Contemporary Mathematics

Volume 107, 1990

Gradient Estimates at the Boundary

for Solutions to Nondivergence

Elliptic Equations

BARTOLOME BARCELO, LUIS ESCAURIAZA, AND EUGENE FABES

Introduction

Questions of existence, uniqueness, and regularity of solutions to the clas-

sical Dirichlet problem for second order elliptic operators in nonvariational

form are well understood when the coefficients of the operator are contin-

uous. This degree of understanding completely changes when the property

of continuity on the coefficients is removed. No satisfactory theory of "well-

posedness" exists for nondivergence form elliptic operators under the mere

assumption of bounded measurable coefficients. The study of a priori esti-

mates remains a basic tool in establishing regularity properties of the possible

solutions to nonvariational problems. The hope is to obtain a rich enough

regularity theory guaranteeing uniqueness. Though this hope remains unre-

alized, recent progress has been made, and the purpose of this report is to

present this progress from a unified point of view, namely from the proper-

ties of Green's functions associated with elliptic operators in nondivergence

form.

We will consider the question of the a priori boundary regularity of solu-

tions

u

to the problem

(0.1)

Lu

=

f

in

D,

ulan=

0,

where D is a smooth bounded domain in Rn and L

=

E~.J=l

aiJ(x)D;ixj

satisfies: a(x)

=

(a;/X)) is a bounded symmetric and uniformly positive

definite matrix; i.e. there exists A , 0 A $ 1 SUCh that for all

X

and

!

in

Rn

1980 Mathematics Subject Classification ( 1985 Revision). Primary 35B45, 35B65, 35J15.

Research partially supported by NSF grant DMS-8421377-03.

©

1990 American Mathematical Society

0271-4132/90 $1.00

+ $.2S

per pqe

http://dx.doi.org/10.1090/conm/107/1066466