EXISTENCE THEOREMS FOR NONLINEAR
PARTIAL DIFFERENTIAL EQUATIONS
FELIX E. BROWDER
PART I
Introduction. It is my purpose in the present discussion to describe the applica-
tion of some recently developed general techniques in nonlinear functional
analysis to the study of the existence theory for solutions of various classes of
nonlinear partial differential equations, specifically nonlinear elliptic boundary
value problems, nonlinear elliptic eigenvalue problems, and nonlinear equations
of evolution of parabolic and wave equation type. Corresponding to these three
classes of problems, we shall consider three related abstract theories: the theory
of nonlinear operators of monotone type from a reflexive Banach space X to its
conjugate space X*, the application of the Lusternik-Schnirelman category in
infinite dimensional spaces, and the theory of semigroups of nonlinear non-
expansive operators in Hilbert and Banach spaces.
Our partial differential equations will be defined for the sake of simplicity on
an open subset G of an Euclidean n-dimensional space
Rn
rather than upon a
manifold and we shall speak for convenience of notation of a single equation
rather than a system of equations. We shall make no use, however, of the
maximum principle for second order scalar elliptic or parabolic equations or of
the Di Giorgi-Nash estimates or their consequences so that there is no conceptual
loss of generality in the restriction.
The points of the open set G will be denoted by x = (xx, ...,x„) and the
elementary differential operators by
Da
= flj= 1
(^/^xjTJ f°r a n
ordered n-tuple
a = (xl9..., a„) of nonnegative integers, with the order of the operator
Da
being
written as |a| = Yj=i
a
/ To write nonlinear partial differential operators in a
convenient form, we introduce the vector space
RSm
whose elements are
£ m
=
{£x |
|a|
^
m} anc*
divide each such £ into two parts £m = ((, rj) where
f
=
{fyl |y5| m 1} e
RSmi
is the lower order part of £m, and ( = {£a| |a| = m}
is the part of £, corresponding to the mth derivatives. In this notation, a general
(nonlinear) partial differential operator of order m is defined by a mapping
F.G x
RSm
-+
R1
and defines an operator on functions u on G which assigns to
each such, w, another function v on G with
v(x) = F(x, £m(u)(x)), xeG,
where £»(* ) =
{Dau(x)\
|a| m}.
The systematic study of boundary value problems for nonlinear partial differen-
tial equations goes back as far at least as the work of Sergei Bernstein in the first
decade of the twentieth century who made a determined and difficult attack
upon second-order nonlinear elliptic equations in the plane using perturbation
l
http://dx.doi.org/10.1090/pspum/016/0269962
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