ∣x∣→∞. For f = 0,g = 0, explicit decay formulas are obtained for the solutions u and p. Finally, the results are compared with the theory of r-generalized solutions, i.e. ∇u∈Lr.      12. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 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}      13. 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.      14. Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities      De-xiang Ma Wei-gao Ge Xue-gang Chen 《应用数学学报(英文版)》2005,21(4):661-670 In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0      15. On a nonlocal problem modelling Ohmic heating in planar domains      Fei Liang  Qi Lin Liu  Yu Xiang Li 《数学学报(英文版)》2013,29(3):523-534 In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.      16. Homogenization of quasi-linear equations with natural growth terms      Valeria Chiadò Piat  Anneliese Defranceschi 《manuscripta mathematica》1990,68(1):229-247 In this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|p. Under these assumptions on a and H we prove that there exists a function H0=H0(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (uε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|p-2u = H0(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.      17. Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity      Da Cheng Cui  Ji Hui Zhang  Ming Wen Fei 《数学学报(英文版)》2012,28(11):2243-2274 In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α 0 > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H 1(?2)-solution u ? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u ? is also established as ? tends to zero.      18. On the Range of the Aluthge Transform   总被引：1，自引：0，他引：1   Guoxing Ji  Yongfeng Pang  Ze Li 《Integral Equations and Operator Theory》2007,57(2):209-215 Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .      19. Removable singularities of some strongly nonlinear elliptic equations      Juan-Luis Vàzquez  Laurent Véron 《manuscripta mathematica》1980,33(2):129-144 Let Ω be some open subset of ?N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L loc ∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?+ and above on ??, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L loc ∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .      20. A Littlewood-Paley type inequality   总被引：2，自引：0，他引：2   Stevo?Stevi?Email author 《Bulletin of the Brazilian Mathematical Society》2003,34(2):211-217 In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that .      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 results for a class of obstacle problems in Heisenberg groups

We study regularity results for solutions u ∈ HW ^1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? ⁿ . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

The Two-Dimensional Exterior Stokes Problem,Existence, Regularity and Decay Properties

The Stokes problem −Δu+∇p = f, div u = g in Ω, u∣_∂Ω = h is investigated for two-dimensional exterior domains Ω. By means of potential theory, existence, uniqueness and regularity results for weak solutions are proved in weighted Sobolev spaces with weights proportional to ∣x∣^δ as ∣x∣→∞. For f = 0,g = 0, explicit decay formulas are obtained for the solutions u and p. Finally, the results are compared with the theory of r-generalized solutions, i.e. ∇u∈L^r.     

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}      13. 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.      14. Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities      De-xiang Ma Wei-gao Ge Xue-gang Chen 《应用数学学报(英文版)》2005,21(4):661-670 In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0      15. On a nonlocal problem modelling Ohmic heating in planar domains      Fei Liang  Qi Lin Liu  Yu Xiang Li 《数学学报(英文版)》2013,29(3):523-534 In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.      16. Homogenization of quasi-linear equations with natural growth terms      Valeria Chiadò Piat  Anneliese Defranceschi 《manuscripta mathematica》1990,68(1):229-247 In this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|p. Under these assumptions on a and H we prove that there exists a function H0=H0(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (uε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|p-2u = H0(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.      17. Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity      Da Cheng Cui  Ji Hui Zhang  Ming Wen Fei 《数学学报(英文版)》2012,28(11):2243-2274 In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α 0 > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H 1(?2)-solution u ? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u ? is also established as ? tends to zero.      18. On the Range of the Aluthge Transform   总被引：1，自引：0，他引：1   Guoxing Ji  Yongfeng Pang  Ze Li 《Integral Equations and Operator Theory》2007,57(2):209-215 Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .      19. Removable singularities of some strongly nonlinear elliptic equations      Juan-Luis Vàzquez  Laurent Véron 《manuscripta mathematica》1980,33(2):129-144 Let Ω be some open subset of ?N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L loc ∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?+ and above on ??, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L loc ∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .      20. A Littlewood-Paley type inequality   总被引：2，自引：0，他引：2   Stevo?Stevi?Email author 《Bulletin of the Brazilian Mathematical Society》2003,34(2):211-217 In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that .

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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

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

On a nonlocal problem modelling Ohmic heating in planar domains

In this paper, we consider the nonlocal problem of the form ut-Δu = (λe-u)/(∫Ωe-udx)2,x ∈Ω, t0 and the associated nonlocal stationary problem -Δv = (λe-v)/(∫Ωe-vdx)2, x ∈Ω,where λ is a positive parameter. For Ω to be an annulus, we prove that the nonlocal stationary problemhas a unique solution if and only if λ 2| Ω| 2 , and for λ = 2|Ω|2, the solution of the nonlocal parabolic problem grows up globally to infinity as t →∞.     

Homogenization of quasi-linear equations with natural growth terms

In this paper we deal with the limit behaviour of the bounded solutions u_ε of quasi-linear equations of the form of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like |ξ|^p. Under these assumptions on a and H we prove that there exists a function H⁰=H⁰(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (u_ε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|^p-2u = H⁰(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties.     

Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger equation $$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$ where 2 < p < ∞, α ₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^?α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H ¹(?²)-solution u _? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u _? is also established as ? tends to zero.     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

Removable singularities of some strongly nonlinear elliptic equations

Let Ω be some open subset of ?^N containing 0 and Ω′=Ω?{0}. If g is a continuous function from ? × ? into ? satisfying some power like growth assumption, then any u∈L _loc ^∞ (Ω′) satisfying $$\begin{array}{*{20}c} { - div (Du \left| {Du} \right|^{p - 2} ) + g(.,u) = 0} & {in \mathcal{D}'(\Omega ')} \\ \end{array} $$ , remains bounded in Ω and satisfies the equation in D'(Ω). We give extensions when the singular set is some compact submanifold of Ω. When g is bounded below on ?⁺ and above on ?^?, then we prove that any subset Σ with 1-capacity zero is a removable singularity for a function u∈L _loc ^∞ (ω?Σ) satisfying $$\begin{array}{*{20}c} { - div \left( {\frac{{Du}}{{\sqrt {1 + \left| {Du} \right|^2 } }}} \right) + g(.,u) = 0} & {in \mathcal{D}'(\Omega - \Sigma )} \\ \end{array} $$ .     

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that