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

Piotr Fijałkowski

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

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

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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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.

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.

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.

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.

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.

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...

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 $N^{n}$ .

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 $ℍ^{n}$ .

A generalization to the Hardy-Sobolev spaces $H^{k, p}$ of an $L^{p}$ - $L^{1}$ logarithmic type estimate

Imed Feki, Ameni Massoudi, Houda Nfata (2018)

Czechoslovak Mathematical Journal

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The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k, p} (G)$ ; $k \in ℕ^{*}$ , $1 \leq p \leq \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$ .

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

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The estimate ${∥D^{k - 1} u∥}_{L^{n / (n - 1)}} \leq {∥A (D) u∥}_{L^{1}}$ is shown to hold if and only if $A (D)$ is elliptic and canceling. Here $A (D)$ is a homogeneous linear differential operator $A (D)$ of order $k$ on $ℝ^{n}$ from a vector space $V$ to a vector space $E$ . The operator $A (D)$ is defined to be canceling if $⋂_{ξ \in ℝ^{n} ∖ {0}} A (ξ) [V] = {0}$ . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous...

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...

Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation

Marcel Guardia, Vadim Kaloshin (2015)

Journal of the European Mathematical Society

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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s > 1$ . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$ -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c > 0$ such that for any $𝒦 ≫ 1$ we find a solution $u$ and a time $T$ such that ${∥ u (T) ∥}_{H^{s}} \geq 𝒦 {∥ u (0) ∥}_{H^{s}}$ . Moreover, the time $T$ satisfies the polynomial bound $0 < T < 𝒦^{C}$ .

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.

On a class of $(p, q)$ -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

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Let $Ω \subset ℝ^{n}$ be a bounded starshaped domain and consider the $(p, q)$ -Laplacian problem $\begin{matrix} - Δ_{p} u - Δ_{q} u = λ (𝐱) {| u |}^{p^{☆} - 2} u + μ {| u |}^{r - 2} u \end{matrix}$ where $μ$ is a positive parameter, $1 < q \leq p < n$ , $r \geq p^{☆}$ and $p^{☆} : = \frac{n p}{n - p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$ -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V (x) \sim {| x |}^{- α}$ , $0 < α < 2$ , and $K (x) \sim {| x |}^{- β}$ , $β > 0$ . Working in weighted Sobolev spaces, the existence of ground states $v_{ε}$ belonging to $W^{1, 2} (ℝ^{N})$ is proved under the assumption that $σ < p < (N + 2) / (N - 2)$ for some $σ = σ_{N, α, β}$ . Furthermore, it is shown that $v_{ε}$ are spikes concentrating at a minimum point of $𝒜 = V^{θ} K^{- 2 / (p - 1)}$ , where $θ = (p + 1) / (p - 1) - 1 / 2$ .

Functions with prescribed singularities

Giovanni Alberti, S. Baldo, G. Orlandi (2003)

Journal of the European Mathematical Society

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The distributional $k$ -dimensional Jacobian of a map $u$ in the Sobolev space $W^{1, k - 1}$ which takes values in the sphere $S^{k - 1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k - 1}$ . In case $M$ is polyhedral, the...

On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Sobolev type inequalities for fractional maximal functions and Riesz potentials in Morrey spaces of variable exponent on half spaces

Yoshihiro Mizuta, Tetsu Shimomura (2023)

Czechoslovak Mathematical Journal

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Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{ℍ, ν} f$ and Riesz potentials $I_{ℍ, α} f$ in weighted Morrey spaces of variable exponent on the half space $ℍ$ . We also obtain Sobolev type inequalities for a $C^{1}$ function on $ℍ$ . As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $Φ (x, t) = t^{p (x)} + {(b (x) t)}^{q (x)}$ , where $p (\cdot)$ and $q (\cdot)$ satisfy log-Hölder conditions, $p (x) < q (x)$ for $x \in ℍ$ , and $b (\cdot)$ is nonnegative and Hölder continuous of order $θ \in (0, 1]$ .

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.

Universal embeddings of $l^{1} (α)$ into the space of continuous functions on a product space

N. Kalamidas, Th. Zachariades (1989)

Colloquium Mathematicae

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Perturbed nonlinear degenerate problems in $ℝ^{N}$

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $d i v (x, \nabla u) + {a (x) | u |}^{p - 2} u = {g (x) | u |}^{p - 2} u + h (x) {| u |}^{s - 1} u$ in $ℝ^{N}$ ⎨ ⎩ u > 0, $l i m_{| x | \to \infty} u (x) = 0$ , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Displaying similar documents to “On the equation x”(t) = F(t, x(t)) in the Sobolev space H 1 ( R ) ”

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