Existence and uniqueness results for a class of obstacle problem via Young’s measure theory

Abstract

The purpose of this article is to prove the existence and uniqueness of weak solutions to the following obstacle problem of p-Laplace-type:

$$\begin{aligned} \displaystyle \int _{\Omega }\sigma _1(z,Du-\mathcal {F}(u)):D(v-u)+\sigma _2(z,Du):(v-u)+ \left\langle u\vert u\vert ^{p-2}, v- u\right\rangle \mathrm {~d}z\ge 0, \end{aligned}$$

with data belonging to the dual of Sobolev spaces. The main result is demonstrated by means of Kinderlehrer and Stampacchia’s Theorem and Young’s measure theory.