TY - CHAP

T1 - Criteria for invariance of convex sets for linear parabolic systems

AU - Kresin, Gershon

AU - Maz’Ya, Vladimir

N1 - Publisher Copyright:
© 2015 G. Kresin, V. Maz’ya.

PY - 2015

Y1 - 2015

N2 - 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 Petrovskiǐ in the layer ℝn × [0,T]. For such systems we obtain necessary and, separately, sufficient conditions for invariance of the closure of an arbitrary convex proper subdomain of ℝm. 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 set and the coefficients of the system. The criterion is concretized for certain classes of invariant convex sets: polyhedral angles, cylindrical and conical bodies.

AB - 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 Petrovskiǐ in the layer ℝn × [0,T]. For such systems we obtain necessary and, separately, sufficient conditions for invariance of the closure of an arbitrary convex proper subdomain of ℝm. 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 set and the coefficients of the system. The criterion is concretized for certain classes of invariant convex sets: polyhedral angles, cylindrical and conical bodies.

UR - http://www.scopus.com/inward/record.url?scp=85050119599&partnerID=8YFLogxK

U2 - 10.1090/conm/653/13188

DO - 10.1090/conm/653/13188

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AN - SCOPUS:85050119599

T3 - Contemporary Mathematics

SP - 227

EP - 241

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -