caa a22 4500 606193847 CHVBK 20210128100911.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s00030-015-0350-1 doi (NATIONALLICENCE)springer-10.1007/s00030-015-0350-1 Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form [Elektronische Daten] [Tomasz Klimsiak, Andrzej Rozkosz] 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. The Author(s), 2015 Semilinear equation nationallicence Dirichlet form and operator nationallicence Measure data nationallicence Renormalized solution nationallicence Feynman-Kac formula nationallicence Klimsiak Tomasz Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warszawa, Poland aut Rozkosz Andrzej Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland aut Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/6(2015-12-01), 1911-1934 1021-9722 22:6<1911 2015 22 30 https://doi.org/10.1007/s00030-015-0350-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00030-015-0350-1 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Klimsiak Tomasz Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956, Warszawa, Poland aut NATIONALLICENCE 700 1- Rozkosz Andrzej Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland aut NATIONALLICENCE 773 0- Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/6(2015-12-01), 1911-1934 1021-9722 22:6<1911 2015 22 30