Unique solvability of the Dirichlet problem for the equation \(\Delta _{p} u=0\) in the exterior of a paraboloid (Q2849091)

The author considers the Dirichlet problem NEWLINE\[NEWLINE -\text{div}(|\nabla u|^{p-2}\nabla u)=0 \text{ in }\Omega,\quad u|_{\partial\Omega}=f NEWLINE\]NEWLINE in the exterior of an \(n\)-dimensional paraboloid, \(p\in(1,n)\). The space of the traces \(u|_{\Gamma}\) on the boundary of the paraboloid for functions \(u\) in the class \(L_p^1\) is described explicitly. This implies necessary and sufficient conditions for the existence and uniqueness of a solution to the Dirichlet problem.