English
Related papers

Related papers: Boundary-driven patterns in elongated convex domai…

200 papers

We address the question of existence of nonconstant stable stationary solution to the heat equation on a class of convex domains subject to nonlinear boundary flux involving a positive parameter. Such solutions which were known to exist in…

Analysis of PDEs · Mathematics 2010-03-16 Arnaldo Simal do Nascimento

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland [20] and Matano [44] states that all stable solutions are constant in convex bounded domains.…

Analysis of PDEs · Mathematics 2021-02-12 Samuel Nordmann

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in…

Analysis of PDEs · Mathematics 2026-03-27 Marta Calanchi , Giulio Ciraolo , Francesca Messina

Higher regularity estimate has been a challenging question for the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in [15] even for symmetric convex…

Analysis of PDEs · Mathematics 2021-03-29 Hongxu Chen , Chanwoo Kim

We investigate the problem $$-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \mbox{ in } \Omega, \quad \frac{\partial u}{\partial \mathbf{n}} = 0 \mbox{ on } \partial \Omega, \leqno{(P_\lambda)} $$ where $\Omega$ is a bounded smooth…

Analysis of PDEs · Mathematics 2016-03-17 Humberto Ramos Quoirin , Kenichiro Umezu

In this paper we study the existence of solutions in parabolic Schauder space of a nonlinear mixed boundary value problem for the heat equation in a perforated domain. From a given regular open set $\Omega\subseteq\mathbb{R}^n$ we remove a…

Analysis of PDEs · Mathematics 2024-06-27 Riccardo Molinarolo

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $\Omega$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the…

Analysis of PDEs · Mathematics 2026-03-31 Xavier Cabre , Neus Consul , Matthias Kurzke

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

Analysis of PDEs · Mathematics 2024-01-22 Humberto Ramos Quoirin , Kenichiro Umezu

Given a smooth bounded domain $\Omega$ in $\mathbb{R}^2$, we study the following anisotropic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=\lambda a(x) u^{p-1}e^{u^p},\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\,…

Analysis of PDEs · Mathematics 2025-02-13 Yibin Zhang

We consider the boundary value problem $-\Delta u + u =\lambda e^u$ in $\Omega$ with Neumann boundary condition, where $\Omega$ is a bounded smooth domain in $\mathbb R^2$, $\lambda>0.$ This problem is equivalent to the stationary…

Analysis of PDEs · Mathematics 2016-03-14 Manuel del Pino , Giusi Vaira , Angela Pistoia

We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $\Omega$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $\Omega$ to be a…

Analysis of PDEs · Mathematics 2013-07-01 Luis Silvestre , Boyan Sirakov

In this article, we investigate the incoming boundary value problem for the stationary linearized Boltzmann equations in $ \Omega \subseteq \mathbb{R}^{3}$. For a $C^2$ bounded domain with boundary of positive Gaussian curvature, the…

Analysis of PDEs · Mathematics 2023-04-19 I-Kun Chen , Ping-Han Chuang , Chun-Hsiung Hsia , Daisuke Kawagoe , Jhe-Kuan Su

We consider the equation $-\epsilon^{2}\Delta u + u = u^ {p}$ in a bounded domain $\Omega\subset\R^{3}$ with edges. We impose Neumann boundary conditions, assuming $1<p<5$, and prove concentration of solutions at suitable points of…

Analysis of PDEs · Mathematics 2015-05-20 Serena Dipierro

In this note we construct smooth bounded domains $\Omega \subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} \Delta u + \lambda u &= 0 &\qquad& \text{ in } \Omega, \newline u &= b…

Analysis of PDEs · Mathematics 2025-09-03 Miles H. Wheeler

We consider the class of semi-stable positive solutions to semilinear equations $-\Delta u=f(u)$ in a bounded domain $\Omega\subset\mathbb R^n$ of double revolution, that is, a domain invariant under rotations of the first $m$ variables and…

Analysis of PDEs · Mathematics 2012-02-07 Xavier Cabre , Xavier Ros-Oton

Let $M$ be a Riemannian manifold and $\Omega$ a smooth domain of $M$. We study the following heat diffusion problem: assume that the initial temperature is equal to $1$, uniformly on $\Omega$, and is $0$ on its complement. Heat will then…

Differential Geometry · Mathematics 2025-07-14 Andrea Bisterzo , Alessandro Savo

Let $M$ be a Riemannian manifold and $\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\Omega$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of…

Differential Geometry · Mathematics 2014-06-12 Alessandro Savo

This paper investigates the existence and regularity of strong solutions to the incompressible Navier-Stokes equations within a bounded domain $\Omega \subset \mathbb{R}^3$, subject to the boundary condition $(u\cdot \vec{n})|_{\partial…

Analysis of PDEs · Mathematics 2023-07-25 Vu Thanh Nguyen

This paper studies the heat equation $u_t=\Delta u$ in a bounded domain $\Omega\subset\mathbb{R}^{n}(n\geq 2)$ with positive initial data and a local nonlinear Neumann boundary condition: the normal derivative $\partial u/\partial n=u^{q}$…

Analysis of PDEs · Mathematics 2018-04-25 Xin Yang , Zhengfang Zhou

We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In…

Analysis of PDEs · Mathematics 2014-11-03 Serena Dipierro , Xavier Ros-Oton , Enrico Valdinoci
‹ Prev 1 2 3 10 Next ›