Order Sobolev Research Articles

Estimate for concentration level of the Adams functional and extremals for Adams-type inequality

This paper is mainly concerned with the existence of extremals for the Adams inequality. We first establish an upper bound for the classical Adams functional along of all concentrated sequences in the higher order Sobolev space with homogeneous Navier boundary conditions WNm,nm(Ω), which in particular includes the classical Sobolev space W0m,nm(Ω), where Ω is a smooth bounded domain in Euclidean n-space. Secondly, based on the Concentration-compactness alternative due to Do Ó and Macedo, we prove the existence of extremals for the Adams inequality under Navier boundary conditions for second order derivatives at least for higher dimensions when Ω is an Euclidean ball.

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.

Effective Mass Theorems with Bloch Modes Crossings

We study a Schrödinger equation modeling the dynamics of an electron in a crystal in the asymptotic regime of small wave-length comparable to the characteristic scale of the crystal. Using Floquet Bloch decomposition, we obtain a description of the limit of time averaged energy densities. We make a rather general assumption assuming that the initial data are uniformly bounded in a high order Sobolev spaces and that the crossings between Bloch modes are at worst conical. We show that despite the singularity they create, conical crossing do not trap the energy and do not prevent dispersion. We also investigate the interactions between modes that can occurred when there are some degenerate crossings between Bloch bands.

Aubin Type Almost sharp Moser–Trudinger Inequality Revisited

We give a new proof of the almost sharp Moser–Trudinger inequality on smooth compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces and generalize it to Riemannian manifolds with continuous metrics and to higher order Sobolev spaces on manifolds with boundary under several boundary conditions. These generalizations can be applied to fourth order Q curvature equations in dimension 4.

New families of fractional Sobolev spaces

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, and they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also uncovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.

Compensated compactness: Continuity in optimal weak topologies

For l-homogeneous linear differential operators A of constant rank, we study the implicationvj⇀vinXAvj→AvinW−lY}⇒F(vj)⇝F(v)inZ, where F is an A-quasiaffine function and ⇝ denotes an appropriate type of weak convergence. Here Z is a local L1-type space, either the space M of measures, or L1, or the Hardy space H1; X,Y are Lp-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of X,Y,Z are sharp. Analogous statements are also given in the case when F(v) is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove Hp-bounds for the sequence (F(vj))j, for appropriate p<1, and new convergence results in the dual of Hölder spaces when (vj) is A-free and lies in a suitable negative order Sobolev space W−β,s. The choice of these Hölder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.

Partial derivatives in the nonsmooth setting

We study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author in [13].Our main results are:i)The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivativesii)A quite complete set of results relating the property f∈W2,2(X×Y) on one side with that of f(⋅,y)∈W2,2(X) and f(x,⋅)∈W2,2(Y) for a.e. y,x respectively on the other. Here X,Y are RCD spaces so that second order Sobolev spaces are well defined. These results are in turn based upon the study of Sobolev regularity, and of the underlying notion of differential, for a map with values in a Hilbert module: we mainly apply this notion to the map x↦dyf(x,⋅) in order to build, under the appropriate regularity requirements, its differential dxdyf.

On a new class of fractional calculus of variations and related fractional differential equations

This paper analyzes a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. This paper establishes the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

Global well-posedness for the derivative nonlinear Schrödinger equation

We study the derivative nonlinear Schr\"odinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.

Second order Sobolev regularity for p-harmonic functions in SU(3)

Second order Sobolev regularity for p-harmonic functions in SU(3)

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.

Convergence of the kinetic annealing for general potentials

The convergence of the kinetic langevin simulated annealing is proven under mild assumptions on the potential U for slow logarithmic cooling schedules, which widely extends the scope of the previous results of [14]. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established. The results are based on an adaptation to non-elliptic non-reversible kinetic settings of a localization/local convergence strategy developed by Fournier and Tardif in [6] in the overdamped elliptic case, and on precise quantitative high order Sobolev hypocoercive estimates.

Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the 3D Jordan–Moore–Gibson–Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model removing the &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L^1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.&lt;/p&gt;

The microlocal irregularity of Gaussian noise

The study of random Fourier series, linear combinations of trigonometric functions whose coefficients are independent (in our case Gaussian) random variables with polynomially bounded means and standard deviations, dates back to Norbert Wiener in one of the original constructions of Brownian motion. A geometric generalization -- relevant e.g.\ to Euclidean quantum field theory with an infrared cutoff -- is the study of random Gaussian linear combinations of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact Riemannian manifold $(M,g)$, Gaussian noise $\Phi$. I will prove that, when our random coefficients are independent Gaussians whose standard deviations obey polynomial asymptotics and whose means obey a corresponding polynomial upper bound, the resultant random $\mathscr{H}^s$-wavefront set $\operatorname{WF}^s(\Phi)$ (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$, depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the behavior of the wavefront set at it. Consequently, the random $C^\infty$-wavefront set $\operatorname{WF}(\Phi)$ is almost surely the entirety of the cosphere bundle. The method of proof is as follows: using Sazonov's theorem and its converse, it suffices to understand which compositions of microlocal cutoffs and inclusions of $L^2$-based fractional order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from general facts about the HS-norms of the elements of the pseudodifferential calculus of Kohn and Nirenberg.

Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1&lt;p_1,p_2,p&lt;\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Helgason-Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we prove in a unified way that the sharp constant in the n−12-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n−12-th order Sobolev constant when n is odd and n≥5. We will also establish a lower bound of the coefficient of the Hardy term for the k-th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension n and k-th order derivatives. As a consequence, we thus show that the sharp constant in the k-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n is strictly less than the best k-th order Sobolev constant for all n≥2k+2. Precise expressions and optimal bounds for Green's functions of the operator −ΔH−(n−1)24 on the hyperbolic space Bn and operators of the product form are given, where (n−1)24 is the spectral gap for the Laplacian −ΔH on Bn. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space in terms of hypergeometric functions. In fact, we will establish the precise formulas of the Green's functions of the operator∏j=0k−1((ν+j)2−(n−1)2/4−ΔH),ν≥0, in terms of hypergeometric function F(a,b;c,z).Our approach to establish the main theorems is using the Helgason-Fourier analysis on hyperbolic spaces and are substantially different from that in dealing with the first order Hardy-Sobolev-Maz'ya inequalities on upper half spaces.

On Lipschitz approximations in second order Sobolev spaces and the change of variables formula

In this paper we study approximations of functions of Sobolev spaces Wp,loc2(Ω), Ω⊂Rn, by Lipschitz continuous functions. We prove that if f∈Wp,loc2(Ω), 1≤p<∞, then there exists a sequence of closed sets {Ak}k=1∞,Ak⊂Ak+1⊂Ω, such that the restrictions f|Ak are Lipschitz continuous functions and capp(S)=0, S=Ω∖⋃k=1∞Ak. Using these approximations we prove the change of variables formula in the Lebesgue integral for mappings of Sobolev spaces Wp,loc2(Ω;Rn) with the Luzin capacity-measure N-property.

Multilevel decompositions and norms for negative order Sobolev spaces

We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H − s H^{-s} for s ∈ ( 0 , 1 ) s\in (0,1) . Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.

Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities

We study the Cauchy problem to the semilinear fourth-order Schrodinger equations: $$\begin{aligned} {\left\{ \begin{array}{ll} i\partial _t u+\partial _x^4u=G\left( \left\{ \partial _x^{k}u\right\} _{k\le \gamma },\left\{ \partial _x^{k}{\bar{u}}\right\} _{k\le \gamma }\right) , &{} t>0, x\in {\mathbb {R}}, u|_{t=0}=u_0\in H^s({\mathbb {R}}), \end{array}\right. }\quad \quad (4\mathrm{NLS}) \end{aligned}$$ where $$\gamma \in \{1,2,3\}$$ and the unknown function $$u=u(t,x)$$ is complex valued. In this paper, we consider the nonlinearity G of the polynomial $$\begin{aligned} G(z)=G(z_1,\ldots ,z_{2(\gamma +1)}) :=\sum _{m\le |\alpha |\le l}C_{\alpha }z^{\alpha }, \end{aligned}$$ for $$z\in {\mathbb {C}}^{2(\gamma +1)}$$ , where $$m,l\in {\mathbb {N}}$$ with $$3\le m\le l$$ and $$C_{\alpha }\in {\mathbb {C}}$$ with $$\alpha \in ({\mathbb {N}}\cup \{0\})^{2(\gamma +1)}$$ is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space $$H^s({\mathbb {R}})$$ or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by Pornnopparath (J Differ Equ, 265:3792–3840, 2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by Bejenaru et al. (Ann Math 173:1443–1506, 2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.

A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic domains in Rd

We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains D of the type (0.1) D ( M ) : = { x ∈ R d : x | x | ∈ M } , M ⊂ S d − 1 . We obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.