Semi-linear boundary value problems with respect to an ideal boundary on a self-adjoint harmonic space (Q801442)

The author has developed an elegant theory of Dirichlet integrals on abstract harmonic spaces [(see \textit{F.-Y. Maeda}, Dirichlet integrals on harmonic spaces, Lect. Notes Math. 803 (1980; Zbl 0426.31001)]. Using this theory among others the notion of normal derivative can be defined and thus boundary value problems involving normal derivatives can be formulated in the context of harmonic spaces. Let (X,\({\mathcal H})\) be a self-adjoint harmonic space and \(\sigma\) a canonical measure representation associated with a symmetric Green function on X, let \(\partial^*X\) denote the ideal boundary of X and N(.) the normal derivative. The semi-linear boundary value problem discussed in the present paper can be formulated very roughly as follows. Let F be a sheaf morphism acting on differences of two continuous superharmonic functions and taking values in the space of signed measures with continuous potentials. Let \(\beta\) be a mapping defined on a space of boundary functions and with values in a space of measures on \(\partial^*X\). Let \(\tau\) be a function on \(\partial^*X\), \(\Lambda \subset \partial^*X\). The problem is to find a function u on X of the form \(u=H_{\phi}+g\), where \(\phi\) is defined on \(\partial^*X\) and g is of potential type on X, such that \[ \sigma (u)+F(u)=0\quad on\quad X,\quad \phi =\tau \quad on\quad \partial^*X-\Lambda,\quad N(u)=\beta (\phi)\quad on\quad \Lambda. \] Before solving this semi-linear problem some relevant linear problems are studied.