Unique solvability of the Dirichlet problem for the equation $\Delta _p u=0$ in the exterior of a paraboloid

Abstract:

The Dirichlet problem \[ -\mathrm {div}(|\nabla u|^{p-2}\nabla u)=0 \ \mbox { in } \ \Omega , \ u\big |_{\partial \Omega }=f, \] is considered in the exterior of an $n$-dimensional paraboloid, $p\in (1,n)$. The space of the traces $u\big |_\Gamma$ 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.