Sobolev Functions Research Articles

On the Space of Locally Sobolev-Slobodeckij Functions

The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature.

Complex manifolds of Sobolev mappings and a Hartogs-type theorem in loop spaces

ABSTRACT We recall the complex structure on the generalized loop spaces , where S is a compact real manifold with boundary and X is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain domains in generalized loop spaces.

Hardy-Sobolev inequalities and boundary growth of Sobolev functions for double phase functionals on the half space

Hardy-Sobolev inequalities and boundary growth of Sobolev functions for double phase functionals on the half space

Asymptotic analysis of deformation behavior in high-contrast fiber-reinforced materials: Rigidity and anisotropy

We identify the restricted class of attainable effective deformations in a model of reinforced composites with parallel, long, and fully rigid fibers embedded in an elastic body. In mathematical terms, we characterize the weak limits of sequences of Sobolev maps whose gradients on the fibers lie in the set of rotations. These limits are determined by an anisotropic constraint in the sense that they locally preserve length in the fiber direction. Our proof of the necessity emerges as a natural generalization and modification of the recently established asymptotic rigidity analysis for composites with layered reinforcements. However, the construction of approximating sequences is more delicate here due to the higher flexibility and connectedness of the soft material component. We overcome these technical challenges by a careful approximation of the identity that is constant on the rigid components, combined with a lifting in fiber bundles for Sobolev functions. The results are illustrated with several examples of attainable effective deformations. If an additional partial second-order regularization is introduced into the material model, only rigid body motions can occur macroscopically.

On equicontinuity of families of mappings between Riemannian surfaces with respect to prime ends

Given a domain of some Riemannian surface,we consider questions related to the possibility of a continuousextension to the boundary of one class of Sobolev mappings. It isproved that such maps have a continuous boundary extension in termsof prime ends, and under some additional restrictions their familiesare equicontinuous at inner and boundary points of the domain. Wehave separately considered the cases of homeomorphisms and mappingswith branching.

Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations

"We study the distortion features of homeomorphisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms."

The Concept of a Trace and the Boundedness of the Trace Operator in Banach-Sobolev Function Spaces

In this paper a Banach function space (b.f.s. in short) on a n-dimensional bounded domain Ω with Lebesgue measure is considered. The Banach function space on the -dimensional surface generated by the norm of the space is defined. The Sobolev function space is defined using the norm of the space as well as the concept of the trace of an arbitrary function from this space on the surface S. Based on this concept, the space of traces is defined and a characterization of this space is given. It was proved that it is boundedly embedded in the space Particular cases of a functional space can be Lebesgue spaces, Lebesgue spaces with variable summability exponents, Morrey space, Orlicz space, grand Lebesgue spaces and etc. The obtained results allow us to consider boundary value problems for differential equations in b.f.s.

A Radon transform on the cylinder

We define a parametric Radon transform R that assigns to a Sobolev function on the cylinder S×R in R3 its mean values along sets Eζ formed by the intersections of planes through the origin and the cylinder. We show that R is a continuous operator, prove an inversion formula, provide a support theorem, as well as a characterization of its null space. We conclude by presenting a formula for the dual transform R⁎. We show that R and its dual R⁎ are related to the right-sided and left-sided Chebyshev fractional integrals. Using this relationship, we characterize the null space of R and R⁎ and provide an inversion formula for R⁎.

Quantitative characterization of traces of Sobolev maps

We give a quantitative characterization of traces on the boundary of Sobolev maps in [Formula: see text], where [Formula: see text] and [Formula: see text] are compact Riemannian manifolds, [Formula: see text]: the Borel-measurable maps [Formula: see text] that are the trace of a map [Formula: see text] are characterized as the maps for which there exists an extension energy density [Formula: see text] that controls the Sobolev energy of extensions from [Formula: see text]-dimensional subsets of [Formula: see text] to [Formula: see text]-dimensional subsets of [Formula: see text].

Thermalisation for Wigner matrices

We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices W and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem [36] from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to eitW for large t, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices.

Diffusion tensor regularization with metric double integrals

AbstractIn this paper, we propose a variational regularization method for denoising and inpainting of diffusion tensor magnetic resonance images. We consider these images as manifold-valued Sobolev functions, i.e. in an infinite dimensional setting, which are defined appropriately. The regularization functionals are defined as double integrals, which are equivalent to Sobolev semi-norms in the Euclidean setting. We extend the analysis of [14] concerning stability and convergence of the variational regularization methods by a uniqueness result, apply them to diffusion tensor processing, and validate our model in numerical examples with synthetic and real data.

Testing the Sobolev property with a single test plan

We prove that on an arbitrary metric measure space the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in t

On the heterogeneous distortion inequality

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n\] for almost every $x \in \mathbb{R}^n$. Here $K \in [1, \infty)$ is a constant and $\sigma \geq 0$ is a function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover the class of $K$-quasiregular mappings when $\sigma \equiv 0$, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp H\"older continuity estimate for all solutions, provided that $\sigma \in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ for some $\varepsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

Boundary growth of Sobolev functions of monotone type for double phase functionals

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).

On the regularity of the maximal function of a BV function

We show that the non-centered maximal function of a BV function is quasicontinuous. We also show that if the non-centered maximal functions of an SBV function is a BV function, then it is in fact a Sobolev function. Using a recent result of Weigt [12] , we are in particular able to show that the non-centered maximal function of a set of finite perimeter is a Sobolev function.

Lifting in compact covering spaces for fractional Sobolev mappings

Let $\pi : \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, $0<s<1$ and $2 \le sp< m$, we prove that every mapping $u \in W^{s, p} (\mathcal{M}, \mathcal{N})$ has a lifting in $W^{s,p}$, i.e., we have $u = \pi \, \circ \, \widetilde{u}$ for some mapping $\widetilde{u} \in W^{s, p} (\mathcal{M}, \widetilde{\mathcal{N}})$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result \emph{settles completely} the question of the lifting in Sobolev spaces over covering spaces. The proof relies on an a priori estimate of the oscillations of $W^{s,p}$ maps with $0<s<1$ and $sp>1$, in dimension $1$. Our argument also leads to the existence of a lifting when $0<s<1$ and $1<sp<2\le m$, provided there is no topological obstruction on $u$, i.e., $u = \pi \, \circ \, \widetilde{u}$ holds in this range provided $u$ is in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$. However, when $0<s<1$, $sp = 1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}(\mathcal{M},\mathcal{N})$ in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$, such that $u = \pi \, \circ \, \widetilde{u}$ does not hold for any $\widetilde{u} \in W^{s, p} ({\mathcal{M}}, \widetilde{\mathcal{N}} )$.

Trace theory for Sobolev mappings into a manifold

We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $\pi_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $\pi_1 (\mathcal{N})$ is finite and $\pi_2 (\mathcal{N}) \simeq \dotsb \simeq \pi_{\lfloor p - 1 \rfloor} (\mathcal{N}) \simeq \{0\}$. We present several open problems connected to the extension problem.

Topological singularities for vector-valued Sobolev maps and applications

Topological singularities for vector-valued Sobolev maps and applications

Real Paley‐Wiener theorem for the octonion Fourier transform

For the octonion Fourier transform, we establish the real Paley‐Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via for any octonion‐valued Sobolev function .

On the relation of the spectral test to isotropic discrepancy and Lq-approximation in Sobolev spaces

This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12]. We show that the isotropic discrepancy of a lattice point set is at most d22(d+1) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.