Double phase obstacle problems with multivalued convection and mixed boundary value conditions

2021/06/29 by Zeng, Shengda, Rădulescu, Vicenţiu D., Winkert, Patrick
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.15422

Abstract

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by \mathcal S the solution set of the mixed boundary value problem and by \mathcal S_n the solution sets of the approximating problems, we establish the following convergence relation ∅≠ w-\limsup_n→∞\mathcal S_n=s-\limsup_n→∞\mathcal S_n⊂ \mathcal S, where w-\limsup_n→∞\mathcal S_n and s-\limsup_n→∞\mathcal S_n stand for the weak and the strong Kuratowski upper limit of \mathcal S_n, respectively.

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