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Potential theory for elliptic systems. (English) Zbl 0854.60062
Summary: The existence and uniqueness theorem is proved for solutions of the Dirichlet boundary value problems for weakly coupled elliptic systems on bounded domains. The elliptic systems are only assumed to have measurable coefficients and have singular coefficients for the lower-order terms. A probabilistic representation theorem for solutions of the Dirichlet boundary value problems is obtained by using the switched diffusion process associated with the system. A strong positivity result for solutions of the Dirichlet boundary value problems is proved. Formulas expressing resolvents and kernel functions for the system by those of the component elliptic operators are also obtained.

60H30 Applications of stochastic analysis (to PDEs, etc.)
60J60 Diffusion processes
35J45 Systems of elliptic equations, general (MSC2000)
Full Text: DOI
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[19] ITHACA, NEW YORK COLUMBIA, MISSOURI 65211 E-mail: zchen@math.cornell.edu E-mail: mathzz@mizzou1.missouri.edu
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