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Homological dimensions for endomorphism algebras of Gorenstein projective modules

Aiping ZhangXueping Lei — 2024

Czechoslovak Mathematical Journal

Let A be a CM-finite Artin algebra with a Gorenstein-Auslander generator E , M be a Gorenstein projective A -module and B = End A M . We give an upper bound for the finitistic dimension of B in terms of homological data of M . Furthermore, if A is n -Gorenstein for 2 n < , then we show the global dimension of B is less than or equal to n plus the B -projective dimension of Hom A ( M , E ) . As an application, the global dimension of End A E is less than or equal to n .

Local boundedness for minimizers of variational integrals under anisotropic nonstandard growth conditions

Zesheng FengAiping ZhangHongya Gao — 2024

Czechoslovak Mathematical Journal

This paper deals with local boundedness for minimizers of vectorial integrals under anisotropic growth conditions by using De Giorgi’s iterative method. We consider integral functionals with the first part of the integrand satisfying anisotropic growth conditions including a convex nondecreasing function g , and with the second part, a convex lower order term or a polyconvex lower order term. Local boundedness of minimizers is derived.

Regularizing effect of the interplay between coefficients in some noncoercive integral functionals

Aiping ZhangZesheng FengHongya Gao — 2024

Czechoslovak Mathematical Journal

We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type 𝒥 ( v ) = Ω j ( x , v , v ) d x + Ω a ( x ) | v | 2 d x - Ω f v d x , v W 0 1 , 2 ( Ω ) , where Ω N , j is a Carathéodory function such that ξ j ( x , s , ξ ) is convex, and there exist constants 0 τ < 1 and M > 0 such that | ξ | 2 ( 1 + | s | ) τ j ( x , s , ξ ) M | ξ | 2 for almost all x Ω , all s and all ξ N . We show that, even if 0 < a ( x ) and f ( x ) only belong to L 1 ( Ω ) , the interplay | f ( x ) | 2 Q a ( x ) implies the existence of a minimizer u W 0 1 , 2 ( Ω ) which belongs to L ( Ω ) .

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