ESTIMATES ON SOLUTIONS TO CERTAIN QUASILINEAR EQUATIONS IN DIVERGENCE FORM

Abstract

The convergence of approximations to solutions of nonlinear elliptic equations is closely related to the structure of the equations. As examples, we examine certain quasilinear elliptic equations with quadratic growth in the gradient defined on bounded domains. $L^{\infty}$ and $H^1$ estimates on approximating solutions are performed to deduce the convergence to a solution in $H^1_0(\Omega) \cap L^{\infty}(\Omega)$. In some cases, $H^1$ a priori bound can be derived without referring to $L^{\infty}$ estimate. Furthermore, a $W^{2,p}(\mathbf{\Omega})$ bound is also established to deduce the existence of strong solutions in $W^{2,p}(\mathbf{\Omega}) \cap W^{1,p}_0(\mathbf{\Omega})$.