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Invariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms

hal.structure.identifierInstitut for Applied Analysis
dc.contributor.authorARENDT, Wolfgang
hal.structure.identifierInstitut for Applied Analysis
dc.contributor.authorDIER, Dominik
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorOUHABAZ, El Maati
dc.date.accessioned2024-04-04T02:22:30Z
dc.date.available2024-04-04T02:22:30Z
dc.date.created2013-02-27
dc.date.issued2014
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189656
dc.description.abstractEnWe consider a non-autonomous form $\fra:[0,T]\times V\times V \to \C$ where $V$ is a Hilbert space which is densely and continuously embedded in another Hilbert space $H$. Denote by $\A(t) \in \L(V,V')$ the associated operator. Given $f \in L^2(0,T, V')$, one knows that for each $u_0 \in H$ there is a unique solution $u\in H^1(0,T;V')\cap L^2(0,T;V)$ of $$\dot u(t) + \A(t) u(t) = f(t), \, \, u(0) = u_0.$$ %\begin{align*} %&\dot u(t) + \A(t)u(t)= f(t)\\ %& u(0)=u_0. %\end{align*} This result by J.~L.~Lions is well-known. The aim of this article is to find a criterion for the invariance of a closed convex subset $\Conv$ of $H$; i.e.\ we give a criterion on the form which implies that $u(t)\in \Conv$ for all $t\in[0,T]$ whenever $u_0\in\Conv$. In the autonomous case for $f = 0$, the criterion is known and even equivalent to invariance by a result proved in \cite{Ouh96} (see also \cite{Ouh05}). We give applications to positivity and comparison of solutions to heat equations with non-autonomous Robin boundary conditions. We also prove positivity of the solution to a quasi-linear heat equation.
dc.description.sponsorshipAux frontières de l'analyse Harmonique - ANR-12-BS01-0013
dc.language.isoen
dc.subject.enSesquilinear forms
dc.subject.ennon-autonomous evolution equations
dc.subject.ennon-linear heat equations
dc.subject.eninvariance of closed convex sets
dc.title.enInvariance of Convex Sets for Non-autonomous Evolution Equations Governed by Forms
dc.typeArticle de revue
dc.identifier.doi10.1112/jlms/jdt082
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halMathématiques [math]/Analyse fonctionnelle [math.FA]
dc.identifier.arxiv1303.1167
bordeaux.journalJ. London Math. Soc
bordeaux.page14
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00797182
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00797182v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=J.%20London%20Math.%20Soc&rft.date=2014&rft.spage=14&rft.epage=14&rft.au=ARENDT,%20Wolfgang&DIER,%20Dominik&OUHABAZ,%20El%20Maati&rft.genre=article


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