Sobolev Slobodeckij Spaces Research Articles

Sobolev embeddings for kinetic Fokker-Planck equations

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.

Multiplication in vector‐valued anisotropic function spaces and applications to non‐linear partial differential equations

AbstractWe study multiplication as well as Nemytskij operators in anisotropic vector‐valued Besov spaces , Bessel potential spaces , and Sobolev–Slobodeckij spaces . Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar‐valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector‐valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs.

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.

Multiplication in Sobolev spaces, revisited

In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, in part motivated by a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev-Slobodeckij spaces when a bounded domain is replaced by Rn. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and carefully distinguish between the case of spaces defined on the all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood-Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in this form. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev-Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for the construction of counter-examples such as the one included here.

On Solvability of Integro-Differential Equations

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

Two notes on the O'Hara energies

<p style='text-indent:20px;'>The O'Hara energies, introduced by Jun O'Hara in 1991, were proposed to answer the question of what is a "good" figure in a given knot class. A property of the O'Hara energies is that the "better" the figure of a knot is, the less the energy value is. In this article, we discuss two topics on the O'Hara energies. First, we slightly generalize the O'Hara energies and consider a characterization of its finiteness. The finiteness of the O'Hara energies was considered by Blatt in 2012 who used the Sobolev-Slobodeckij space, and naturally we consider a generalization of this space. Another fundamental problem is to understand the minimizers of the O'Hara energies. This problem has been addressed in several papers, some of them based on numerical computations. In this direction, we discuss a discretization of the O'Hara energies and give some examples of numerical computations. Particular one of the O'Hara energies, called the Möbius energy thanks to its Möbius invariance, was considered by Kim-Kusner in 1993, and Scholtes in 2014 established convergence properties. We apply their argument in general since the argument does not rely on Möbius invariance.

Optimal approximation order of piecewise constants on convex partitions

We prove that the error of the best nonlinear Lp-approximation by piecewise constants on convex partitions is O(N−2d+1), where N is the number of cells, for all functions in the Sobolev space Wq2(Ω) on a cube Ω⊂Rd, d⩾2, as soon as 2d+1+1p−1q⩾0. The approximation order O(N−2d+1) is achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev–Slobodeckij spaces Wqr(Ω) embedded in Lp(Ω), some of which also improve the standard estimate O(N−1d) known to be optimal on isotropic partitions.

Characterization of Sobolev-Slobodeckij spaces using geometric curvature energies

We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a family of curvature energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in L^p and the appropriate energy is finite.

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

Abstract In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in W s , p ⁢ ( ℝ N ) {W^{s,p}(\mathbb{R}^{N})} . Define ∥ u ∥ 1 , τ = ( [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ) 1 p for any ⁢ τ &gt; 0 . \lVert u\rVert_{1,\tau}=\bigl{(}[u]^{p}_{W^{s,p}(\mathbb{R}^{N})}+\tau\lVert u% \rVert_{p}^{p}\bigr{)}^{\frac{1}{p}}\quad\text{for any }\tau&gt;0. There holds sup u ∈ W s , p ⁢ ( ℝ N ) , ∥ u ∥ 1 , τ ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) &lt; + ∞ , \sup_{u\in W^{s,p}(\mathbb{R}^{N}),\lVert u\rVert_{1,\tau}\leq 1}\int_{\mathbb% {R}^{N}}\Phi_{N,s}\bigl{(}\alpha\lvert u\rvert^{\frac{N}{N-s}}\bigr{)}&lt;+\infty, where s ∈ ( 0 , 1 ) {s\in(0,1)} , s ⁢ p = N {sp=N} , α ∈ [ 0 , α * ) {\alpha\in[0,\alpha_{*})} and Φ N , s ⁢ ( t ) = e t - ∑ i = 0 j p - 2 t j j ! . \Phi_{N,s}(t)=e^{t}-\sum_{i=0}^{j_{p}-2}\frac{t^{j}}{j!}. Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation: ( - Δ ) p s ⁢ u ⁢ ( x ) + V ⁢ ( x ) ⁢ | u ⁢ ( x ) | p - 2 ⁢ u ⁢ ( x ) = f ⁢ ( x , u ) + ε ⁢ h ⁢ ( x ) in ⁢ ℝ N , (-\Delta)_{p}^{s}u(x)+V(x)\lvert u(x)\rvert^{p-2}u(x)=f(x,u)+\varepsilon h(x)% \quad\text{in }\mathbb{R}^{N}, where V ⁢ ( x ) {V(x)} has a positive lower bound, f ⁢ ( x , t ) {f(x,t)} behaves like e α ⁢ | t | N / ( N - s ) {e^{\alpha\lvert t\rvert^{N/(N-s)}}} , h ∈ ( W s , p ⁢ ( ℝ N ) ) * {h\in(W^{s,p}(\mathbb{R}^{N}))^{*}} and ε &gt; 0 {\varepsilon&gt;0} . Moreover, we also derive a weak solution with negative energy.

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincare map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev–Slobodeckij spaces and explicitly obtain a formal linearization of the Poincare map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.

On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces

We give a contribution to the problem of finding the optimal exponent in the Moser–Trudinger inequality in the fractional Sobolev–Slobodeckij space \widetilde{W}^{s,p}_0(\Omega) , where \Omega \subset \mathbb R^N is a bounded domain, s \in (0,1) , and sp = N . We exhibit an explicit exponent \alpha^*_{s,N}&gt;0 , which does not depend on \Omega , such that the Moser–Trudinger inequality does not hold true for \alpha \in (\alpha^*_{s,N},+\infty) .

A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one

We discuss some recent results by Parini and Ruf on a Moser–Trudinger type inequality in the setting of Sobolev–Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole \mathbb R and we give an answer to one of their open questions.

THE ENERGY SPACES OF THE TANGENT POINT ENERGIES

In this note, we will give a necessary and sufficient condition under which the tangent point energies introduced by von der Mosel and Strzelecki in [J. Geom. Anal., pp. 1–55 (2011), J. Knot Theory Ramifications21 (2012) 1250044] are bounded. We show that an admissible submanifold has bounded 𝔈q-energy if and only if it is injective and locally agrees with the graph of functions that belong to Sobolev–Slobodeckij space [Formula: see text]. The known Morrey embedding theorems of von der Mosel and Strzelecki can then be interpreted as standard Morrey embedding theorems for these spaces. Especially, this shows that the Hölder exponents for the embeddings in [J. Geom. Anal., pp. 1–55 (2011)] are sharp.

A note on integral Menger curvature for curves

AbstractFor continuously differentiable embedded curves γ, we will give a necessary and sufficient condition for the boundedness of integral versions of the Menger curvature. We will show that for p &gt; 3 the integral Menger curvature \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {M}_p(\gamma )$\end{document} is finite if and only if γ belongs to the Sobolev Slobodeckij space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{2-\frac{2}{p},p}(\mathbb {R} / \mathbb {Z}, \mathbb {R}^n)$\end{document}. The quantity \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak {J}_p (\gamma )$\end{document} – defined by taking the supremum of the p‐th power of Menger curvature with respect to one variable and then integrating over the remaining two – is finite for p &gt; 2, if and only if γ belongs to the space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^{2-\frac{1}{p},p}$\end{document}.

Sharp boundedness and regularizing effects of the integral Menger curvature for submanifolds

We show that embedded and compact C1 manifolds have finite integral Menger curvature if and only if they are locally graphs of functions belonging to certain Sobolev–Slobodeckij spaces. Furthermore, we prove that for some intermediate energies of integral Menger type a similar characterization of objects with finite energy can be given.

BOUNDEDNESS AND REGULARIZING EFFECTS OF O'HARA'S KNOT ENERGIES

In this paper, we will give a necessary and sufficient condition under which O'Hara's Ej,p-energies are bounded. We show that a regular curve has bounded Ej,p-energy if and only if it is injective and belongs to a certain Sobolev–Slobodeckij space.

Jump processes and nonlinear fractional heat equations on metric measure spaces

AbstractJump processes on metric‐measure spaces are investigated by using heat kernels. It is shown that the heat kernel corresponding to a σ ‐stable type process decays at a polynomial rate rather than at an exponential rate as a Brownian motion. The domain of the Dirichlet form associated with the jump process is a Sobolev–Slobodeckij space, and the embedding theorems for this space are derived by using the heat kernel technique. As an application, we investigate nonlinear fractional heat equations of the form with non‐negative initial values on a metric‐measure space F , and show the non‐existence of non‐negative global solution if 1 &lt; p ≤ 1 + $ { {\sigma \beta} \over {\alpha} } $ , where α is the Hausdorff dimension of F whilst β is the walk dimension of F . (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

The trace of Sobolev-Slobodeckij spaces on Lipschitz domains

In this paper we give some well-known trace theorems for Sobolev-Slobodeckij spaces under minimal regularity assumptions on the domain.