Sobolev Estimates Research Articles

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given m∈N∖{0} and a compact Riemannian manifold N, we construct for every map u in the critical Sobolev space Wm/(m+1),m+1(Sm,N), a map U:B1m+1→N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

Weighted fractional Poincaré inequalities via isoperimetric inequalities

Our main result is a weighted fractional Poincaré–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré–Sobolev estimate with Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_p$$\\end{document} weights of Fabes–Kenig–Serapioni by means of a fractional type result in the spirit of Bourgain–Brezis–Mironescu. Examples are given to show that the corresponding Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-versions of weighted Poincaré inequalities do not hold for p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}.

Some Results of Stochastic Differential Equations

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory.

Subellipticity, compactness, $H^{\epsilon}$ estimates and regularity for $\bar{\partial}$ on weakly $q$-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain \Omega in a Stein manifold X of dimension n , we give a sufficient condition for subelliptic estimates for the \bar{\partial} -Neumann problem. Moreover, we study the compactness of the \bar{\partial} -Neumann operator N on \Omega . Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L^{2} -setting and the Sobolev estimates of N on \Omega for any \bar{\partial} -closed (r, k) -form with k \geqslant q (resp. k \leqslant q ). Furthermore, we study the \bar{\partial} -problem with support conditions in \Omega for forms of type (r, k) , with values in a holomorphic vector bundle. Applications to the \bar{\partial}_{b} -problem for smooth forms on boundaries of \Omega are given.

A quantitative second order estimate for (weighted) p-harmonic functions in manifolds under curvature-dimension condition

We build up a quantitative second-order Sobolev estimate of ln⁡w for positive p-harmonic functions w in Riemannian manifolds under Ricci curvature bounded from below and also for positive weighted p-harmonic functions w in weighted manifolds under the Bakry-Émery curvature-dimension condition.

Weighted Sobolev estimates of the truncated Beurling operator

AbstractGiven a bounded planar domain with boundary, , and a weight , we show that the corresponding truncated Beurling operator is a bounded operator sending into itself. Weighted Sobolev estimates for other Cauchy‐type integrals are also obtained.

A sharpened energy‐Strichartz inequality for the wave equation

AbstractWe consider the sharp Strichartz estimate for the wave equation on in the energy space, due to Bez and Rogers. We show that it can be refined by adding a term proportional to the distance from the set of maximizers, in the spirit of the classical sharpened Sobolev estimate of Bianchi and Egnell.

Sobolev estimates for fractional parabolic equations with space-time non-local operators

We obtain $$L_p$$ estimates for fractional parabolic equations with space-time non-local operators $$\begin{aligned} \partial _t^\alpha u - Lu + \lambda u= f \quad \text {in } (0,T) \times {\mathbb {R}}^d , \end{aligned}$$ where $$\partial _t^\alpha u$$ is the Caputo fractional derivative of order $$\alpha \in (0,1]$$ , $$T\in (0,\infty )$$ , and $$\begin{aligned} Lu(t,x) {:}{=} \int _{{\mathbb {R}}^d} \bigg ( u(t,x+y)-u(t,x) - y\cdot \nabla _xu(t,x)\chi ^{(\sigma )}(y)\bigg )K(t,x,y)\,dy \end{aligned}$$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel K with respect to t and y. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $$\alpha = 1$$ , which extend the previous results in Mikulevičius and Pragarauskas (J Differ Equ 256(4):1581–1626, 2014) and Zhang (Annales l’IHPAnalyse Nonlinéaire 30:573–614, 2013) by using a quite different method.

Existence theorems for the dbar equation and Sobolev estimates on $ q $-convex domains

Existence theorems for the dbar equation and Sobolev estimates on $ q $-convex domains

Improved Regularity of Harmonic Diffeomorphic Extensions on Quasihyperbolic Domains

Let \(\mathbb{X}\) be a Jordan domain satisfying certain hyperbolic growth conditions. Assume that φ is a homeomorphism from the boundary \(\partial \mathbb{X}\) of \(\mathbb{X}\) onto the unit circle. Denote by h the harmonic diffeomorphic extension of φ from \(\mathbb{X}\) onto the unit disk. We establish the optimal Orlicz-Sobolev regularity and weighted Sobolev estimate of h. These generalize the Sobolev regularity of h in [A. Koski, J. Onninen, Sobolev homeomorphic extensions, J. Eur. Math. Soc. 23 (2021) 4065–4089, Theorem 3.1].

On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights

Abstract In this paper, we unify and improve some of the results of Bourgain, Brezis, and Mironescu and the weighted Poincaré–Sobolev estimate by Fabes, Kenig, and Serapioni. More precisely, we get weighted counterparts of the Poincaré–Sobolev-type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behavior of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincaré inequalities.

A variational approach to first order kinetic mean field games with local couplings

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the well-posedness theory of mean field games with control on the acceleration assume either that the running and final costs are regularizing functionals of the density variable, or the presence of noise, i.e. a second-order system. In this article we construct global in time weak solutions to a first order mean field games system involving kinetic transport operators, where the costs are local (hence non-regularizing) functions of the density variable with polynomial growth. We show the uniqueness of these solutions on the support of the agent density. This is achieved by characterizing solutions through two convex optimization problems in duality. As part of our approach, we develop tools for the analysis of mean field games on a non-compact domain by variational methods. We introduce a notion of ‘reachable set’, built from the initial measure, that allows us to work with initial measures with or without compact support. In this way we are able to obtain crucial estimates on minimizing sequences for merely bounded and continuous initial measures. These are then carefully combined with L 1-type averaging lemmas from kinetic theory to obtain pre-compactness for the minimizing sequence. Finally, under stronger convexity and monotonicity assumptions on the data, we prove higher order Sobolev estimates of the solutions.

Global Stability for Nonlinear Wave Equations with Multi-Localized Initial Data

In this paper, we initiate the study of the global stability of nonlinear wave equations with initial data that are not required to be localized around a single point. More precisely, we allow small initial data localized around any finite collection of points which can be arbitrarily far from one another. Existing techniques do not directly apply to this setting because they require norms with radial weights away from some center to be small. The smallness we require on the data is measured in a norm which does not depend on the scale of the configuration of the data. Our method of proof relies on a close analysis of the geometry of the interaction between waves originating from different sources. We prove estimates on the bilinear forms encoding the interaction, which allow us to show improved bounds for the energy of the solution. We finally apply a variant of the vector field method involving modified Klainerman–Sobolev estimates to prove global stability. As a corollary of our proof, we are able to show global existence for a class of data whose \(H^1\) norm is arbitrarily large.

JKO estimates in linear and non-linear Fokker–Planck equations, and Keller–Segel: $L^p$ and Sobolev bounds

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L^p and L^\infty norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher-order results, and in particular to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called “five gradients inequality” allows us to recover some estimates that can be deduced from the Bakry–Émery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller–Segel chemotaxis model.

Global subrepresentation formulas in chain domains with irregular boundaries

For a large class of chain domains with rough boundaries, we derive global subrepresentation formulas (i.e., pointwise inequalities reminiscent of part of the Fundamental Theorem of Calculus). The results are new even for John and Boman domains. We also show how they lead to global first order Poincaré–Sobolev estimates in a large collection of Φ-John domains. The restriction on Φ is expressed as an integral condition on the number of chaining balls of various specific sizes.

Wave and scattering operators for the nonlinear Klein-Gordon equation on a quarter-plane

We consider the initial-boundary value problem for the nonlinear Klein-Gordon equation posed on a quarter-plane with Dirichlet non zero boundary conditions. We construct the wave and scattering operators for this model in the scattering-supercritical regime. We study the influence of the boundary data on the asymptotic behavior of solutions. We show that if the boundary data has sufficiently slow decay rate, it affects the order of the nonlinearity for which we are able to construct the scattering operator. This suggests a modification for the critical value of the nonlinear term due to the presence of non zero boundary condition. In order to prove our results, we construct an explicit formula for the solutions to corresponding linear Klein-Gordon equation on a quarter-plane. Using this explicit formula, we obtain weighted Sobolev estimates for the solution to the complete nonlinear problem. In order to control the long-time behavior of the solution we find an operator J suitable for our settings, analogous to operator x+it∂x in the case of NLS equation.

A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

We study stable solutions to the equation (−Δ)1/2u=f(u), posed in a bounded domain of Rn. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions n≤4. This result, which was known only for n=1, follows from a new interior Hölder estimate that is completely independent of the nonlinearity f.A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabré, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the Hölder bound for n≤4, a universal H1/2 estimate in all dimensions.Our L∞ bound is expected to hold for n≤8, but this has been settled only in the radial case or when f(u)=λeu. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when f(u)=λeu, even in the radial case.

On k ‐point configuration sets with nonempty interior

We give conditions for $k$-point configuration sets of thin sets to have nonempty interior, applicable to a wide variety of configurations. This is a continuation of our earlier work \cite{GIT19} on 2-point configurations, extending a theorem of Mattila and Sj\"olin \cite{MS99} for distance sets in Euclidean spaces. We show that for a general class of $k$-point configurations, the configuration set of a $k$-tuple of sets, $E_1,\,\dots,\, E_k$, has nonempty interior provided that the sum of their Hausdorff dimensions satisfies a lower bound, dictated by optimizing $L^2$-Sobolev estimates of associated generalized Radon transforms over all nontrivial partitions of the $k$ points into two subsets. We illustrate the general theorems with numerous specific examples. Applications to 3-point configurations include areas of triangles in $\mathbb R^2$ or the radii of their circumscribing circles; volumes of pinned parallelepipeds in $\mathbb R^3$; and ratios of pinned distances in $\mathbb R^2$ and $\mathbb R^3$. Results for 4-point configurations include cross-ratios on $\mathbb R$, triangle area pairs determined by quadrilaterals in $\mathbb R^2$, and dot products of differences in $\mathbb R^d$.

Covering-monopole map and higher degree in non-commutative geometry

We analyze the monopole map over the universal covering space of a compact four-manifold. We induce the property of local properness of the covering-monopole map under the condition of closedness of the Atiyah–Hitchin–Singer (AHS) complex. In particular, we construct a higher degree of the covering-monopole map when the linearized equation is isomorphic. This induces a homomorphism between the K -groups of the group C^* -algebra. We apply a non-linear analysis on the covering space, which is related to L^p cohomology. We also obtain various Sobolev estimates on the covering spaces. By applying the Singer conjecture on L^2 cohomology, we propose a conjecture of an aspherical version of the \frac{10}{8} -inequality. This is satisfied for a large class of four-manifolds, including some complex surfaces of general type.

Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.