In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.
The continouity and compactness of embedding operators in in Sobolev-Lions type spaces are derived. By applying this result separability properties of degenerate anisotropic differential operator equations, well-posedeness and Strichartz type estimates for solution of corresponding parabolic problem are established
We obtain an improved Sobolev inequality in H^s spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in H^s obtained in [P. Gerard, ESAIM 1998] using the abstract approach of dislocation spaces developed in [K. Tintarev & K. H. Fieseler, Imperial College Press 2007]. We also analyze directly the local defect of compactness of the Sobolev embedding in term…
For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.
In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sobolev embedding constant on $\mathbb{R}^d$. As an application, we prove local and global Moser--Trudinger inequalities.
In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( Ω,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this spaces. Moreover, we discuss the existence of weak solutions for weighted Dirichlet problem of $p(.)$-Laplacian equation \begin{equation*} \left\{ \begin{array}{cc} -\text{div}\left( \vartheta (x)\left\vert \nabla f\right\vert ^{p(x)-2}\nabla f\right) =\vartheta _{0}(x)\left\vert f\right\vert ^{q(x)-2}f & x\in Ω\\ f=0 & x\in \partial Ω\end{array} \right. \end{equati…
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domains.
We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this embedding is (almost) always characterized by a multiplicative norm inequality. When a,b,c>-N and attention is confined to smooth functions with compact support, these inequalities coincide with the Caffarelli-Kohn-Nirenberg inequalities. Variants for radially symmetric functions, or when r<=min{p,q}, are also obtained along the way.
We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
Let $G$ be a noncompact connected Lie group, denote with $ρ$ a right Haar measure and choose a family of linearly independent left-invariant vector fields $\mathbf{X}$ on $G$ satisfying Hörmander's condition. Let $χ$ be a positive character of $G$ and consider the measure $μ_χ$ whose density with respect to $ρ$ is $χ$. In this paper, we introduce Sobolev spaces $L^p_α(μ_χ)$ adapted to $\mathbf{X}$ and $μ_χ$ ($1<p<\infty$, $α\geq 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some n…
In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of some fourth order elliptic equation involving the critical Sobolev exponent.
This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $Ω$ with respect to the norm: $$\|f\|_{QH^{1,p}(v,μ;Ω)} = \|f\|_{L^p_v(Ω)} + \|\nabla f\|_{\mathcal{L}^p_Q(μ;Ω)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $Ω$. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition …
We study the removability of a singular set for elliptic equations involving weight functions and variable exponents. We consider the case where the singular set satisfies conditions related to some generalization of upper Minkowski content or a net measure, and give sufficient conditions for removability of this singularity for equations in the weighted variable exponent Sobolev spaces.
New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.
The purpose of this paper is to extend the embedding theorem of Sobolev spaces involving general kernels and we provide a sharp critical exponent in these embeddings. As an application, solutions for equations driven by a general integro-differential operator, with homogeneous Dirichlet boundary conditions, is established by using the Mountain Pass Theorem.
Linear interpolation inequalities that combine Hardy's inequality with sharp Sobolev embedding are obtained using classical arguments of Hardy and Littlewood (Bliss lemma). Such results are equivalent to Caffarelli-Kohn-Nirenberg inequalities with sharp constants. A one-dimensional convolution inequality for the exponential density is derived as an application of these methods.
In this paper, we study the critical Sobolev embeddings $W^{1,p(x)}(Ω)\subset L^{p^*(x)}(Ω)$ for variable exponent Sobolev spaces from the point of view of the $Γ$-convergence. More precisely we determine the $Γ$-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant.
We derive sharp Sobolev inequalities for Sobolev spaces on metric spaces. In particular, we obtain new sharp Sobolev embeddings and Faber-Krahn estimates for Hörmander vector fields.
In this paper we establish an optimal Lorentz space estimate for the Riesz potential acting on curl-free vectors: There is a constant $C=C(α,d)>0$ such that \[ \|I_αF \|_{L^{d/(d-α),1}(\mathbb{R}^d;\mathbb{R}^d)} \leq C \|F\|_{L^1(\mathbb{R}^d;\mathbb{R}^d)} \] for all fields $F \in L^1(\mathbb{R}^d;\mathbb{R}^d)$ such that $\operatorname*{curl} F=0$ in the sense of distributions. This is the best possible estimate on this scale of spaces and completes the picture in the regime $p=1$ of the well-established results for $p>1$.
We introduce the non-homogeneous analogs of Van Schaftingen's classes. We show that these classes refine the embedding $W^{1,n}\subset bmo$. The analogous results established on bounded Lipschitz domains and Riemannian manifolds with bounded geometry.
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
We prove Sobolev embedding Theorems with weights for vector bundles in a complete riemannian manifold. We also get general Gaffney's inequality with weights. As a consequence, under a "weak bounded geometry" hypothesis, we improve classical Sobolev embedding Theorems for vector bundles in a complete riemannian manifold. We also improve known results on Gaffney's inequality in a complete riemannian manifold.
In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations. In this case, the fundamental solution is invariant with respect to a non-Euclidean translation group and the usual convolut…
In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.
In this paper, we consider a maximizing problem associated with the Sobolev type embedding on the space of bounded variation. We show that, although the maximizing problem suffers from both of the non-compactness of vanishing and concentrating phenomena, there exists a maximizer for some range of the exponents. Furthermore, we show that any maximizer must be given by a characteristic function on a ball.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
In the present paper, we deal with a new continuous and compact embedding theorems for the fractional Orlicz-Sobolev spaces, also, we study the existence of infinitely many nontrivial solutions for a class of non-local fractional Orlicz-Sobolev Schrödinger equations whose simplest prototype is $$(-\triangle)^{s}_{m}u+V(x)m(u)=f(x,u),\ x\in\mathbb{R}^{d},$$ where $0<s<1$, $d\geq2$ and $(-\triangle)^{s}_{m}$ is the fractional $M$-Laplace operator. The proof is based on the variant Fountain theorem established by Zou.
In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' "vanishing" Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of n…
We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms. Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains, the latter for weights equal to powers of the distance to the boundary. …
Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embeddi…
The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.
We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding theorems in different contexts. In particular we include a detailed study of Gaussian measures as well as probablity measures between Gaussian and exponential. We show a kind of reverse Polya-Szego principle that allows us to obtain continuity as a self improvement from boundedness, using symetrization inequalities. Our methods also allow for precise estimat…
A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangementinvariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into space…
Fractional Sobolev embeddings on noncommutative torus
School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Institute of Mathematics and Mathematical Modeling（数学与数学建模研究所）；School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）
In this paper we study the existence of solutions and their concentration phenomena of a singularly perturbed semilinear Schrodinger equation with the presence of the critical Sobolev exponent.
We offer a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on $\mathbb{R}^n$. We use this idea to extend the result to real hyperbolic spaces $\mathbb{H}^n$.
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.
This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H…
The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the fractional Hardy inequality and the iso-capacitary type inequality. Secondly, we will strengthen fractional Sobolev type inequalities in Besov spaces via capacitary Lorentz spaces associated with Besov capacities. For this purpose, we first study the embedding of the associated capacitary Lorentz space to the classical Lorentz space. Then, the embedding of th…
In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $λ_1(Ω)\|u\|_{L^1(\partialΩ)} \le \|u\|_{W^{1,1}(Ω)}$ that are independent of $Ω$. This estimates generalize those of \cite{BS} concerning the $p$-Laplacian to the case $p=1$. We apply our results to prove existence of an extremal for this embedding. We then study an optimal design problem related to $λ_1$, and eventually compute the shape derivative of the functional $Ω\toλ_1(Ω)$. As a consequence, we obtain that a ball of $\R^n$ of radius $n$ is critical for volume-preserving deformations.
The aim of this paper is to give an extension of the improved Sobolev embedding theorem for single-valued functions to the case of vector-valued functions which is involved with the three-dimensional massless Dirac operator together with the three- or two-dimensional Weyl--Dirac (or Pauli) operator, the Cauchy--Riemann operator and also the four-dimensional Euclidian Dirac operator.
