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• W3120646987 abstract "We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle. En employant la approche théorique de la mesure, nous présentons une étude de la régularité locale de supersolutions faibles aux équations elliptiques et paraboliques dans la forme divergeante. Dans la première partie, nous montrons que les supersolutions faibles deviennent semi-continues inférieurement après redéfinition sur un ensemble de mesure zèro. La preuve repose sur un principe général, à savoir le lemme de type De Giorgi, qui offre une approche unifiée d'une large classe d'équations elliptiques et paraboliques, y compris une équation elliptique anisotrope, l'équation parabolique associé au p-Laplacien et l'équation des milieux poreux. Dans une seconde partie, nous établissons que dans le cas des problémes paraboliques, le représentant semi-continu inférieurement à un instant provient en chaque point de la «ess lim inf» du temps écoulé. Nous montrons également qu'il peut être récupéré en limite par une valeur moyenne sur le passé. La preuve s'appuie sur l'expansion de la positivité des supersolutions faibles. Nos résultats établissent des propriétés structurelles d'équations aux dérivées partielles, indépendantes de tout type de principe de comparaison." @default.
• W3120646987 created "2021-01-18" @default.
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• W3120646987 date "2021-03-01" @default.
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• W3120646987 title "Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior" @default.
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• W3120646987 doi "https://doi.org/10.1016/j.matpur.2021.01.008" @default.
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