Abstract
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic differential operators which do not necessarily have the maximum principle and are non-symmetric in general. Our method is probabilistic. It turns out that we need to solve a class of backward stochastic differential equations with singular coefficients, which is of independent interest itself. The theory of Dirichlet forms also plays an important role.
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Acknowledgments
This work is partly supported by National Natural Science Foundation of China (No.11671372, No.11431014, No.11401557).
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Yang, S., Zhang, T. Backward Stochastic Differential Equations and Dirichlet Problems of Semilinear Elliptic Operators with Singular Coefficients. Potential Anal 49, 225–245 (2018). https://doi.org/10.1007/s11118-017-9654-6
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DOI: https://doi.org/10.1007/s11118-017-9654-6
Keywords
- Dirichlet boundary value problem
- Semilinear second order elliptic differential equations
- Dirichlet forms
- Backward stochastic differential equations