EXISTENCE OF WEAK SOLUTIONS FOR OBSTACLE PROBLEMS WITH VARIABLE GROWTH IN ORLICZ-SOBOLEV SPACES

This article delves into the exploration of weak solutions’ existence within the context of obstacle problems associated with the following inequality: \(\varvec{\displaystyle \int _{\Omega }\mathcal {V}(z,w,Dw):D(v-w)~\textrm{d} z \ge \left\langle g, v- w\right\rangle }\) ${\displaystyle {\mathbf{\int }}_{\mathbf{\Omega }}\mathcal{V}\mathbf{(}\mathit{z}\mathbf{,}\mathit{w}\mathbf{,}\mathit{D}\mathit{w}\mathbf{)}\mathbf{:}\mathit{D}\mathbf{(}\mathit{v}\mathbf{-}\mathit{w}\mathbf{)}\text{d}\mathit{z}\mathbf{\ge }〈\mathit{g},\mathbf{,},\mathit{v},\mathbf{-},\mathit{w}〉}$ , where \(\varvec{v}\) $\mathit{v}$ lies in a convex set \(\varvec{\mathcal {F}_{\mathcal {O},\mathcal {B} }}\) ${\mathcal{F}}_{\mathcal{O}\mathbf{,}\mathcal{B}}$ . The primary methodology employed in this investigation involves the utilization of Young’s measure theory, complemented by a theorem originating from Kinderlehrer and Stampacchia, specifically tailored for reflexive Orlicz-Sobolev spaces.