Classical Sobolev Spaces Research Articles

Continuous imbedding theorems in Musielak spaces

We provide continuous Sobolev-type imbedding results for Musielak spaces. In the first result the target space is a Musielak space bigger than the considered one, while the second is that of bounded and continuous functions. Continuous imbeddings into a particular Musielak space built upon Sobolev conjugate functions generate serious difficulties and does not holds in general for arbitrary Musielak functions, as it is well known in variable exponent Sobolev spaces. We overcome this problem by imposing only a regularity assumption on the Musielak functions without resorting to other restrictions such as the Δ2/∇2 conditions. Nevertheless, the continuous embedding in the space of bounded continuous functions is obtained by using the convex envelope, which then allows us to use a result already obtained in Orlicz spaces. The results we obtain extend and unify those obtained in classical Sobolev spaces, variable exponent Sobolev spaces as well as those obtained by Donaldson-Trudinger in the setting of Orlicz spaces.

On solvability of polyharmonic Dirichlet problem in symmetric Sobolev spaces

Let be a bounded domain with sufficiently smooth boundary and be a symmetric space defined on the measure space . We consider a Dirichlet problem for ‐th order polyharmonic equation, and we establish its solvability (in strong sense) in Sobolev space generated by the norm of . Such spaces include classical Sobolev spaces, Orlicz–Sobolev spaces, grand Sobolev spaces, and Marcinkiewicz–Sobolev spaces. The obtained results are new for these special cases, too.

Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization

AbstractEmbeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.

Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications

Abstract We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.

Remainder terms of a nonlocal Sobolev inequality

AbstractIn this note, we study a nonlocal version of the Sobolev inequality where is the best constant, denotes the standard convolution and denotes the classical Sobolev space with respect to the norm . By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak ‐norm of above inequality for all with .

Fractional Sobolev Spaces with Kernel Function on Compact Riemannian Manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p(·,·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\cdot , \\cdot )$$\\end{document}-Laplacian operator. As one of the main tools, topological degree theory is applied.

Discrete approximation of nonlocal-gradient energies

Abstract We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.

Convergence and error estimates for pseudo-polyharmonic div-curl and elastic interpolation on a bounded domain

This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.

Long‐term stability of periodic smooth irrotational flows to the electrons Euler–Poisson system in three dimensions

We are concerned with the one‐fluid Euler–Poisson system for the electrons on the cubes in three dimensions. With the bootstrap estimates, we prove long‐term stability of periodic solutions of this system with small smooth irrotational perturbations of a constant background. Our main result is that the solutions are well‐posed for a time at least , where is the size of the initial data, the sufficiently large is the side length of the cubes, and is the index of the classical Sobolev spaces .

New Brezis–Van Schaftingen–Yung–Sobolev type inequalities connected with maximal inequalities and one parameter families of operators

Motivated by the recent characterization of Sobolev spaces due to Brezis–Van Schaftingen–Yung we prove new weak-type inequalities for one parameter families of operators connected with mixed norm inequalities. The novelty of our approach comes from the fact that the underlying measure space incorporates the parameter as a variable. We also show that our framework can be adapted to treat related characterizations of Sobolev spaces obtained earlier by Bourgain–Nguyen. The connection to classical and fractional order Sobolev spaces is shown through the use of generalized Riesz potential spaces and the Caffarelli–Silvestre extension principle. Higher order inequalities are also considered. We indicate many examples and applications to PDE's and different areas of Analysis, suggesting a vast potential for future research. In a different direction, and inspired by methods originally due to Gagliardo and Garsia, we obtain new maximal inequalities which combined with mixed norm inequalities are applied to obtain Brezis–Van Schaftingen–Yung type inequalities in the context of Calderón–Campanato spaces. In particular, Log versions of the Gagliardo–Brezis–Van Schaftingen–Yung spaces are introduced and compared with corresponding limiting versions of Calderón–Campanato spaces, resulting in a sharpening of recent inequalities due to Crippa–De Lellis and Brué–Nguyen.

Critical Kirchhoff equations involving the p-sub-Laplacians operators on the Heisenberg group

In this paper, we deal with a class of Kirchhoff-type critical elliptic equations involving the [Formula: see text]-sub-Laplacians operators on the Heisenberg group of the form [Formula: see text] where [Formula: see text] is the [Formula: see text]-sub-Laplacian, [Formula: see text], [Formula: see text] is the horizontal Sobolev space on [Formula: see text]. And [Formula: see text] is the homogeneous dimension of [Formula: see text], [Formula: see text] is a real parameter, [Formula: see text] is the critical Sobolev exponent on the Heisenberg group. Under some proper assumptions on the Kirchhoff function [Formula: see text], the potential function [Formula: see text] and [Formula: see text], together with the mountain pass theorem and the concentration-compactness principles for classical Sobolev spaces on the Heisenberg group, we prove the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases on the Heisenberg group. The results of this paper extend or complete recent papers and are new in several directions for the non-degenerate and degenerate critical Kirchhoff equations involving the [Formula: see text]-Laplacian type operators on the Heisenberg group.

Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in the higher order Sobolev space with homogeneous Navier boundary conditions WNm,nm(Ω), which in particular includes the classical Sobolev space W0m,nm(Ω), where Ω is a smooth bounded domain in Euclidean n-space. Secondly, based on the Concentration-compactness alternative due to Do Ó and Macedo, we prove the existence of extremals for the Adams inequality under Navier boundary conditions for second order derivatives at least for higher dimensions when Ω is an Euclidean ball.

On the limiting behaviour of some nonlocal seminorms: a new phenomenon

In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.

A unified divergent approach to Hardy–Poincaré inequalities in classical and variable Sobolev spaces

We present a unified strategy to derive Hardy–Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy–Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. We extend the idea to the more general context of variable exponent Sobolev spaces. Surprisingly, despite the well-known counterexamples of Fan et al. (2005) [28], we show that a modular form of the Poincaré inequality is actually possible provided one restricts to the class of functions in u∈Cc∞(Ω) such that |u|⩽1. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.

On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.

Lower bounds for integration and recovery in L2

Function values are, in some sense, “almost as good” as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space into L2 are square-summable. Here we mainly prove new lower bounds. In particular we prove that the sampling numbers behave worse than the approximation numbers for Sobolev spaces with small smoothness. Hence there can be a logarithmic gap also in the case where the singular numbers of the embedding are square-summable. We first prove new lower bounds for the integration problem, again for rather classical Sobolev spaces of periodic univariate functions.

Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on ℝ^{𝕕}

We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space onRd\mathbb {R}^d, ford≥1d \geq 1. Both spaces are Hilbert spaces involvingweight functions, which determine the behaviour as different variables tend to±∞\pm \infty, andweight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space onRd\mathbb {R}^dwas initially introduced by Nichols and Kuo in 2014 to analyse the error ofquasi-Monte Carlo(QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo and Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands withkinksorjumpscoming from option pricing problems. In this same setting, Griewank, Kuo, Leövey and Sloan in 2018 subsequently extended these ideas by developing a practicalsmoothing by preintegrationtechnique to approximate integrals of such functions with kinks or jumps.We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation usingNNpoints converges at a rate close to1/N1/N.

Solutions of Navier-Stokes Equations with Coriolis Force in the Rotational Framework

In this article, for 0 ≤m<∞ and the index vectors q=(q_1,q_2 ,q_3 ),r=(r_1,r_2,r_3) where 1≤q_i≤∞,1<r_i<∞ and 1≤i≤3, we study new results of Navier-Stokes equations with Coriolis force in the rotational framework in mixed-norm Sobolev-Lorentz spaces H ̇^(m,r,q) (R^3), which are more general than the classical Sobolev spaces. We prove the existence and uniqueness of solutions to the Navier-Stokes equations (NSE) under Coriolis force in the spaces L^∞([0, T]; H ̇^(m,r,q) ) by using topological arguments, the fixed point argument and interpolation inequalities. We have achieved new results compared to previous research in the Navier-Stokes problems.

The well-posedness, blow-up, and traveling waves for a two-component Fornberg–Whitham system

In this paper, the two-component Fornberg–Whitham system is studied. We first investigate the local well-posedness in classical Sobolev space and establish a blow-up criterion by a local-in-time priori estimate, and then, we give some sufficient conditions on the initial data to lead to wave breaking. Finally, we discuss analytically the existence of periodic traveling waves by the bifurcation theorem.

Solitary wave solutions of a Whitham–Boussinesq system

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and a good approximation to the incompressible Euler equations. In subsequent papers (Dinvay, 2019; Dinvay et al., 2019) the initial-value problem was studied and well-posedness in classical Sobolev spaces was proved. Here we prove existence of solitary wave solutions and provide their asymptotic description. Our proof relies on a variational approach and a concentration-compactness argument. The main difficulties stem from the fact that in the considered Euler–Lagrange equation we have a non-local operator of positive order appearing both in the linear and non-linear parts. Our approach allows us to obtain solitary waves for a particular Boussinesq system as well.