Criteria for well-posedness of degenerate elliptic and parabolic problems

Abstract

We provide explicit criteria for uniqueness or nonuniqueness of solutions to a wide class of second order elliptic and parabolic problems. The operator coefficients may be unbounded or vanish, or not to have a limit when approaching some part of the boundary, referred to as singular boundary. We discuss whether boundary conditions should be imposed on such a part to ensure well-posedness. The answer depends on the dimension of the singular boundary, and possibly on the behavior of coefficients near it. © 2008 Elsevier Masson SAS. All rights reserved

A note on the strong maximum principle

“…Similar methods have been used to study the wellposedness of boundary value problems for elliptic and parabolic equations with singular coefficients (see [18,20] and the references therein). Observe that, by the connection between the operator −u + cu, with c unbounded, and the degenerate elliptic operator − 1 c u , equation (1.2) is the counterpart of the first exit time equation (1.1).…”

Numerical treatment of degenerate diffusion equations via Feller's boundary classification, and applications

“…A recent analytical work [16] has been conducted on existence and uniqueness of solutions to degenerate elliptic and parabolic equations on open bounded connected domains in R n . The authors discuss when BCs have to be imposed, but not which ones they should or could be.…”

“…[15] and references therein, and very recently [16]. On the other hand, in recent years, the numerical treatment of linear (and nonlinear) degenerate parabolic PDEs, in one or more space dimensions, has been considered in connection with the so-called Black-Scholes models, which started playing an important role in financial mathematics in 1973.…”

Uniqueness and support properties of solutions to singular quasilinear parabolic equations on surfaces of revolution

“…see the proof of Theorem 2.18 in [14]), we extend V to the whole of M so that it is a supersolution to Eq. (4.3) satisfying (4.6) and (4.7).…”