Title: Stability of solutions of varying degenerate elliptic equations

Authors: Li Gongbao and O. Martio

Issue: Volume 47 (1998), Issue 3, 873-892

Abstract: Let $1 < p_0 < \infty$, $s > p_0$, and $p_i \to p_0$. For fixed $\psi$, $\theta \in W^{1,s}(\Omega)$, consider the solution $u_i$ to the obstacle problem associated with a second order quasilinear degenerate elliptic equation $\nabla \cdot A_{p_i} (x,\nabla u) = 0$ with obstacle $\psi$ and boundary values $\theta$. Here $|A_{p_i}(x,\xi)| \approx |\xi|^{p_i}$, and the functions $A_{p_i}$ have uniformly bounded structure constants. If for a.e. $x \in \Omega$, $A_{p_i}(x,\xi) \to A_{p_0}(x,\xi)$ uniformly on compact subsets of $\mathhbb{R}^n$, then it is shown that $u_i \to u_0$ in $W^{1,t}(\Omega)$, where $u_0$ is the corresponding solution to $\nabla \cdot A_{p_0} (x,\nabla u) = 0$ and $t > p_0$ depends on $n$, $s$, $p_0$, the structure constants, and the regularity of $\Omega$.