Random relaxed Dirichlet problems. (English) Zbl 0673.60067

Summary: We investigate sequences of relaxed Dirichlet problems of the form: \[ - \Delta u_ h+\mu_ hu_ h=0 \] where \(\mu_ h\) are random Borel measures belonging to a suitable class \({\mathcal M}_ 0\). By means of a variational approach, necessary and sufficient conditions for the convergence in probability of the sequence \(u_ h\) toward the solution of a deterministic relaxed Dirichlet problem are given. Some applications to Dirichlet problems in random perturbated domains and to a Schrödinger equation with random singular potentials are considered.


60H25 Random operators and equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory
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