Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems

IF 1.3 3区 数学 Q4 AUTOMATION & CONTROL SYSTEMS Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-01-24 DOI:10.1051/cocv/2022050
Antonio Giuseppe Grimaldi, Erica Ipocoana
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引用次数: 3

Abstract

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form \begin {gather*} \min \biggl\{ \int_{\Omega} F(x,w,Dw) d x \ : \ w \in \mathcal{K}_{\psi}(\Omega) \biggr\}, \end {gather*} with $F$ double phase functional of the form \begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end {equation*} where $\Omega$ is a bounded open subset of $\mathbb{R}^n$ , $\psi \in W^{1,p}(\Omega)$ is a fixed function called \textit { obstacle } and $\mathcal{K}_{\psi}(\Omega)= \{ w \in W^{1,p}(\Omega) : w \geq \psi \ \text{a.e. in} \ \Omega \}$ is the class of admissible functions . Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property .
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由于Besov空间具有较高的可微性,使得Besov空间可扩展为一类双相障碍问题
我们研究了形式为\begin {gather*} \min \biggl\{\int_{\Omega} F(x,w,Dw) d x \的变分障碍问题解梯度的高分数可微性:\ w \in \mathcal{K}_{\psi}(\Omega) \biggr\}, \end {gather*}与$F$双相泛函的形式为\begin {equation*} F(x,w,z)=b(x,w)(|z|^p+a(x)|z|^q), \end {equation*}其中$\Omega$是$\mathbb{R}^n$的有界开放子集,$\psi \in w ^{1,p}(\Omega)$是一个固定函数,称为\ texttit {obstacle}和$\mathcal{K}_{\psi}(\Omega)= \{w \in w ^{1,p}(\Omega): w \geq \psi \ \text{a.e。in} \ \ \ \}$是可容许函数的类。假设障碍物的梯度属于一个合适的Besov空间,我们能够证明解的梯度保持一定的分数可微性。
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来源期刊
Esaim-Control Optimisation and Calculus of Variations
Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics
自引率
7.10%
发文量
77
期刊介绍: ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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