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 =
in the unit disk D of
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
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 =
in the unit ball of
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 {+»}
l + log+ (jl , ) , ifd = 3
if d ^ 4.
For every Borel subset F of 9D, we define the boundary capacity of T by
(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
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