Cauchy fluxes and Gauss–Green formulas for divergence-measure fields over general open sets
| Autor: | Chen, G, Comi, G, Torres, M |
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| Přispěvatelé: | Chen, Gui-Qiang G., Comi, Giovanni Eugenio, Torres, Monica |
| Rok vydání: | 2019 |
| Předmět: |
Open set
Classification of discontinuities 01 natural sciences Mathematics (miscellaneous) Mathematics - Analysis of PDEs Settore MAT/05 - Analisi Matematica Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics Mathematics Continuum mechanics Mechanical Engineering 010102 general mathematics Gauss Mathematical analysis 28C05 26B20 28A05 26B12 35L65 35L67 76A02 (Primary) 28A75 28A25 26B05 26B30 26B40 35D30 (Secondary) Analysi Cauchy distribution 16. Peace & justice Lipschitz continuity Functional Analysis (math.FA) 010101 applied mathematics Mathematics - Functional Analysis Mathematics - Classical Analysis and ODEs Bounded function Gravitational singularity Analysis Analysis of PDEs (math.AP) |
| Popis: | We establish the interior and exterior Gauss-Green formulas for divergence-measure fields in $L^p$ over general open sets, motivated by the rigorous mathematical formulation of the physical principle of balance law via the Cauchy flux in the axiomatic foundation, for continuum mechanics allowing discontinuities and singularities. The method, based on a distance function, allows to give a representation of the interior (resp. exterior) normal trace of the field on the boundary of any given open set as the limit of classical normal traces over the boundaries of interior (resp. exterior) smooth approximations of the open set. In the particular case of open sets with continuous boundary, the approximating smooth sets can explicitly be characterized by using a regularized distance. We also show that any open set with Lipschitz boundary has a regular Lipschitz deformable boundary from the interior. In addition, some new product rules for divergence-measure fields and suitable scalar functions are presented, and the connection between these product rules and the representation of the normal trace of the field as a Radon measure is explored. With these formulas at hand, we introduce the notion of Cauchy fluxes as functionals defined on the boundaries of general bounded open sets for the rigorous mathematical formulation of the physical principle of balance law, and show that the Cauchy fluxes can be represented by corresponding divergence-measure fields. 81 pages, 1 figure; to be published in Arch. Ration. Mech. Anal. 2019 |
| Databáze: | OpenAIRE |
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