XIV
INTRODUCTION
Our results. The main objective of this work is to give a positive answer to
the questions raised in Problem 1 and Problem 2, in the particular case of equation
Au u2. In order to use the tools provided by the Brownian snake approach, we
have restricted ourselves to the equation Au =
u2
instead of trying to tackle the
general equation Lu ip(u). For technical reasons, we shall consider only domains
D that are bounded and of class
C4.
However, we are reasonably confident as far as
the generalization of our results and some of our methods to more general equations
and domains is concerned.
Equivalence between UY and capd(T). We emphasize above that UY = 0 if and
only if cap^(r) = 0. We investigate more closely the connections between ur and
cap
a
(r).
RESULT
0.1 (See Chapter 2, Section 3.3.1). There exist two positive continuous
functions C\ and C2 defined on D such that
Ci(x)capd(T) ^ UY{X) ^ C2{x)capd(T)
for every Borel set Y C dD and every x G D.
We give specific possible values for C\ and C2 in Theorem 3.15 (for C2) and
Proposition 3.16 (for C\).
ur = ur and r = r. We prove the following theorem which provides a positive
answer to Problem 2.
RESULT 0.2 (See Chapter 3, Theorem 3.1). For every Borel set T C dD, we
have
ur = ur-
In particular, ur is a-moderate.
It immediately follows from this theorem that the two fine topologies defined
by Dynkin and Kuznetsov coincide.
RESULT
0.3 (See Chapter 3, Corollary 3.5). We have the identity
T = T.
Classification results and probabilistic representation. We now give a detailed
statement of our main theorem.
RESULT 0.4 (See Chapter 5, Theorem 5.1). Let d^ 3 be an integer, let D be a
bounded domain of class C4 in M.d and let u be a nonnegative solution of Au 4u2
in D such that
tr(u) = (T,v).
Then
u
UY
® uv.
In probabilistic terms, for every x G D,
u(x) = Nx
(£D
n r ^ 0) + Nx (l£D
n r = 0
(1 - exp(-^))) .
A nonnegative solution is uniquely determined by its fine trace.
As a consequence of this theorem, we give a positive answer to Problem 1.
RESULT 0.5 (See Chapter 5, Theorem 5.2). All nonnegative solutions of Au =
u2
in a bounded domain of class
C4
in
Rd,
d^ 3, are a-moderate.
Note that Theorem 0.5 is deduced from Theorem 0.4 even though a direct proof
of Theorem 0.5 would imply Theorem 0.4.
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