Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form
| Autor: | Andrzej Rozkosz, Tomasz Klimsiak |
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| Rok vydání: | 2015 |
| Předmět: |
Class (set theory)
Dirichlet form Applied Mathematics Operator (physics) Mathematical analysis Mathematics::Analysis of PDEs Feynman–Kac formula Function (mathematics) Parabolic partial differential equation Measure (mathematics) Nonlinear system Mathematics - Analysis of PDEs FOS: Mathematics Primary: 35D99 Secondary: 35J61 35K58 60H30 Analysis Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Nonlinear Differential Equations and Applications NoDEA. 22:1911-1934 |
| ISSN: | 1420-9004 1021-9722 |
| DOI: | 10.1007/s00030-015-0350-1 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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