Estimates on the approximation of solutions to certain quasilinear elliptic equations

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∞ and H1 estimates on approximating solutions are performed to deduce the convergence to a solution in H01(Ω)∩L∞(Ω). In some cases, H1 a priori bound can be derived without referring to L∞ estimate. Furthermore, a W2,p(Ω) bound is also established to deduce the existence of strong solutions in W2,p(Ω)∩W01,p(Ω).