Abstract:
We consider two classes of problems for a strongly elliptic second-order 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 $(n-1)$-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 Neumann-to-Dirichlet operator $N$ and the Dirichlet-to-Neumann operator $D$ on parts $\Gamma_1$ and $\Gamma_2$ of $\Gamma$.
The operators $N_1$ and $D_2$ are connected with Poincaré–Steklov-type 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 non-selfadjoint case they form a complete system. If $\Gamma$ is almost smooth (smooth outside a closed subset of zero measure), then the eigenvalues of self-adjoint problems have natural asymptotics.
2. Problems with boundary or transmission conditions on a non-closed 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 non-closed screen. The results are similar to those indicated above. The corresponding operators are restrictions $A_S$ to $S$ of the single layer potential-type 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.
