On continuity properties of monotone operators beyond the natural domain of definition

Abstract

We study boundary value problems associated to a nonlinear elliptic system of partial differential equations. The leading second order elliptic operator provides L2-coerciveness and has at most linear growth with respect to the gradient. Incorporating properties of Lipschitz approximations of Sobolev functions we are able to show that the problems in consideration are well-posed, in the sense of Hadamard, in W1,p for all \({p \in (p_0,2]}\) for certain \({p_0 \in [3/2,2)}\) . For simplicity we restrict ourselves to the Dirichlet problem.