The third boundary value problem in potential theory for domains with a piecewise smooth boundary

Medková, Dagmar

Czechoslovak Mathematical Journal, Tome 47 (1997) no. 4, pp. 651-679.

Résumé

The paper investigates the third boundary value problem $\frac{\partial u}{\partial n}+\lambda u=\mu $ for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where $\nu $ is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure ${T}\nu $. Denote by ${T}\:\nu \rightarrow {T}\nu $ the corresponding operator on the space of signed measures on the boundary of the investigated domain $G$. If there is $\alpha \ne 0$ such that the essential spectral radius of $(\alpha I-{T})$ is smaller than $|\alpha |$ (for example, if $G\subset R^3$ is a domain “with a piecewise smooth boundary” and the restriction of the Newtonian potential ${\mathcal U}\lambda $ on $\partial G$ is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition $\mu \in $ for which $\mu (\partial G)=0$.

Détail

Classification : 31B20, 35J05, 35J25