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Invariance of convex sets for non‐autonomous evolution equations governed by forms
Author(s) -
Arendt Wolfgang,
Dier Dominik,
Ouhabaz El Maati
Publication year - 2014
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdt082
Subject(s) - hilbert space , regular polygon , mathematics , operator (biology) , space (punctuation) , boundary (topology) , combinatorics , pure mathematics , heat equation , mathematical physics , mathematical analysis , geometry , computer science , biochemistry , chemistry , repressor , transcription factor , gene , operating system
We consider a non‐autonomous form a : [ 0 , T ] × V × V → C where V is a Hilbert space which is densely and continuously embedded in another Hilbert space H . Denote by A ( t ) ∈ L ( V , V ' )the operator associated with a ( t , · , · ) . Given f ∈ L 2 ( 0 , T , V ' ) , one knows that for eachu 0 ∈ H there is a unique solution u ∈ H 1 ( 0 , T ; V ' ) ∩ L 2 ( 0 , T ; V )ofu ˙ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = u 0 .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 C of H ; that is, we give a criterion on the form which implies that u ( t ) ∈ C for all t ∈ [ 0 , T ] wheneveru 0 ∈ C . In the autonomous case for f = 0 , the criterion is known and even equivalent to invariance by a result proved by Ouhabaz ‘Invariance of closed convex sets and domination criteria for semigroups’, Potential Anal . 5 (1996) 611–625. See also Ouhabaz ‘ Analysis of heat equations on domains ’, London Mathematical Society Monographs. Princeton University Press, Princeton, NJ, 2005. 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.