No Lavrentiev gap for some double phase integrals

IF 1.3 3区数学 Q1 MATHEMATICS

Advances in Calculus of Variations Pub Date : 2022-08-30 DOI:10.1515/acv-2021-0109

Filomena De Filippis, F. Leonetti

引用次数: 5

Abstract

Abstract We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ⁢ ( u ) ≔ ∫ Ω f ⁢ ( x , D ⁢ u ⁢ ( x ) ) ⁢ 𝑑 x , \mathcal{F}(u)\coloneqq\int_{\Omega}f(x,Du(x))\,dx, where the density f ⁢ ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {x\in\Omega\subset\mathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ⁢ ( x , z ) ⩽ L ⁢ ( 1 + | z | q ) , \lvert z\rvert^{p}\leqslant f(x,z)\leqslant L(1+\lvert z\rvert^{q}), where 1 < p < q < p ⁢ ( n + α n ) {1

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对于某些双相积分没有Lavrentiev隙

摘要证明了非自治函子＿＿ (u)是∫Ω f ＿ (x)，D ＿ (u) ＿ (x)) ＿𝑑x, \mathcal{F} (u) \coloneqq\int ＿ {\Omega} f(x,Du(x))\，dx，其中f ＿ (x, z) {f(x,z)}对于x∈Ω是α-Hölder连续的，∧∈Ω {x\in\Omega\subset\mathbb{R} ^ {n}}满足(p，q) {(p,q)} -生长条件| z | p≤f(x,z)≤L(1+ | z | q)， \lvert z \rvert ^ {p}\leqslant f(x,z) \leqslant L(1+ \lvert z \rvert ^ {q})，其中1 < p < q < p≠(n + α n) {1

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来源期刊

Advances in Calculus of Variations MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

3.90

自引率

5.90%

发文量

审稿时长

>12 weeks

期刊介绍： Advances in Calculus of Variations publishes high quality original research focusing on that part of calculus of variation and related applications which combines tools and methods from partial differential equations with geometrical techniques.

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