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

作者：Karagiorgos Yiannis^*; Yannakakis Nikos

来源：Advances in Calculus of Variations, 2016, 9(1): 65-76.

DOI：10.1515/acv-2014-0003

• Neumann problem
• variable exponent
• p(x)-Laplacian
• viscosity solution
• infinity Laplacian
• infinity harmonic function

In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-Laplacian is involved and p = infinity in a subdomain. By considering a suitable sequence p(k) of bounded variable exponents such that p(k) -> 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 (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 = infinity. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.

• 出版日期2016-1