We are concerned with the nonnegative solutions of Au = u2 in a bounded
and smooth domain in Rd. We prove that they are uniquely determined by their
fine trace on the boundary as defined in [DK98a], thus answering a major open
question of [Dy02]. A probabilistic formula for a solution in terms of its fine trace
and of the Brownian snake is also provided. A major role is played by the solutions
which are dominated by a harmonic function in D. The latters are called moderate
in Dynkin's terminology We show that every nonnegative solution of Au =
D is the increasing limit of moderate solutions.
Received by the editor 31 Dec 2002.
2000 Mathematics Subject Classification. 35J65 (35J60 35C99 60J80).
Key words and phrases. Elliptic partial differential equations, Representation of solutions,
Branching processes.
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