p(x)-Harmonic functions with unbounded exponent in a subdomain

Details p(x)-Harmonic functions with unbounded exponent in a subdomain p(x)-Harmonic functions with unbounded exponent in a subdomain

2008-09-16

arxiv.org/abs/0809.2731v4

Mathematics

Analysis of PDEs

Corrected typos; updated references; section 4. is new. This is the final version, accepted for publication in Ann. Inst. H. P

Scientific paper

We study the Dirichlet problem $-\div(|\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 $\Omega$.

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