

Sixth International Conference on Differential and Functional Differential Equations DFDE2011
August 19, 2011 12:00, Moscow






Mixed problems and cracktype problems for strongly elliptic secondorder systems in domains with Lipschitz boundaries
M. S. Agranovich^{} ^{} Moscow Institute of Electronics and Mathematics, Russia

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Abstract:
We consider two classes of problems for a strongly elliptic secondorder system in a bounded $n$dimensional domain with Lipschitz boundary, $n\ge 2$. For simplicity, we assume that the domain $\Omega=\Omega^+$ lies on the standard torus $\mathbb T^n$ and that the Dirichlet and Neumann problems in $\Omega^+$ and in the complementary domain $\Omega^$ are uniquely solvable.
1. Mixed problems. In the simplest case, the boundary $\Gamma$ of $\Omega$ is divided into two parts $\Gamma_1$ and $\Gamma_2$ by a closed Lipschitz $(n1)$dimensional Lipschitz surface, with the Dirichlet and Neumann conditions on $\Gamma_1$ and on $\Gamma_2$ respectively. The problem is uniquely solvable in the simplest spaces $H^s$ (with the solution in $H^1(\Omega))$ and (the regularity result) in some more general Bessel potential spaces $H_p^s$ and Besov spaces $B_p^s$. Equations on $\Gamma$ are obtained equivalent to the problem. For this, we use analogs $N_1$ and $D_2$ of the NeumanntoDirichlet operator $N$ and the DirichlettoNeumann operator $D$ on parts $\Gamma_1$ and $\Gamma_2$ of $\Gamma$.
The operators $N_1$ and $D_2$ are connected with Poincaré–Steklovtype spectral problems with spectral parameter on a part of $\Gamma$. In the selfadjoint case, the eigenfunctions form a basis in the corresponding spaces, and in the nonselfadjoint case they form a complete system. If $\Gamma$ is almost smooth (smooth outside a closed subset of zero measure), then the eigenvalues of selfadjoint problems have natural asymptotics.
2. Problems with boundary or transmission conditions on a nonclosed surface $S$, which is a part of a closed Lipschitz surface $\Gamma$. In elasticity problems, $S$ is a crack, and in problems of acoustics and electrodynamics, it is a nonclosed screen. The results are similar to those indicated above. The corresponding operators are restrictions $A_S$ to $S$ of the single layer potentialtype operator $A$ and $H_S$ to $S$ of the hypersingular operator $H$ on $\Gamma$. For the corresponding spectral problems, the results are similar to those indicated above.

