Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains
The double layer electrostatics operator corresponding to the pseudostress conormal derivative and the double layer hydrostatics operator corresponding to the stress conormal derivative are studied. Let \(K\) be one of those operators, and let \(\Omega\) be a bounded Lipschitz domain in \(\mathbb R^n\) with the Lipschitz character \(char (\Omega)\). It is proved that there exists a constant \(C_0\) such that if \(char (\Omega) < C_0\) then for any \(\beta\in\mathbb C\), \(|\beta|\geq 1\), \(\beta\neq 1\), the operator \((\beta I - K)\) is invertible on \(L^2(\partial\Omega)\) where \(I\) is the identity operator. Moreover, in the electrostatics case the operator \(I - K\) is Fredholm with index zero on \(L^2(\partial\Omega)\) and \(I - K^{\ast}\) is invertible on \(L^2_0(\partial\Omega)\), the subspace of functions with zero mean value on \(\partial\Omega\); in the hydrostatics case again the operator \(I - K\) is Fredholm with index zero on \(L^2(\partial\Omega)\) but \(I - K^{\ast}\) is invertible now on \(L^2_{\psi}(\partial\Omega)\), the subspace of functions whose weighted mean values, with certain weights, are zero. For \(\Omega\) being a bounded, simply connected Lipschitz polyhedron \(\mathcal P\) in \(\mathbb R^3\), the same conclusions are drawn under certain size restrictions for polyhedron's dihedral angles.
scientific article; zbMATH DE number 1340864