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

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

2010-02-16

arxiv.org/abs/1002.3071v2

Mathematics

Analysis of PDEs

This paper has been withdrawn by the author due to a scaling error which invalidate the ilustration provided for the convex do

Scientific paper

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.

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