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 spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.
Let $Ω\subseteq\mathcal{R}^2$ be a domain, let $X$ be a rearrangement invariant space and let $f\in W^{1}X(Ω,\mathcal{R}^2)$ be a homeomorphism between $Ω$ and $f(Ω)$. Then there exists a sequence of diffeomorphisms $f_k$ converging to $f$ in the space $W^{1}X(Ω,\mathcal{R}^2)$.
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
We analyze the Steiner rearrangement in any codimension of Sobolev and $BV$ functions. In particular, we prove a Pólya-Szegő inequality for a large class of convex integrals. Then, we give minimal assumptions under which functions attaining equality are necessarily Steiner symmetric.
Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron-Martin space is denoted by $H$. Let $ν=e^{-U}μ$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \[λu-L_νu=f,\] where $λ>0$, $f\in L^2(X,ν)$ and $L_ν$ is the self-adjoint operator associated with the quadratic form \[(ψ,φ)\mapsto \int_X\left\langle\nabla_Hψ,\nabla_Hφ\right\rangle_Hdν\qquadψ,φ\in…
We study Sobolev spaces on the $n-$dimensional unit ball $B^n(1)$ endowed with a parameter-depending Finsler metric $F_a$, $a\in [0,1],$ which interpolates between the Klein metric $(a=0)$ and Funk metric $(a=1)$, respectively. We show that the standard Sobolev space defined on the Finsler manifold $(B^n(1),F_a)$ is a vector space if and only if $a\in [0,1).$ Furthermore, by exploiting variational arguments, we provide non-existence and existence results for sublinear elliptic problems on $(B^n(1),F_a)$ involving the Finsler-Laplace operator whenever $a\in [0,1).$
We review some topics in the theory of symmetric decreasing rearrangements with a particular focus on Lieb's fundamental contributions. Topics covered include the Brascamp--Lieb--Luttinger theorem, the sharp Young and Hardy--Littlewood--Sobolev inequalities, as well as the continuity of the rearrangement map on Sobolev spaces.
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S.
Let D be a bounded domain in n-dimensional Euclidean space, where n>2, and let 1<p< (2n)/(n-2). We prove a reverse-Holder inequality for functions realizing equality in the Sobolev inequality, which finds a lower bound for their (p-1)-norm in terms of their p-norm. This inequality is sharp, and it is an equality if and only if the domain is a round ball. Our result generalizes a theorem of Payne and Rayner and our proof relies on integral rearrangements and an analysis of the ODE corresponding to the radial case.
We define the notion of colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. If $p \ge 1$ and $M$ and $N$ are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev space $\dot{W}^{1, p} (M, N)$. This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various i…
We investigate properties of function spaces induced by the inner product Sobolev spaces $H^{s}(Ω)$ over a bounded Euclidean domain $Ω$ and by an elliptic differential operator $A$ on $\overlineΩ$. The domain and the coefficients of $A$ are of the class $C^{\infty}$. These spaces consist of all distributions $u\in H^{s}(Ω)$ such that $Au\in H^λ(Ω)$ and are endowed with the corresponding graph norm, with $s,λ\in\mathbb{R}$. We prove an interpolation formula for these spaces and discuss their application to elliptic boundary-value problems.
Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$. The associated Cameron--Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $ν=e^{-U}μ$ and $Ω=G^{-1}(-\infty,0]$. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \begin{align}\label{Probelma in abstract} λu-L_{ν,Ω} u=f, \end{align} where $λ>0$, $f\in L^2(Ω,ν)$ and $L_{ν,Ω}$ is the self-adjoint operator associated with the quadratic fo…
In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces at $G^{-1}(0)$ of the elements of $W^{1,p}(O, μ)$ for $p>1$, as elements of $L^1(G^{-1}(0), ρ)$ where $ρ$ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in $L^q(G^{-1}(0), ρ)$ for $1\leq q<p$ and even in $L^p(G^{-1}(0), ρ)$ under further assumptions. If $O$ is a suitable halfspace, the range is characterized …
Let $Γ$ be a graph endowed with a reversible Markov kernel $p$, and $P$ the associated operator, defined by $Pf(x)=\sum_y p(x,y)f(y)$. Denote by $\nabla$ the discrete gradient. We give necessary and/or sufficient conditions on $Γ$ in order to compare $\Vert \nabla f \Vert_{p}$ and $\Vert (I-P)^{1/2}f \Vert_{p}$ uniformly in $f$ for $1<p<+\infty$. These conditions are different for $p<2$ and $p>2$. The proofs rely on recent techniques developed to handle operators beyond the class of Calderón-Zygmund operators. For our purpose, we also prove Littlewood-Paley inequalities and interpolation resul…
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 note, we establish a $L^p-$version of the Poincaré--Sobolev inequalities in the hyperbolic spaces $\mathbb H^n$. The interest of this result is that it relates both the Poincaré (or Hardy) inequality and the Sobolev inequality with the sharp constant in $\mathbb H^n$. Our approach is based on the comparison of the $L^p-$norm of gradient of the symmetric decreasing rearrangement of a function in both the hyperbolic space and the Euclidean space, and the sharp Sobolev inequalities in Euclidean spaces. This approach also gives the proof of the Poincaré--Gagliardo--Nirenberg and Poincaré--…
Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $μ$ and let $w$ be a positive function on $X$ such that $w\in W^{1,s}(X,μ)$ and $\log w\in W^{1,t}(X,μ)$ for some $s>1$ and $t>s'$. In the present paper we introduce and study Sobolev spaces with respect to the weighted Gaussian measure $ν:=wμ$. We obtain results regarding the divergence operator (i.e. the adjoint in $L^2$ of the gradient operator along the Cameron--Martin space) and the trace of Sobolev functions on hypersurfaces $\{x\in X\,|\, G(x) = 0\}$, where $G$ is a suitable version of a Sobolev …
We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.
We define the notion of higher-order colocally weakly differentiable maps from a manifold $M$ to a manifold $N$. When $M$ and $N$ are endowed with Riemannian metrics, $p\ge 1$ and $k\ge 2$, this allows us to define the intrinsic higher-order homogeneous Sobolev space $\dot{W}^{k,p}(M,N)$. We show that this new intrinsic definition is not equivalent in general with the definition by an isometric embedding of $N$ in a Euclidean space; if the manifolds $M$ and $N$ are compact, the intrinsic space is a larger space than the one obtained by embedding. We show that a necessary condition for the dens…
In this paper we develop a theory of Besov and Triebel--Lizorkin spaces on general noncompact Lie groups endowed with a sub-Riemannian structure. Such spaces are defined by means of hypoelliptic sub-Laplacians with drift, and endowed with a measure whose density with respect to a right Haar measure is a continuous positive character of the group. We prove several equivalent characterizations of their norms, we establish comparison results also involving Sobolev spaces of recent introduction, and investigate their complex interpolation and algebra properties.
We consider Backus's problem in geophysics. This consists in reconstructing a harmonic potential outside the Earth when the intensity of the related field is measured on the Earth's surface. Thus, the boundary condition is (severely) nonlinear. The gravitational case is quite understood. It consists in the local resolution near a monopole, i.e. the potential generated by a point mass. In this paper, we consider the geomagnetic case. This consists in linearizing the field's intensity near the so-called dipole, a harmonic function which models the solenoidal potential of a magnet. The problem is…
We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or compact without boundary. The main result is that for $s > \dim M/2 + 1$, the group $\mathcal D^s(M)$ is geodesically and metrically complete with a surjective exponential map. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching.
This paper establishes isomorphisms for the Laplace operator in weighted Sobolev spaces (WSS). These spaces are similar to standard Sobolev spaces, but they are endowed with weights prescribing functions growth or decay at infinity. Although well established in the whole space, these weighted results do not apply in the specific hypothesis of periodicity. This kind of problem appears when studying singularly perturbed domains (roughness, sieves, porous media, etc). When zooming on a single perturbation pattern, one often ends with a periodic problem set on an infinite strip. We present a unifi…
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchheim's theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen-Koskela's concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with $p$-integrable upper gradient i…
In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s,p}$. The case $p = 2$ of their result implies the nonlocal isoperimetric inequality \[ \frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}},\ \ \ \ \ \ \ 0<s<1/2, \] where $P_s$ indicates the fractional $s$-perimeter, and $B_1$ is the unit ball in $\RN$. In this note we explicitly compute the best constant, and show that for any $0<s<1/2$, one has \[ \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}} = \frac{N π^{\frac N2 + s} \G(1-2s)}{s \G(\frac N2+1)^{\frac{2s}N} \G(1-s)\G(\…
Let $(X,d,μ)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carré du champ". Assume that $(X,d,μ,\E)$ supports a scale-invariant $L^2$-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse Hölder inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalised Bakry-É…
