Sobolev Spaces Of Order Research Articles

Quadratic forms and Sobolev spaces of fractional order

We study quadratic functionals on $L^2(\mathbb{R}^d)$ that generate seminorms in the fractional Sobolev space $H^s(\mathbb{R}^d)$ for $0 < s < 1$. The functionals under consideration appear in the study of Markov jump processes and, independently, in recent research on the Boltzmann equation. The functional measures differentiability of a function $f$ in a similar way as the seminorm of $H^s(\mathbb{R}^d)$. The major difference is that differences $f(y) - f(x)$ are taken into account only if $y$ lies in some double cone with apex at $x$ or vice versa. The configuration of double cones is allowed to be inhomogeneous without any assumption on the spatial regularity. We prove that the resulting seminorm is comparable to the standard one of $H^s(\mathbb{R}^d)$. The proof follows from a similar result on discrete quadratic forms in $\mathbb{Z}^d$, which is our second main result. We establish a general scheme for discrete approximations of nonlocal quadratic forms. Applications to Markov jump processes are discussed.

Strichartz type estimates in mixed Besov spaces with application to critical nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equation with power nonlinearity in the fractional order Sobolev space Hs(Rn). We consider the case where the nonlinear term is Hs-critical and the differentiability thereof is less than s. We prove the existence of time global solutions of the Cauchy problem for small initial data. To this end, we prove Strichartz type estimates for linear inhomogeneous Schrödinger equations in mixed Besov spaces.

Schrödinger-type equations in Gelfand-Shilov spaces

We study the initial value problem for Schrödinger-type equations with initial data presenting a certain Gevrey regularity and an exponential behavior at infinity. We assume the lower order terms of the Schrödinger operator depending on (t,x)∈[0,T]×Rn and complex valued. Under a suitable decay condition as |x|→∞ on the imaginary part of the first order term and an algebraic growth assumption on the real part, we derive global energy estimates in suitable Sobolev spaces of infinite order and prove a well posedness result in Gelfand-Shilov type spaces. We also discuss by examples the sharpness of the result.

Loss of Regularity for the Continuity Equation with Non-Lipschitz Velocity Field

We consider the Cauchy problem for the continuity equation in space dimension $${d \ge 2}$$ . We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $$W^{1,p}$$ , for $$1 \le p<\infty $$ , and a smooth compactly supported initial datum such that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time. We also construct velocity fields in $$W^{r,p}$$ , with $$r>1$$ , and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $$W^{r,p}$$ does not embed in the space of Lipschitz functions. Our constructions are based on examples of optimal mixers from the companion paper Exponential self-similar mixing by incompressible flows (Alberti et al. in J Am Math Soc 32(2):445–490, 2019), and have been announced in Exponential self-similar mixing and loss of regularity for continuity equations (Alberti et al. in Comptes Rendus Math Acad Sci Paris 352(11):901–906, 2014).

On a certain property of a regular difference operator with variable coefficients

We consider a regular difference operator with variable coefficients in a bounded domain. It will be proved that this operator maps continuously and bijectively the Sobolev space of order k with the homogeneous Dirichlet boundary conditions to the subspace of Sobolev space of order k with nonlocal boundary conditions on the shifts of the boundary. This allows to apply the results obtained for nonlocal elliptic problems to the investigation of elliptic differential-difference equations.

Doubly Feller Property of Brownian Motions with Robin Boundary Condition

In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.

A note on homogeneous Sobolev spaces of fractional order

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.

Periodic Solutions and Torsional Instability in a Nonlinear Nonlocal Plate Equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.

On the Convergence in $H^1$-Norm for the Fractional Laplacian

We consider the numerical solution of the fractional Laplacian of index $s \in (1/2, 1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space $\widetilde{H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the $\widetilde{H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.

On variational solutions for whole brain serial-section histology using a Sobolev prior in the computational anatomy random orbit model.

This paper presents a variational framework for dense diffeomorphic atlas-mapping onto high-throughput histology stacks at the 20 μm meso-scale. The observed sections are modelled as Gaussian random fields conditioned on a sequence of unknown section by section rigid motions and unknown diffeomorphic transformation of a three-dimensional atlas. To regularize over the high-dimensionality of our parameter space (which is a product space of the rigid motion dimensions and the diffeomorphism dimensions), the histology stacks are modelled as arising from a first order Sobolev space smoothness prior. We show that the joint maximum a-posteriori, penalized-likelihood estimator of our high dimensional parameter space emerges as a joint optimization interleaving rigid motion estimation for histology restacking and large deformation diffeomorphic metric mapping to atlas coordinates. We show that joint optimization in this parameter space solves the classical curvature non-identifiability of the histology stacking problem. The algorithms are demonstrated on a collection of whole-brain histological image stacks from the Mouse Brain Architecture Project.

Regularity of fractional maximal functions through Fourier multipliers

We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n≥2. We also show that the spherical fractional maximal function maps Lp into a first order Sobolev space in dimensions n≥5.

Stackelberg–Nash exact controllability for the Kuramoto–Sivashinsky equation

This paper deals with a hierarchical control problem for the Kuramoto–Sivashinsky equation following a Stackelberg–Nash strategy. We assume that there is a main control, called the leader, and two secondary controls, called the followers. The leader tries to drive the solution to a prescribed target and the followers intend to be a Nash equilibrium for given functionals. It is known that this problem is equivalent to a null controllability result for an optimality system consisting of three non-linear equations. One of the novelties is a new Carleman estimate for a fourth-order equation with right-hand sides in Sobolev spaces of negative order, which allows to relax some geometric conditions for the observation sets for the followers.

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p-Laplacian in the whole \({\mathbb {R}}^n\).

Pointwise Characterizations of Even Order Sobolev Spaces via Derivatives of Ball Averages

AbstractLet $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$. In this article, the authors establish several equivalent characterizations of Sobolev spaces $W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any $\ell \in \mathbb{N}$ and $k\in \{0,\ldots ,\ell -1\}$, $\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$.

Global Well-Posedness and Analytic Smoothing Effect for the Dissipative Nonlinear Schrödinger Equations

We study the global Cauchy problem for the nonlinear Schrodinger equations in the Sobolev space of fractional order. In particular, we show the global well-posedness and the analytic smoothing effect for global solutions to a dissipative nonlinear Schrodinger equation for large data by applying a priori estimate in the Sobolev space of fractional order.

New Characterizations for the Weighted Fock Spaces

It is known that the standard weighted Bergman spaces over the complex ball can be characterized by means of Lipschitz type conditions. It is also known that the same spaces can be characterized, except for a critical case, by means of integrability conditions of double integrals associated with difference quotients of Bergman functions. In this paper we obtain characterizations of similar type for the class of weighted Fock spaces whose weights grow or decay polynomially at $$\infty $$ . In particular, our result for double-integrability characterization shows that there is no critical case for the Fock spaces under consideration. As applications we also obtain similar characterizations for the corresponding weighted Fock–Sobolev spaces of arbitrary real orders.

Higher order Adams’ inequality with the exact growth condition

Adams’ inequality is the complete generalization of the Trudinger–Moser inequality to the case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space [Formula: see text] served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams’ inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams’ inequality with the exact growth to higher order Sobolev spaces.

Approximation in higher-order Sobolev spaces and Hodge systems

Let d≥2 be an integer, 1≤l≤d−1 and φ be a differential l-form on Rd with W˙1,d coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on Rd with coefficients in L∞∩W˙1,d such that dφ=dψ. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space W˙2,d/2 or more generally in the fractional Sobolev spaces W˙s,p with sp=d. We give a positive answer to this question, provided that d−κ≤l≤d−1, where κ is the largest positive integer such that κ<min⁡(p,d). The proof relies on an approximation result (interesting in its own right) for functions in W˙s,p by functions in W˙s,p∩L∞, even though W˙s,p does not embed into L∞ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.

Spaces Associated with Weighted Sobolev Spaces on the Real Line

A complete description of spaces associated with weighted Sobolev spaces of the first order on the real line with identical summation parameters for a function and its derivative is presented.