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

Mikil Foss; Antonia Passarelli di Napoli; Anna Verde

Displaying similar documents to “Morrey regularity and continuity results for almost minimizers of asymptotically convex integrals”

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.

On the global regularity of $N$ -dimensional generalized Boussinesq system

Kazuo Yamazaki (2015)

Applications of Mathematics

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We study the $N$ -dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.

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.

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity,...

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

Regularity of Lipschitz free boundaries for the thin one-phase problem

Daniela De Silva, Ovidiu Savin (2015)

Journal of the European Mathematical Society

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We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E (u, Ω) = \int_{Ω} {| \nabla u |}^{2} d X + ℋ^{n} ({u > 0} \cap {x_{n + 1} = 0}), Ω \subset ℝ^{n + 1},$ among all functions $u \geq 0$ which are fixed on $\partial Ω$ .

Time regularity and functions of the Volterra operator

Zoltán Léka (2014)

Studia Mathematica

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Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and $| | T^{n + 1} - T ⁿ | | ≍ (n \geq 1)$ . This answers Zemánek’s question on the time regularity property.

The radius of close-to-convexity of $V_{k} (ϱ)$

Prakash G. Umarani (1983)

Annales Polonici Mathematici

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Differentiation of n-convex functions

H. Fejzić, R. E. Svetic, C. E. Weil (2010)

Fundamenta Mathematicae

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The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n - 1)} = f_{(n - 1)}$ except on a countable set. Moreover $f_{(n - 1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

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We consider the maximal function $| | (S^{a} f) [x] {| |}_{L^{\infty} [- 1, 1]}$ where $(S^{a} f) {(t)}^{\land} (ξ) = e^{{i t | ξ |}^{a}} f ̂ (ξ)$ and 0 < a < 1. We prove the global estimate $| | S^{a} f {| |}_{L ² (ℝ, L^{\infty} [- 1, 1])} {\leq C | | f | |}_{H^{s} (ℝ)}$ , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

On some results about convex functions of order $α$

M. Obradović, S. Owa (1986)

Matematički Vesnik

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A characterization of almost continuity and weak continuity

Chrisostomos Petalas, Theodoros Vidalis (2004)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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It is well known that a function $f$ from a space $X$ into a space $Y$ is continuous if and only if, for every set $K$ in $X$ the image of the closure of $K$ under $f$ is a subset of the closure of the image of it. In this paper we characterize almost continuity and weak continuity by proving similar relations for the subsets $K$ of $X$ .

A regularity criterion for the 2D MHD and viscoelastic fluid equations

Zhuan Ye (2015)

Annales Polonici Mathematici

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This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies $\nabla B, \nabla F \in L^{q} (0, T; L^{p} (ℝ ²))$ for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems

Christoph Hamburger (2007)

Bollettino dell'Unione Matematica Italiana

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We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $\operatorname{div}A(x,u,Du)+B(x,u,DU)=0$ , under natural polynomial growth of the coefficient functions $A$ and $B$ . We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

Convex integration with constraints and applications to phase transitions and partial differential equations

Stefan Müller, Vladimír Šverák (1999)

Journal of the European Mathematical Society

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We study solutions of first order partial differential relations $D u \in K$ , where $u : Ω \subset ℝ^{n} \to ℝ^{m}$ is a Lipschitz map and $K$ is a bounded set in $m \times n$ matrices, and extend Gromov’s theory of convex integration in two ways. First, we allow for additional constraints on the minors of $D u$ and second we replace Gromov’s $P$ −convex hull by the (functional) rank-one convex hull. The latter can be much larger than the former and this has important consequences for the existence of ‘wild’ solutions to elliptic systems. Our...

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.

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.

Induced almost continuous functions on hyperspaces

Alejandro Illanes (2006)

Colloquium Mathematicae

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For a metric continuum X, let C(X) (resp., $2^{X}$ ) be the hyperspace of subcontinua (resp., nonempty closed subsets) of X. Let f: X → Y be an almost continuous function. Let C(f): C(X) → C(Y) and $2^{f} : 2^{X} \to 2^{Y}$ be the induced functions given by $C (f) (A) = c l_{Y} (f (A))$ and $2^{f} (A) = c l_{Y} (f (A))$ . In this paper, we prove that: • If $2^{f}$ is almost continuous, then f is continuous. • If C(f) is almost continuous and X is locally connected, then f is continuous. • If X is not locally connected, then there exists an almost continuous function f: X → [0,1]...