On the solvability of the Dirichlet problem for nonlinear elliptic equations (Q914969)

This paper is concerned with the Dirichlet problem \[ Lu+f(u)=h(x)\text{ in } Q,\quad u(x)=\phi (x)\text{ on } \partial Q, \] where Q is a bounded domain in \(R_ n\), \(\phi \in L^ 2(\partial Q)\), and L denotes a self- adjoint elliptic operator. The results obtained here are new in the sense that \(h\in L^ 2(Q)\), while results of earlier works were limited to the case where \(h\in C^{\alpha}(Q).\) But the author remarks that one can not find a solution in \(W^{1,2}(Q)\), because not every function in \(L^ 2(Q)\) is a trace of an element from \(W^{1,2}(Q)\). The weighted Sobolev space \(\tilde W^{1,2}(Q)\) is a suitable space for the Dirichlet problem with \(L^ 2\)-boundary data.