Sturm–Liouville problems in weighted spaces in domains with non-smooth edges. II

Abstract

Abstract. We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain D of R n for a second order elliptic differential operator A(x, ∂). The differential operator is assumed to be of divergent form in D and the boundary operator B(x, ∂) is of Robin type on ∂D. The boundary of D is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset Y ⊂ ∂D and control the growth of solutions near Y . We prove that the pair (A, B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set Y . Moreover, we prove the completeness of root functions related to L.

Finding Eigenvalues and Eigenfunctions of the Zaremba Problem for the Circle

“…In this case the problem (2.2) is called non-coercive. However, in many situations one may successfully use non-coercive forms to study boundary value problems (see, for instance, [11], [19], [20]). We follow these examples.…”

“…), see, for instance, [5], [6], [2], [12], [3], [4], [8], [22] and many others. Recently the approach was adopted to a wide class of non-coercive mixed boundary problems, see [19], [20].…”

On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions

“…for almost all x ∈ D (see, for instance, [14]). For example, one could take the standard non-negative self-adjoint square root D(x) = A(x) of the matrix A(x).…”