Removability of polar sets for solutions of semi-linear equations on a harmonic space (Q1061900)

Let (X,\({\mathcal U})\) be a harmonic space, \({\mathcal R}\) the sheaf of functions which are locally expressible as differences of continuous superharmonic functions. It is supposed that X has a countable base and \(1\in {\mathcal R}\). Let \({\mathcal M}\) be the sheaf of (signed) Radon measures on X, \(\sigma\) : \({\mathcal R}\to {\mathcal M}\) the measure representation, \({\mathcal M}_{\sigma}\) the image sheaf of \(\sigma\). Further let \(F: {\mathcal R}\to {\mathcal M}\) be a sheaf morphism which satisfies two conditions (monotonicity and local Lipschitz condition). The semi-linear equation (1) \(\sigma (u)+F(u)=0\), is considered in the paper. For an open set U denote \({\mathcal H}^ F_ B(U)=\{u\in {\mathcal R}(U):\) u satisfies (1) and is bounded on \(U\}\). The following theorem is proved in the paper: Any closed polar set e in X is \({\mathcal H}^ F_ B\)-removable, i.e., for any open set U and for any \(u\in {\mathcal H}^ F_ B(U-e)\) there is \(\tilde u\in {\mathcal H}^ F_ B(U)\) such that \(\tilde u|_{(U-e)}=u.\) Further the removability of polar sets with respect to solutions of (1) with finite Dirichlet integral in the case of self-adjoint harmonic space is considered.