Known and unknown results on elliptic boundary problems

Gerd Grubb
2006 Bulletin of the American Mathematical Society  
In a survey article by W. N. Everitt and L. Markus [EM05] in the October 2005 issue of Bulletin AMS, the authors raise some questions on boundary problems for elliptic partial differential operators. The article is primarily concerned with ODE where the choice of boundary conditions is finite dimensional, but a secondary purpose is to extend their points of view to elliptic PDE where the possible boundary conditions range in infinite-dimensional spaces, and here the authors present a number of
more » ... hat they consider open problems. The purpose of this note is to show how answers to the problems, as well as much more extensive results, can be found in the existing literature. Section 2.2 of [EM05] presents the partial differential operator to be considered, applied in the distribution sense, the authors consider several operators acting like A (realizations of A): • T 0 is the minimal operator, with domain D(T 0 ) = • W 2 (Ω), • T 1 is the maximal operator, with domain D(T 1 ) = {u ∈ L 2 (Ω) | Au ∈ L 2 (Ω)}, • T Dir is the Dirichlet realization, with domain D(T Dir ) = W 2 (Ω) ∩ • W 1 (Ω). It has been known for many years that T Dir is selfadjoint in L 2 (Ω) with a positive lower bound, that T 0 is closed, densely defined and symmetric with the same lower bound, and that T 1 = T * 0 . T 1 has an infinite-dimensional nullspace consisting of the harmonic functions in L 2 (Ω), In Section 2.3, the authors introduce one more realization, T Har , that they find mysterious; its domain is and it is selfadjoint. They prove by examples, referring to [EMP05] for details, that D(T Har ) is not contained in W 1 (Ω) and that there are elements of D(T Har ) that do not have pointwise radial limits for r 1. They claim on p. 480: "It is an unsolved problem as to whether the operator T Har , as a self-adjoint extension of the Laplace operator in the unit disk, is unique in some noteworthy
doi:10.1090/s0273-0979-06-01114-1 fatcat:jpijhwoivbbutdvd75g6gs4sii