Boundary value problems in oscillating cuspidal wedges
This paper deals with pseudodifferential boundary value problems in domains with cuspidal wedges. A simple example of singular spaces studied here is the so-called canonical wedge \[ W= \{(\varphi(z) x,y,z)\in \mathbb{R}^3;\,|x|\leq 1,\,y\in \mathbb{R},\,z\geq 0\}\quad\text{in }\mathbb{R}^3, \] where \(\varphi\in C^\infty(\mathbb{R}^+)\), \(\varphi(z)> 0\) for \(z> 0\), \(\varphi(0)= 0\). The behaviour of \(\varphi(z)\) near \(z= 0\) specifies the singularity of \(W\) along the edge \(\{(0,y,0)\in \mathbb{R}^3\}\). As in general \(\varphi'(z)\) need not have any limit when \(z\to 0\), hence \(W\) may oscillate close to the edge. The results of this paper are based on an interplay between some classes of operator-valued symbols and the ``order reduction'' symbol \(\Lambda_B\), \(B\) being a \(C^\infty\) compact manifold. In the case of cuspidal wedges the authors develop a calculus of pseudodifferential operators where the behaviour of the corresponding symbols is controlled by a special operator valued function \(\lambda(t,\tau)\) depending on the variables \(t,\tau\in\mathbb{R}\). Moreover, the symbols under investigation vary slowly close to singularities. In Theorem 17.1 a criterion for the Fredholm property of differential operators on manifolds with oscillating cuspidal edges is shown. Part IV of the paper is devoted to boundary value problems including Dirichlet and Neumann ones in domains with cuspidal wedges.
manifolds with edges and cuspidal wedges
scientific article; zbMATH DE number 2143235
scientific article