Boundary-contact problems for domains with edge singularities
Let \(G\) be a bounded domain in the Euclidean space of the form \(G= G_+\cup G_-\cup S\) for open subdomains \(G_+\), \(G_-\) of \(G\), such that \(\overline G_+\cap\overline G_-= S\) is an interface of codimension 1. Assume \(G_+\) is inside \(G\), namely \(\partial G_+= S\), \(S\cap\partial G= \emptyset\) and therefore \(\partial G_-= S\cup\partial G\). Starting from a pair of elliptic partial differential operators \(A_+\), \(A_-\) in \(G_+\), \(G_-\) respectively, the authors consider the boundary value problem: \(A_+ u_+= f_+\) in \(G_+\), \(A_- u_-= f_-\) in \(G_-\), \(Tu_-= h\) on \(\partial G\), \(T_+ u_++ T_- u_-= g\) on \(S\), where \(T\), \(T_+\), \(T_-\) are suitable Shapiro-Lopatinsky trace operators. The main focus of the paper is the case when \(G_+\) and \(G_-\) are manifolds with edges \(Y\subset S\); in this case \(S\) itself is a closed manifold with edge singularity \(Y\). A precise study of the problem is given in terms of pseudo-differential calculus, by using the techniques of \textit{B.-W. Schulze} [Boundary value problems and singular pseudo-differential operators. Wiley, Chichester (1998; Zbl 0907.35146)].
scientific article; zbMATH DE number 5132866
scientific article