A Neumann problem involving the $p(x)$-Laplacian with $p=\infty$ in a subdomain

Yiannis Karagiorgos; Nikos Yannakakis

doi:10.1515/acv-2014-0003

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arXiv · 1310.5173

A Neumann problem involving the $p(x)$-Laplacian with $p=\infty$ in a subdomain

Yiannis Karagiorgos·Nikos Yannakakis

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Abstract

In this paper we study a non-homogeneous Neumann problem, where the $p(x)$-Laplacian is involved and $p=\infty$ in a subdomain. By considering a suitable sequence $p_k$ of bounded variable exponents such that $p_k \to p$ and replacing $p$ with $p_k$ in the original problem, we prove the existence of a solution $u_k$ for each of those intermediate ones. We show that the limit of the $u_k$ exists and after giving a variational characterization of it, in the part of the domain where $p$ is bounded, we show that it is a viscosity solution in the part where $p=\infty$. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.

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BibTeX RIS

Yiannis Karagiorgos, Nikos Yannakakis. 2014-12-12. A Neumann problem involving the $p(x)$-Laplacian with $p=\infty$ in a subdomain. https://doi.org/10.1515/acv-2014-0003

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