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General edge asymptotics of solutions of second-order elliptic boundary value problems. I. (English) Zbl 0791.35032

Authors’ abstract: This is the first of two papers (see below) in which we study the singularities of solutions of second-order linear elliptic boundary value problems at the edges of piecewise analytic domains in \(\mathbb{R}^ 3\). When the opening angle at the edge is variable, there appears the phenomenon of “crossing” of the exponents of singularities. For this case, we introduce the appropriate combinations of the simple tensor product singularities that allow us to give estimates in ordinary and weighted Sobolev spaces for the regular part of the solution and for the coefficients of the singularities. These combinations appear in a natural way as sections of an analytic bundle above the edge. Their behaviour is described with the help of divided differences of powers of the distance to the edge. The class of operators considered includes second-order elliptic operators with analytic complex-valued coefficients with mixed Dirichlet, Neumann or oblique derivative conditions. With our description of the singularities we are able to remove some restrictive hypotheses that were previously made in other works. In this first part, we prove the basic facts in a simplified framework. Nevertheless the tools we use are essentially the same in the general situation.
Reviewer: H.Ding (Beijing)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35A20 Analyticity in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs

Citations:

Zbl 0791.35035
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References:

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