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首页> 外文期刊>The Journal of the London Mathematical Society >Invariance of convex sets for non-autonomous evolution equations governed by forms
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Invariance of convex sets for non-autonomous evolution equations governed by forms

机译：非自治演化方程受形式控制的凸集的不变性

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摘要

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 each u_0 ∈ H there is a unique solution u ∈ H1(0, T; V') ∩ L~2(0, T; V) of u (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] whenever u_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.
机译：我们考虑一个非自治形式a：[0，T]×V×V→C其中V是一个希尔伯特空间，它密密麻麻并连续地嵌入另一个希尔伯特空间H。用A（t）∈L（V，V表示'）与a（t，·，·）相关的运算符。给定f∈L〜2（0，T，V'），人们知道对于每个u_0∈H都有唯一解u∈H1（0，T; V'）∩L〜2（0，T; V）的u（t）+ A（t）u（t）= f（t），u（0）= u_0.JL Lions的结果众所周知。本文的目的是找到H的闭合凸子集C不变性的判据。也就是说，我们给出一个形式为的准则，该准则意味着无论何时u_0∈C，所有t∈[0，T]的u（t）∈C。在f = 0的自治情况下，该准则是已知的，甚至等于由Ouhabaz“封闭凸集的不变性和半群的控制标准”证明的结果不变。 5（1996）611–625。另请参见Ouhabaz的“域热方程分析”，伦敦数学协会专论。普林斯顿大学出版社，新泽西州普林斯顿，2005年。我们提供了正性应用以及具有非自治Robin边界条件的热方程解的比较。我们还证明了拟线性热方程解的正性。

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