On a family of torsional creep problems. (English) Zbl 0701.35015

Kawohl, Bernhard

J. Reine Angew. Math. 410, 1-22 (1990).

The subject of the paper is the dependence of solutions u to \[ -\Delta_ pu=-div(| \nabla u|^{p-2}\nabla u)=1\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega \] on p and the limiting behaviour as \(p\in (1,\infty)\) tends to \(\infty\) and 1 respectively. For \(p\to \infty\) the solution \(u_ p\) converges to d(x,\(\partial \Omega)\), the distance function to \(\partial \Omega\). For \(p\to 1\) the situation is more complex. In fact if \(\Omega \subset {\mathbb{R}}^ N\) is a ball of radius \(R\leq N\), the family \(u_ p\) goes to zero uniformly in \(\Omega\), while for \(R>N\) it goes to infinity everywhere in \(\Omega\). This behaviour is linked to the isoperimetric inequality. The proofs use variational arguments.

Reviewer: B.Kawohl

MSC:

35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J70	Degenerate elliptic equations
35J65	Nonlinear boundary value problems for linear elliptic equations
74B20	Nonlinear elasticity

Keywords:

p-Laplacian; distance function; isoperimetric inequality