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

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
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