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.