Mouad Allalou, Abderrahmane Raji, Farah Balaadich

The Existence and Uniqueness of Weak Solutions to Obstacle Problems with Variable Growth and Weak Monotonicity

abstract:
In this paper, we show the existence and uniqueness of weak solutions to obstacle problem
$$ \int\limits_{\Omega}\sigma(x,Du):D(v-u)+ \big\langle u|u|^{p(x)-2}, v- u\big\rangle\,\mathrm{d}x\geq 0, $$
for $v$ belonging to the convex set $\mathcal{K}_{\psi, \theta}$. The main tool used here is the Young measure theory and a theorem of Kinderlehrer and Stampacchia combined with the theory of Sobolev spaces with variable exponent.

Mathematics Subject Classification: 35J88, 28A33, 46E35

Key words and phrases: Obstacle problem, Kinderlehrer and Stampacchia, Young measures, $p(x)$-variable exponents