DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

SMOOTH SOLUTIONS OF ELLIPTIC EQUATIONS
IN NONSMOOTH DOMAINS
GARY M. LIEBERMAN
We discuss some situations in which the solution of an elliptic boundary value problem
is smoother than the regularity of the boundary.
Copyright © 2006 Gary M. Lieberman. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this lecture, we examine a special regularity result for solutions of second-order elliptic
equations. We ask when the solution of a boundary value problem for such an equation
is smoother than the boundary of the domain in which the problem is posed.
To see the significance of this question, we first recall that classical Schauder theory
(see [6, Chapter 6]) says that if ∂Ω ∈ C k,α for some integer k ≥ 2andsomeα ∈ (0,1), then
solutions of any of the standard boundary value problems (i.e., the Dirichlet problem, the
Neumann problem, or the oblique derivative problem) with sufficiently smooth data are
also in C k,α . Specifically, if the elliptic operator L has the form
Lu = a ij D ij u + b i D i u + cu (1.1)
with a ij , b i ,andc all in C k−2,α (Ω), then any solution of the Dirichlet problem
Lu = f in Ω, u = ϕ on ∂Ω (1.2)
with f ∈ C k−2,α (Ω)andϕ ∈ C k,α (∂Ω)isinC k,α (Ω) with similar results for the Neumann
and oblique derivative problems. In addition, if we only assume that ∂Ω ∈ C 1,α ,then
any solution of one of the boundary value problems (with appropriate smoothness on
the other data) is also in C 1,α (see [5] for a precise statement of the result in the case
of Dirichlet data and [9] in the case of Neumann or oblique derivative data). Moreover,
this result is optimal in the following sense: given k and α, we can find a domain Ω with
∂Ω ∈ C k,α and Dirichlet data ϕ ∈ C k,α such that the solution of
Δu = 0 inΩ, u = ϕ on ∂Ω, (1.3)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 677–682