The Annals of Probability

Potential theory for elliptic systems

Z. Q. Chen and Z. Zhao

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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.

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Ann. Probab., Volume 24, Number 1 (1996), 293-319.

First available in Project Euclid: 15 January 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 35J45
Secondary: 60J60: Diffusion processes [See also 58J65]

Weakly coupled elliptic system weak solution Dirichlet boundary value problem Dirichlet space switched diffusion irreducibility resolvent kernel function


Chen, Z. Q.; Zhao, Z. Potential theory for elliptic systems. Ann. Probab. 24 (1996), no. 1, 293--319. doi:10.1214/aop/1042644718.

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