Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators. (English) Zbl 0907.35039

Nadirashvili, Nikolai

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 3, 537-550 (1997).

Let \(L:= \sum a_{ij} (x)\partial_i \partial_j\) denote an elliptic operator on \(\mathbb{R}^n\) with measurable coefficients \(a_{ij}= a_{ji}\). Following M. C. Cerutti, L. Escauriaza and E. B. Fabes [Ann. Mat. Pura Appl., IV. Ser. 163, 161-180 (1993; Zbl 0799.35055)] a ‘good solution’ of the Dirichlet problem \(Lu=0\) in the smooth bounded domain \(\Omega \subset \mathbb{R}^n\), \(u=g\) on \(\partial \Omega\) is a limit of a sequence of classical solutions \(u_k\) to the Dirichlet problems \(L_ku_k =0\) in \(\Omega\), \(u_k=g\) on \(\partial \Omega\), where the operators \(L_k\) have a uniform ellipticity constant and their smooth coefficients tend to those of \(L\) almost everywhere. The author constructs an operator \(L\) such that for the \(n\)-dimensional unit ball \(\Omega\) \((n\geq 3)\) the Dirichlet problem above has at least two good solutions. According to this construction, nonuniqueness holds for a corresponding martingale problem related to \(L\).

Reviewer: K.J.Witsch (Essen)

MSC:

35J25	Boundary value problems for second-order elliptic equations
60G44	Martingales with continuous parameter
35R05	PDEs with low regular coefficients and/or low regular data

Citations:

Zbl 0799.35055

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