-Harmonic functions with unbounded exponent in a subdomain
Abstract
We study the Dirichlet problem $-÷(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \to \infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x) \wedge n$, in particular, with $p_n = n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $\Omega$.
- Publication:
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Annales de L'Institut Henri Poincare Section (C) Non Linear Analysis
- Pub Date:
- November 2009
- DOI:
- 10.1016/j.anihpc.2009.09.008
- arXiv:
- arXiv:0809.2731
- Bibcode:
- 2009AIHPC..26.2581M
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35J20;
- 35J60;
- 35J70
- E-Print:
- Corrected typos