viii INTRODUCTION

According to Dynkin, the idea of describing solutions of (2) by measures on the

boundary had been suggested to him by Brezis (see [Dy02], Chapter 10, Notes).

The rough trace. In 1993[LG93b], Le Gall establishes a one-to-one correspon-

dence between all (nonnegative) solutions of Au =

u2

in the unit disk D of

M2

and

pairs (K, v) where K is a compact subset of dD and v is a Radon measure on the

relatively open set O = dD\K (see [LG93b]). The set K may be seen as a set of

singularities on the boundary. More precisely, y G K if and only if

limsup dist(x, dD)2u(x) 0.

x — y, x e D

The measure v is then defined as a vague limit of measures lo(y)u(ry)a(dy) as r | 1

where a is the Lebesgue measure on dD. In [LG97], Le Gall proves similar results

for all smooth domains of

R2.

He also provides a probabilistic representation for a

solution in terms of the associated pair (K, v).

In 1996, Marcus and Veron extend these results by purely analytic methods to

the equation

Au =

ua

in the unit ball of

Rd,

when a 1 and d ® _ j (the so-called subcritical case).

The name "trace" is suggested in [MV96] and proofs can be found in [MV98a].

The first general definition of the trace (called later the "rough trace" by Dynkin

in [Dy02] in contrast with the "fine trace" defined below) is given in 1995 by

Dynkin and Kuznetsov (see [DK98a]). To every solution it, there corresponds a

pair TR(w) = (T,u) as above, where T C dD is closed and v is a Radon measure

on dD\T (see Section 1.3.8).

However it follows from a counterexample in [LG96] that except in dimension

2 (subcritical case), a solution is not uniquely determined by its rough trace.

From now on, we restrict ourselves to dimensions d ^ 3.

Moderate solutions. Let /c^(x,y), x G D, y G dD, be the Poisson kernel of D.

The Poisson integral representation

(4) hv(x) = / kD{x,y)v(dy), x G D

establishes a one-to-one correspondence between nonnegative harmonic functions

in D and the finite measures v on the boundary dD (see Section 1.2.1).

For every finite measure v on dD, we define the energy of v by

£{u) = J v{dy) J v{dy')Sd{\y - y'\) G K+ U {+»}

where

l + log+ (jl , ) , ifd = 3

\u\3~d,

if d ^ 4.

For every Borel subset F of 9D, we define the boundary capacity of T by

cap

a

(r) = inf (E(v))

where the infimum is taken over the set V(T) of all probability measures v on T

and where, by convention, oo

_ 1

= 0. When d ^ 4, cap^(r) is nothing but the