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