Criteria for Invariance of Convex Bodies for Linear Parabolic Systems

Gershon Kresin; Vladimir Maz'ya

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arXiv · 1402.5552

Criteria for Invariance of Convex Bodies for Linear Parabolic Systems

Gershon Kresin·Vladimir Maz'ya

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Abstract

We consider systems of linear partial differential equations, which contain only second and first derivatives in the $x$ variables and which are uniformly parabolic in the sense of Petrovski\v{\i} in the layer ${\mathbb R}^n\times [0,T]$. For such systems we obtain necessary and, separately, sufficient conditions for invariance of a convex body. These necessary and sufficient conditions coincide if the coefficients of the system do not depend on $t$. The above mentioned criterion is formulated as an algebraic condition describing a relation between the geometry of the invariant convex body and coefficients of the system. The criterion is concretized for certain classes of invariant convex sets: polyhedral angles, cylindrical and conical bodies.

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Gershon Kresin, Vladimir Maz'ya. 2014-02-22. Criteria for Invariance of Convex Bodies for Linear Parabolic Systems. https://arxiv.org/abs/1402.5552

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