Weak Solutions for Obstacle Problems Via Young Measure in variable-exponent Sobolev space

Weak Solutions for Obstacle Problems Via Young Measure

Abstract

We first discuss the existence and uniqueness of weak solution for the obstacle problem \\
$\displaystyle\int_{\Omega}\phi_1\bigl(y,Dw-\mathcal{T}(w)\bigl):D(\nu-w)+\phi_2(y,Dw):(\nu-w)+ \left\langle w\vert w\vert^{r(y)-2}, \nu- w\right\rangle\mathrm{~d}y\geq0 $ with variable exponent, where $u: \Omega \rightarrow \mathbb{R}^m$ is a vector-valued function and $\Omega$ is a bounded open domain in $\mathbb{R}^n(n \geq 2)$ , the existence is proved by means of the Young measure and under assumptions on $\phi_1$, $\phi_2$ and a theorem of Kinderlehrer and Stampacchia.