The effect of curvature and symmetry on stable stationary solutions in convex domains

TL;DR

This paper investigates the existence and stability of nonconstant solutions to the heat equation in convex domains, showing that domain shape and symmetry influence solution behavior and that convexity alone does not prevent such solutions.

Contribution

It demonstrates the existence of nonconstant stable solutions in certain convex domains and analyzes how domain shape and symmetry affect solution stability and properties.

Findings

Nonconstant stable solutions exist in smoothed convex domains derived from cubes.

Convexity alone does not guarantee the nonexistence of such solutions.

Symmetry inheritance by local minimizers is established.

Abstract

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 dumbbell-type domains and not to exist in $N$-dimensional balls are shown to exist in some convex domains obtained by smoothing out, in a convenient way, the edges and corners of a cube. Therefore convexity of the domain is not a necessary condition for nonexistence of this type of solutions. If the parameter is small enough we prove that the only equilibria are the constant ones by using the Implicit Function Theorem in a special setting. Symmetry inheritance by local minimizers from symmetry properties of the domain is also addressed.