Mixed boundary value problems for the Laplace–Beltrami equation

Abstract

We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth bounded surface C with a smooth boundary in non-classical setting in the Bessel potential spaceTo the initial BVP we apply a quasi-localization and obtain a model BVP for the Laplacian. The model mixed BVP on the half plane is reduced to an equivalent system of Mellin convolution equation (MCE) in Sobolev-Slobodečkii space (potential method). MCE is ivestigated in both Bessel potential and Sobolev-Slobodečkii spaces. The symbol of the obtained system is written explicitly and is responsible for the Fredholm properties and the index of the system. An explicit criterion for the unique solvability of the initial BVP in the non-classical setting is derived as well.

Mixed impedance boundary value problems for the Laplace–Beltrami equation

“…In [15,19] a different approach was suggested, which allows investigation of boundary value problems for elliptic partial differential equations in the nonclassical Lebesgue space setting (3.3). The investigation is based on the boundary integral equation method and the solvability results for Mellin convolution equations in the Bessel potential spaces.…”

“…of (5.1), (5.2) and (5.3)) understand the invertibility, Fredholmness and local invertibility of the corresponding boundary integral operators (of the operators in (5.12), (6.5) and in (6.9), respectively). Let us recall the main results about the mixed Dirichlet-Neumann BVP (5.1), obtained in [19].…”

“…Throughout the paper we will consider c = 0 since otherwise our problem would be simply the Dirichlet-Neumann boundary value problem already analysed in [23, Theorem 14], in [27, § 5.1] in the classical Hilbert space setting (see (3.1) below) and in [19] in the nonclassical Lebesgue space setting (see (3.3)…”

Laplace-Beltrami equation on a hypersurface with Lipschitz boundary

“…even for a hypersurface with the smooth boundary and for Dirichlet (5.2) and Neumann (5.3) BVPs for a hypersurface with the Lipshitz boundary and the non-classical setting (1.5), the solvability conditions change dramatically (cf. [13,14,10]).…”

Regularized integral formulation of mixed Dirichlet-Neumann problems

“…After the initial contribution by Zaremba, early works concerning the Zaremba boundary value problem include results by Signorini [25] (1916: solution of the Zaremba problem in the upper half plane using complex variable methods); Giraud [17] (1934: existence of solution of Zaremba problems for general elliptic operators); Fichera [13, 14] (1949, 1952: regularity studies at Zaremba points, Zaremba-type problem for the elasticity equations in two spatial dimensions); Magenes [22] (1955: proof of existence and uniqueness, single layer potential representation); Lorenzi [21] (1975: Sobolev regularity around a corner which is also a Dirichlet-Neumann junction); and Wigley [29,30] (1964,1970: explicit asymptotic expansions around Dirichlet-Neumann junctions), amongst others. More recent contributions in this area include reference [28], which provides a valuable review in addition to a study of Zaremba singularities and theoretical results concerning Galerkin-based computational approaches; reference [7], which considers the Zaremba problem for the biharmonic equation; references [10,11], which study Zaremba boundary value problems for Helmholtz and Laplace-Beltrami equations; reference [8], which discusses the solvability of the Zaremba problem from the point of view of pseudo-differential calculus and Sobolev regularity theory; reference [18] which introduces a certain inverse preconditioning technique to reduce the number of linear algebra iterations for the iterative numerical solution of this problem and which gives rise to high-order convergence; and finally, reference [4], which successfully applies the method of difference potentials to the variable-coefficient Zaremba problem, with convergence order approximately equal to one.…”