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

Singular limits for 2-dimensional elliptic problem involving exponential nonlinearity with nonlinear gradient term

Given a bounded open regular set W ì \mathbbR²{\Omega \subset \mathbb{R}^2} and x₁, x₂, ?, x_m ? W{x_1, x_2, \ldots, x_m \in \Omega}, we give a sufficient condition for the problem

Highly time-oscillating solutions for very fast diffusion equations

This work is concerned with the fast diffusion equation

On the global well-posedness of a class of Boussinesq�CNavier�CStokes systems

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems

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

Gradient estimates for the elliptic and parabolic Lichnerowicz equations on compact manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3. Denote D_g=-div_g?{\Delta_g=-{\rm div}_g\nabla} the Laplace–Beltrami operator. We establish some local gradient estimates for the positive solutions of the Lichnerowicz equation

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

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}      9. Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration      O. M. Boldovskaya 《Ukrainian Mathematical Journal》2010,62(7):1040-1060 We consider the Neumann initial boundary-value problem for the equation ut = \textdiv( um - 1| Du |l- 1Du ) - up {u_t} = {\text{div}}\left( {{u^{m - 1}}{{\left| {Du} \right|}^{\lambda - 1}}Du} \right) - {u^p}      10. Renormalized solution for Stefan type problems: existence and uniqueness      Noureddine Igbida  Karima Sbihi  Petra Wittbold 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):69-93 We consider a class of nonlinear degenerate problems of Stefan type: $u_t- \Delta w -\nabla F(u,w)= g(\cdot,u), \ w\in \beta(u)$ where β is a maximal monotone graph in ${\mathbb{R}^2,}We consider a class of nonlinear degenerate problems of Stefan type: ut- Dw -?F(u,w) = g(·,u),  w ? b(u)u_t- \Delta w -\nabla F(u,w)= g(\cdot,u), \ w\in \beta(u)      11. A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}      Boris Andreianov  Mohamed Maliki 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):109-118 We study the Cauchy problem in \mathbbRN{\mathbb{R}^N} for the parabolic equation ut+div F(u)=Dj(u),u_t+{\rm div}\,F(u)=\Delta\varphi(u),      12. Large solutions to an anisotropic quasilinear elliptic problem      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《Annali di Matematica Pura ed Applicata》2010,189(4):689-712 In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      13. 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)      14. 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,      15. Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth      Christoph Scheven 《Journal of Evolution Equations》2010,10(3):597-622 For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems ?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime      Martial Agueh  Adrien Blanchet  José A. Carrillo 《Journal of Evolution Equations》2010,10(1):59-84 We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.      17. 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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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

Removability of an isolated singularity of solutions of the Neumann problem for quasilinear parabolic equations with absorption that admit double degeneration

We consider the Neumann initial boundary-value problem for the equation

Renormalized solution for Stefan type problems: existence and uniqueness

We consider a class of nonlinear degenerate problems of Stefan type: $u_t- \Delta w -\nabla F(u,w)= g(\cdot,u), \ w\in \beta(u)$ where β is a maximal monotone graph in ${\mathbb{R}^2,}We consider a class of nonlinear degenerate problems of Stefan type:

ut- Dw -?F(u,w) = g(·,u),  w ? b(u)u_t- \Delta w -\nabla F(u,w)= g(\cdot,u), \ w\in \beta(u)      11. A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}      Boris Andreianov  Mohamed Maliki 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(1):109-118 We study the Cauchy problem in \mathbbRN{\mathbb{R}^N} for the parabolic equation ut+div F(u)=Dj(u),u_t+{\rm div}\,F(u)=\Delta\varphi(u),      12. Large solutions to an anisotropic quasilinear elliptic problem      Jorge García-Melián  Julio D. Rossi  José C. Sabina de Lis 《Annali di Matematica Pura ed Applicata》2010,189(4):689-712 In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      13. 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)      14. 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,      15. Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth      Christoph Scheven 《Journal of Evolution Equations》2010,10(3):597-622 For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems ?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime      Martial Agueh  Adrien Blanchet  José A. Carrillo 《Journal of Evolution Equations》2010,10(1):59-84 We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.      17. 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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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

A note on uniqueness of entropy solutions to degenerate parabolic equations in {\mathbb{R}^N}

We study the Cauchy problem in \mathbbR^N{\mathbb{R}^N} for the parabolic equation

Large solutions to an anisotropic quasilinear elliptic problem

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem ${\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)$ in a bounded domain ${\Omega \subset \mathbb{R}^N \times \mathbb{R}^M}In this paper we consider existence, asymptotic behavior near the boundary and uniqueness of positive solutions to the problem

divx (|?x u|p-2?xu)(x,y) + divy (|?y u|q-2?y u) (x, y) = ur(x, y){\rm div}_x (|\nabla_x u|^{p-2}\nabla_xu)(x,y) + {\rm div}_y (|\nabla_y u|^{q-2}\nabla_y u) (x, y) = u^r(x, y)      13. 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)      14. 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,      15. Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth      Christoph Scheven 《Journal of Evolution Equations》2010,10(3):597-622 For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems ?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime      Martial Agueh  Adrien Blanchet  José A. Carrillo 《Journal of Evolution Equations》2010,10(1):59-84 We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.      17. 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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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 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)      14. 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,      15. Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth      Christoph Scheven 《Journal of Evolution Equations》2010,10(3):597-622 For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems ?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime      Martial Agueh  Adrien Blanchet  José A. Carrillo 《Journal of Evolution Equations》2010,10(1):59-84 We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.      17. 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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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

Non-linear Calderón–Zygmund theory for parabolic systems with subquadratic growth

For vector-valued solutions of parabolic systems $\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)$ with polynomial growth of rate ${p\in\Big(\frac{2n}{n+2},2\Big)}For vector-valued solutions of parabolic systems

?tu-div a(x,t,Du)=div(|F|p-2F)\partial_tu-{\rm div}\, a(x,t,Du)={\rm div}\left(|F|^{p-2}F\right)      16. Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime      Martial Agueh  Adrien Blanchet  José A. Carrillo 《Journal of Evolution Equations》2010,10(1):59-84 We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation rt=div(|?rm |p-2 ?(rm)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRn{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L 1-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.      17. 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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

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.      18. Large solutions for equations involving the p-Laplacian and singular weights      Jorge García-Melián 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,31(2):594-607 In this paper we consider the boundary blow-up problem Δpu = a(x)uq in a smooth bounded domain Ω of \mathbbRN{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here Dpu = div(|?u|p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.      19. Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets      Ricardo A. Sáenz 《Journal of Fourier Analysis and Applications》2012,18(2):240-265 In this paper we study the boundary limit properties of harmonic functions on ℝ+×K, the solutions u(t,x) to the Poisson equation \frac?2 u?t2 + Du = 0,\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,      20. Energy decay rates for solutions of the wave equation with boundary damping and source term      Jong Yeoul Park  Tae Gab Ha  Yong Han Kang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,9(1):235-265 In this paper, we consider the wave equation u" - Du = |u|r uu' - \Delta u = |u|^\rho u      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

Large solutions for equations involving the p-Laplacian and singular weights

In this paper we consider the boundary blow-up problem Δ_pu = a(x)u^q in a smooth bounded domain Ω of \mathbbR^N{\mathbb{R}}^N, with u = +∞ on ∂Ω. Here D_pu = div(|?u|^p-2?u)\Delta_{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u) is the well-known p-Laplacian operator with p > 1, q > p − 1, and a(x) is a nonnegative weight function which can be singular on ∂Ω. Our results include existence, uniqueness and exact boundary behavior of positive solutions.     

Nontangential Limits and Fatou-Type Theorems on Post-Critically Finite Self-Similar Sets

In this paper we study the boundary limit properties of harmonic functions on ℝ₊×K, the solutions u(t,x) to the Poisson equation

Energy decay rates for solutions of the wave equation with boundary damping and source term

In this paper, we consider the wave equation