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General interface problems. I. (English) Zbl 0824.35014
The transmission problems for elliptic operators of order \(2m\) with general boundary and interface conditions are considered, and the weak forms of these problems are described. Along the external boundary the authors impose classical boundary conditions satisfying the so-called Shapiro-Lopastinskii conditions while on the interfaces they define general transmission conditions and introduce a new covering condition. These allow them to use Agranovitch-Visik’s results to get the solvability, the regularity and the asymptotics of the solutions in weighted Sobolev spaces. The assertion that the weak solution of a transmission problem admit a decomposition into regular and singular parts in weighted Sobolev spaces is proved. Finally, some numerical examples for the location of the singular exponents are given at the end of the paper.
Reviewer: H.Ding (Beijing)

MSC:
35B65 Smoothness and regularity of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35J40 Boundary value problems for higher-order elliptic equations
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