On a necessary condition in the calculus of variations in Sobolev spaces with variable exponent

Elhoussine Azroul; Meryem El Lekhlifi; Badr Lahmi; Abdelfattah Touzani

Displaying similar documents to “On a necessary condition in the calculus of variations in Sobolev spaces with variable exponent”

On the equation x”(t) = F(t, x(t)) in the Sobolev space $H^{1} (R)$

Piotr Fijałkowski (1991)

Annales Polonici Mathematici

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Sharp embedding results for spaces of smooth functions with power weights

Martin Meyries, Mark Veraar (2012)

Studia Mathematica

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We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $ℝ^{d}$ , equipped with power weights $w (x) = {| x |}^{γ}$ , γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.

A new characterization of the Sobolev space

Piotr Hajłasz (2003)

Studia Mathematica

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The purpose of this paper is to provide a new characterization of the Sobolev space $W^{1, 1} (ℝ ⁿ)$ . We also show a new proof of the characterization of the Sobolev space $W^{1, p} (ℝ ⁿ)$ , 1 ≤ p < ∞, in terms of Poincaré inequalities.

On limiting embeddings of Besov spaces

V. I. Kolyada, A. K. Lerner (2005)

Studia Mathematica

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We investigate the classical embedding $B_{p, θ}^{s} \subset B_{q, θ}^{s - n (1 / p - 1 / q)}$ . The sharp asymptotic behaviour as s → 1 of the operator norm of this embedding is found. In particular, our result yields a refinement of the Bourgain, Brezis and Mironescu theorem concerning an analogous problem for the Sobolev-type embedding. We also give a different, elementary proof of the latter theorem.

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

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We prove that a function belonging to a fractional Sobolev space $L^{α, p} (ℝ ⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m, λ} (ℝ ⁿ)$ , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

On spaces $L^{p (x)}$ and $W^{k, p (x)}$

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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Sharp $L^{1}$ estimates for singular transport equations

Sergiu Klainerman, Igor Rodnianski (2008)

Journal of the European Mathematical Society

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We provide $L^{1}$ estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates is of a more general interest.

Sobolev-Besov spaces of measurable functions

Hans Triebel (2010)

Studia Mathematica

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The paper deals with spaces $L_{p}^{s} (ℝ ⁿ)$ of Sobolev type where s > 0, 0 < p ≤ ∞, and their relations to corresponding spaces $B_{p, q}^{s} (ℝ ⁿ)$ of Besov type where s > 0, 0 < p ≤ ∞, 0 < q ≤ ∞, in terms of embedding and real interpolation.

A subelliptic Bourgain–Brezis inequality

Yi Wang, Po-Lam Yung (2014)

Journal of the European Mathematical Society

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We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{N L}}^{1, Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for on the Heisenberg group $ℍ^{n}$ .

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

Hidemitsu Wadade (2014)

Studia Mathematica

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We establish the embedding of the critical Sobolev-Lorentz-Zygmund space $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p} (ℝ ⁿ)$ into the generalized Morrey space $ℳ_{Φ, r} (ℝ ⁿ)$ with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in $H_{p, q, λ ₁, . . ., λ ₘ}^{n / p + 1} (ℝ ⁿ)$ . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.

Variable exponent trace spaces

Lars Diening, Peter Hästö (2007)

Studia Mathematica

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The trace space of $W^{1, p (\cdot)} (ℝ ⁿ \times [0, \infty))$ consists of those functions on ℝⁿ that can be extended to functions of $W^{1, p (\cdot)} (ℝ ⁿ \times [0, \infty))$ (as in the fixed-exponent case). Under the assumption that p is globally log-Hölder continuous, we show that the trace space depends only on the values of p on the boundary. In our main result we show how to define an intrinsic norm for the trace space in terms of a sharp-type operator.

On a Sobolev type inequality and its applications

Witold Bednorz (2006)

Studia Mathematica

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Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T : = B_{| | \cdot | |} (0, r)$ , r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $s u p_{s, t \in T} | f (s) - f (t) | \leq 6 A B (\int_{0}^{r} ψ (1 / A ε^{n - 1}) ε^{n - 1} d ε + 1 / (n | B_{| | \cdot | |} (0, 1) |) \int_{T} φ (1 / B | | \nabla f (u) | | ⁎) d u)$ , where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on...

Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron Kerman, Luboš Pick (2008)

Studia Mathematica

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We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m, ϱ} (Ω)$ be compactly imbedded into the rearrangement-invariant space $L_{σ} (Ω)$ , where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{ϱ} (0, | Ω |)$ into $L_{σ} (0, | Ω |)$ . The results are illustrated with examples in which ϱ and σ are both...

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.

Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

Rémi Arcangéli, Juan José Torrens (2013)

Studia Mathematica

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We collect and extend results on the limit of $σ^{1 - k} {(1 - σ)}^{k} {| v |}_{l + σ, p, Ω}^{p}$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and ${| \cdot |}_{l + σ, p, Ω}$ is the intrinsic seminorm of order l+σ in the Sobolev space $W^{l + σ, p} (Ω)$ . In general, the above limit is equal to $c {[v]}^{p}$ , where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

The Bohr-Pál theorem and the Sobolev space $W ₂^{1 / 2}$

Vladimir Lebedev (2015)

Studia Mathematica

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The well-known Bohr-Pál theorem asserts that for every continuous real-valued function f on the circle there exists a change of variable, i.e., a homeomorphism h of onto itself, such that the Fourier series of the superposition f ∘ h converges uniformly. Subsequent improvements of this result imply that actually there exists a homeomorphism that brings f into the Sobolev space $W ₂^{1 / 2} ()$ . This refined version of the Bohr-Pál theorem does not extend to complex-valued functions. We show that if...

Composition operator and Sobolev-Lorentz spaces $W L^{n, q}$

Stanislav Hencl, Luděk Kleprlík, Jan Malý (2014)

Studia Mathematica

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Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator $T_{f} : u \mapsto u \circ f$ maps the Sobolev-Lorentz space $W L^{n, q} (Ω^{'})$ to $W L^{n, q} (Ω)$ for some q ≠ n then f must be a locally bilipschitz mapping.

A characterization of Sobolev spaces via local derivatives

David Swanson (2010)

Colloquium Mathematicae

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Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function $f \in W^{k, p} (Ω)$ possesses an $L^{p}$ derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space $W^{k, p} (Ω)$ . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Strong density for higher order Sobolev spaces into compact manifolds

Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen (2015)

Journal of the European Mathematical Society

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Given a compact manifold $N^{n}$ , an integer $k \in ℕ_{*}$ and an exponent $1 \leq p < \infty$ , we prove that the class $C^{\infty} ({\bar{Q}}^{m}; N^{n})$ of smooth maps on the cube with values into $N^{n}$ is dense with respect to the strong topology in the Sobolev space $W^{k, p} (Q^{m}; N^{n})$ when the homotopy group $π_{⌊ k p ⌋} (N^{n})$ of order $⌊ k p ⌋$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m - ⌊ k p ⌋ - 1$ , without any restriction on the homotopy group of $N^{n}$ .

The polynomial approximation of continuous functions defined on $(- \infty, \infty)$

Leetsch C. Hsu (1959)

Czechoslovak Mathematical Journal

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Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

B. Bojarski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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For a function $f \in L_{l o c}^{p} (ℝ ⁿ)$ the notion of p-mean variation of order 1, $₁^{p} (f, ℝ ⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1, p} (ℝ ⁿ)$ in terms of $₁^{p} (f, ℝ ⁿ)$ is directly related to the characterisation of $W^{1, p} (ℝ ⁿ)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

Operator $e^{- λ s^{α}}$

A. Takači (1978)

Matematički Vesnik

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