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Non-negative solutions of \(\Delta u=u^ 2\) in the unit disc. (Les solutions positives de \(\Delta u=u^ 2\) dans le disque unité.) (French. Abridged English version) Zbl 0791.60049

Summary: We use a probabilistic approach to derive a general representation theorem for nonnegative solutions of the nonlinear equation \(\Delta u=u^ 2\) in the unit disc \(D\). We establish a one-to-one correspondence between nonnegative solutions and pairs \((K,\nu)\), where \(K\) is a compact subset of \(\partial D\) and \(\nu\) is a Radon measure on \(\partial D \backslash K\). The set \(K\) is the set of points in \(\partial D\) where the solution \(u\) explodes badly, say like the inverse of the square of the distance to the boundary. The measure \(\nu\) can be interpreted as the “boundary value” of \(u\) on \(\partial D \backslash K\). Conversely, the solution \(u\) is written in terms of \((K,\nu)\) by an explicit probabilistic formula.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
35R60 PDEs with randomness, stochastic partial differential equations
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