Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data

Abstract

We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation

$$\begin{aligned} -\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) -\sum _{i=1}^{N}\partial _{i}\phi _{i}( u)=\mu , \end{aligned}$$

where the right hand side \(\mu \) belongs to \(L^{1}(\Omega )+ W^{-1, \vec {p'}}(\Omega )\). The operator \(-\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) \) is a Leray–Lions anisotropic operator and \(\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})\).

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Correspondence to Abdelhafid Salmani.

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Salmani, A., Akdim, Y. & Redwane, H. Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data. Ricerche mat 69, 121–151 (2020). https://doi.org/10.1007/s11587-019-00452-0

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Keywords

  • Entropy solutions
  • Anisotropic elliptic equations
  • Anisotropic Sobolev space

Mathematics Subject Classification

  • 35J60
  • 35J87
  • 35J66