Global solvability of the Laplace equation in weighted Sobolev spaces

Abstract

We consider a non-local boundary value problem for the Laplace equation in unbounded studding the weak and strong solvability of that problem in the framework of the weighted Sobolev space $W_{\nu }^{1,p}$, with a Muckenhoupt weight. We proved that if any weak solution belongs to the space $W_{\nu }^{2,p}$, then it is also a strong solution and satisfies the corespding boundary conditions. It should be noted that such problems do not fit into the general theory of elliptic equations and require a special technique.

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