Abstract
We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function u is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition if and only if u is its probabilistic solution, i.e. u can be represented by a suitable nonlinear Feynman–Kac functional. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.
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This work was supported by NCN Grant No. 2012/07/B/ST1/03508.
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Klimsiak, T., Rozkosz, A. Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form. Nonlinear Differ. Equ. Appl. 22, 1911–1934 (2015). https://doi.org/10.1007/s00030-015-0350-1
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DOI: https://doi.org/10.1007/s00030-015-0350-1