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Minimal regularity of the solutions of some transmission problems. (English) Zbl 1035.35035

Let \(\Omega \) be a two-dimensional polygonal domain of \(\mathbb{R}^2\), which is decomposed in a finite union of disjoint non-empty polygonal domains \(\Omega_j,\) \(j \in I\). The topic of the paper is the regularity of the solution of the transmission problem for the Laplace operator, this is the following problem \[ p_i\Delta u_i = f_i \text{ in } \Omega_i, \] where \(p_i\) are given positive material constants and \(f_i\) are given functions, with boundary conditions on the exterior boundary and transmission conditions on the interfaces. The main result is to prove minimal regularity of the solutions with or without conditions on the material constants \(p_i\). Numerical examples are presented showing the sharpness of the bounds.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs
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[1] Costabel, Mathematical Modelling and Numerical Analysis M2AN 33 pp 627– (1999)
[2] Nicaise, Mathematical Models and Methods in Applied Sciences 9 pp 855– (1999)
[3] Die Regulariät der L??sungen von Interface-Problemenen in Gebieten mit singulären punkten. Diplomthesis, University Rostock, 1992.
[4] Regularity results for interface problems in 2D. Preprint No. 655, Berlin, 2000.
[5] Elliptic Problems in Nonsmooth Domains. Pitman: London, 1985. · Zbl 0695.35060
[6] The Finite Element Method for Elliptic Problems (St. Math. Appl.), vol. 4. North-Holland: Amsterdam, 1978. · doi:10.1016/S0168-2024(08)70178-4
[7] Kondratiev, Transactions of the Moscow Mathematical Society 16 pp 227– (1967)
[8] Elliptic Boundary Value Problems in Corner Domains. Smoothness And Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341. Spinger: Berlin, 1988. · Zbl 0668.35001 · doi:10.1007/BFb0086682
[9] Polygonal Interface Problems, Series ?Methoden und Verfahren der Mathematischen Physik?, vol. 39. Peter Lang Verlag: 1993.
[10] Nicaise, Mathematical Methods in Applied Science 17 pp 395– (1994)
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