A free boundary problem for $\infty$–Laplace equation

Abstract.

We consider a free boundary problem for the p-Laplacian

$\Delta_pu={\rm div} (\vert\nabla u\vert^{p-2}\nabla u),$

describing nonlinear potential flow past a convex profile K with prescribed pressure $|\nabla u(x)| =a(x)$ on the free stream line. The main purpose of this paper is to study the limit as $p\to\infty$ of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the $\infty$-Laplacian

$\Delta_\infty u=\nabla^2u\nabla u\cdot\nabla u,$

in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case $a(x)\equiv a_0>0$ the limit is given by the distance function.