INTRODUCTION ix
classical Newtonian capacity of T, if T is viewed as a subset of a (d— 1)-dimensional
manifold.
A major role in what follows is played by the (nonnegative) solutions of (3)
which are dominated by a harmonic function. Following Dynkin and Kuznetsov in
[DK96b], such solutions are called moderate. A moderate solution has a minimal
harmonic majorant h which is itself associated with a measure v on the boundary via
Formula (4). According to [LG95] (see also [DK96b] in a more general framework
or Propositions 1.16 and 1.19 in Chapter 1), this construction establishes a one-
to-one correspondence between moderate solutions u of (3) and finite measures v
on the boundary which do not charge sets of zero boundary capacity. If uv is a
moderate solution of (3) associated with an admissible measure v on dD, the rough
trace of uv is given by
T R K ) = (0,i/).
We put v G M\ if v is a finite measure on dD which does not charge sets of
boundary capacity zero. Every v G A/i is an increasing limit of measures of finite
energy. We set v G A/o if v is a E-finite measure on 3D which does not charge sets
of boundary capacity zero. (Recall that a E-finite measure is a countable sum of
finite measures). If v G A/o, there exists a sequence vn G A/i such that vn \ v. One
can then define without ambiguity a solution uv by
uv lim | uUn.
(See Section 1.3.6). An element of
W0 := {uy ; v G J\f0}
is called a cr-moderate solution of (3). The u-moderate solutions may also be
defined as the increasing limits of moderate solutions.
Removable boundary singularities. The sets of boundary capacity zero are ex-
ceptional sets which play a crucial role. It is shown by Le Gall ([LG95]) and, in
a more general setting, by Dynkin and Kuznetsov in [DK96b] and Marcus and
Veron in [MV98a] and [MV98b], that they correspond to removable boundary
singularities for equation (3) (see Theorem 1.25). A removable boundary singular-
ity is a compact subset T C dD with the following property: if equation (3) holds
for u in D with u\QD\r 0, then u = 0. The same notion makes sense for the
general equation
Au =
ua,
a 1
in a smooth and bounded domain in
Rd.
There exists a nontrivial removable bound-
ary singularity if and only if d ^ _ j (supercritical case) whereas the empty set
is the only removable boundary singularity in the subcritical case d a _ j (see
[GmV]). In particular, as far as equation (3) is concerned, there exists a nontrivial
removable boundary singularity if and only if o f ^ 3.
If K C dD is compact, we define UK as the maximal nonnegative solution of
Au =
4u2
in D
u
\dD\K = 0.
(See Section 1.3.7). It is easy to see that UK = 0 if and only if K is a removable
boundary singularity, which holds if and only if c&pd(K) 0 by the remarks above.
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