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

Zesheng Feng; Aiping Zhang; Hongya Gao

Displaying similar documents to “Local boundedness for minimizers of variational integrals under anisotropic nonstandard growth conditions”

Gradient regularity for minimizers of functionals under $p$ - $q$ subquadratic growth

F. Leonetti, E. Mascolo, F. Siepe (2001)

Bollettino dell'Unione Matematica Italiana

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Si prova la maggior sommabilità del gradiente dei minimi locali di funzionali integrali della forma $\int_{Ω} f (D u) d x,$ dove $f$ soddisfa l'ipotesi di crescita ${|ξ|}^{p} - c_{1} \leq f (ξ) \leq c (1 + {|ξ|}^{q}),$ con $1 < p < q \leq 2$ . L'integrando $f$ è $C^{2}$ e $D D f$ ha crescita $p - 2$ dal basso e $q - 2$ dall'alto.

Partial regularity of minimizers of quasiconvex integrals with subquadratic growth: the general case

Menita Carozza, Giuseppe Mingione (2001)

Annales Polonici Mathematici

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We prove partial regularity for minimizers of the functional $\int_{Ω} f (x, u (x), D u (x)) d x$ where the integrand f(x,u,ξ) is quasiconvex with subquadratic growth: $| f (x, u, ξ) | \leq L (1 + | ξ |^{p})$ , p < 2. We also obtain the same results for ω-minimizers.

Regularity for minimizers of non-autonomous non-quadratic functionals in the case $1<p<2$ : an a priori estimate

Andrea Gentile (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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We establish an a priori estimate for the second derivatives of local minimizers of integral functionals of the form $\mathcal{F}(\nu,\Omega)=\int_{\Omega}f(x,D\nu(x))\,dx$ with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the $x$ variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

Morrey regularity and continuity results for almost minimizers of asymptotically convex integrals

Mikil Foss, Antonia Passarelli di Napoli, Anna Verde (2008)

Applicationes Mathematicae

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In a recent paper [Forum Math., 2008] the authors established some global, up to the boundary of a domain Ω ⊂ ℝⁿ, continuity and Morrey regularity results for almost minimizers of functionals of the form $u \mapsto \int_{Ω} g (x, u (x), \nabla u (x)) d x$ . The main assumptions for these results are that g is asymptotically convex and that it satisfies some growth conditions. In this article, we present a specialized but significant version of this general result. The primary purpose of this paper is provide several applications of this...

Functionals with $p (x)$ growth and regularity

Emilio Acerbi, Giuseppe Mingione (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We consider the integral functional $\int f (x, D u) d x$ under non standard growth assumptions of $(p, q)$ -type: namely, we assume that ${|z|}^{p (x)} \leq f (x, z) \leq L (1 + {|z|}^{p (x)})$ , a relevant model case being the functional $\int {|D u|}^{p (x)} d x$ . Under sharp assumptions on the continuous function $p (x) > 1$ we prove regularity of minimizers both in the scalar and in the vectorial case, in which we allow for quasiconvex energy densities. Energies exhibiting this growth appear in several models from mathematical physics.

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In this Note II we continue the analysis of the phenomenon of mechanical twinning that we began in a preceding Note I $(^{1})$ . Furthermore, we point out some fundamental properties useful in the study of growth twins, for which a fully comprehensive thermoelastic theory is not yet available.

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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New extension of the variational McShane integral of vector-valued functions

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

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We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$ -dimensional Euclidean space $ℝ^{m}$ . It is a “natural” extension of the variational McShane integral (the strong McShane integral) from $m$ -dimensional closed non-degenerate intervals to open and bounded subsets of $ℝ^{m}$ . We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study...

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Local analysis of a cubically convergent method for variational inclusions

Steeve Burnet, Alain Pietrus (2011)

Applicationes Mathematicae

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This paper deals with variational inclusions of the form 0 ∈ φ(x) + F(x) where φ is a single-valued function admitting a second order Fréchet derivative and F is a set-valued map from $ℝ^{q}$ to the closed subsets of $ℝ^{q}$ . When a solution z̅ of the previous inclusion satisfies some semistability properties, we obtain local superquadratic or cubic convergent sequences.

Convex approximation of an inhomogeneous anisotropic functional

Giovanni Bellettini, Maurizio Paolini (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The numerical minimization of the functional $F (u) = \int_{Ω} ϕ (x, ν_{u}) |D u| + \int_{\partial Ω} μ u d H^{n - 1} - \int_{Ω} κ u d x$ , $u \in B V (Ω; \{- 1, 1\})$ is addressed. The function $ϕ$ is continuous, has linear growth, and is convex and positively homogeneous of degree one in the second variable. We prove that $F$ can be equivalently minimized on the convex set $B V (Ω; [- 1, 1])$ and then regularized with a sequence ${\{F_{ϵ} (u)\}}_{ϵ}$ , of stricdy convex functionals defined on $B V (Ω; [- 1, 1])$ . Then both $F$ and $F_{ϵ}$ , can be discretized by continuous linear finite elements. The convexity property of the functionals on $B V (Ω; [- 1, 1])$ is useful in the numerical...

Continuity of the non-convex play operator in the space of rectifiable curves

Jana Kopfová, Vincenzo Recupero (2023)

Applications of Mathematics

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We prove that the vector play operator with a uniformly prox-regular characteristic set of constraints is continuous with respect to the $B V$ -norm and to the $B V$ -strict metric in the space of rectifiable curves, i.e., in the space of continuous functions of bounded variation. We do not assume any further regularity of the characteristic set. We also prove that the non-convex play operator is rate independent.