Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary-driven patterns in elongated convex domains

Maicon Sonego

Abstract

We consider the heat equation in a smooth bounded convex domain $Ω\subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_νu = λ(u - u^3)$. Stable non-constant stationary solutions do not exist when $Ω$ is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of $Ω$ is fixed and its diameter is sufficiently large, then there exists $λ>0$ for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.

Boundary-driven patterns in elongated convex domains

Abstract

We consider the heat equation in a smooth bounded convex domain with nonlinear Neumann boundary condition . Stable non-constant stationary solutions do not exist when is a ball. We show that this behavior is not a consequence of convexity alone. More precisely, if the inradius of is fixed and its diameter is sufficiently large, then there exists for which the problem admits such a solution. The result reveals a geometric mechanism for the emergence of stable non-constant stationary solutions in elongated convex domains.
Paper Structure (4 sections, 4 theorems, 42 equations, 2 figures)

This paper contains 4 sections, 4 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['con']}
  4. Concluding Remarks

Key Result

Theorem 1

Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded convex domain such that $\mathrm{inr}(\Omega)=r$ and $D(\Omega)=D$. Then if $D$ is large enough, there exists $\lambda>0$ for which problem con admits a pattern.

Figures (2)

  • Figure 1: Convex domain $\Omega \subset \mathbb{R}^2$ with inradius $r$ and diameter $D$.
  • Figure 2: $Q_{\delta}$, $\Omega_j$, $d$ and $D(\Omega_j)$ ($j=l,r$) in $\Omega \subset \mathbb{R}^2$.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • Remark 1