Order Sobolev Research Articles

A note on Sobolev isometric immersions below W2,2 regularity

This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W2,2 setting. We show that the Hessian of each coordinate function of a W2,p, p<2, isometric immersion satisfies a low rank property in the almost everywhere sense, in particular, its Gaussian curvature vanishes almost everywhere. Meanwhile, we provide an example of a W2,p, p<2, isometric immersion from a bounded domain of R2 into R3 that has multiple singularities.

Analysis and Approximation of a Fractional Cahn--Hilliard Equation

We derive a fractional Cahn--Hilliard equation (FCHE) by considering a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in [0,1]$, where the choice $\alpha=1$ corresponds to the classical Cahn--Hilliard equation while the choice $\alpha=0$ recovers the Allen--Cahn equation. The existence of a unique solution is established and it is shown that the equation preserves mass for all positive values of fractional order $\alpha$ and that it indeed reduces the free energy. We then turn to the delicate question of the $L_\infty$ boundedness of the solution and establish an $L_\infty$ bound for the FCHE in the case where the nonlinearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier--Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semidiscrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order $\alpha$. It is...

Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms

In this paper we consider the cubic Schrödinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in $H^{s}$ for $s&gt;\frac{131}{416} $. We also obtain improved growth bounds for higher order Sobolev norms.

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that \(\sqrt {Q\left( x \right)} \) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q∇u) = b, we find sufficient conditions in order that \(\sqrt Q \nabla u\) has some first order smoothness. Specifically, if Ω is a bounded open set in Rn, we study when the components of \(\sqrt Q \nabla u\) belong to the first order Sobolev space WQ1,2(Ω) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of Σi=1nXi′Xiu + b = 0. We do not assume that {Xi} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.

Sobolev functions on varifolds

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to satisfy a uniform lower density bound and a dimensionally critical summability condition on its mean curvature, the following statements hold. Firstly, continuous and compact embeddings of Sobolev spaces into Lebesgue spaces and spaces of continuous functions are available. Secondly, the geodesic distance associated to the varifold is a continuous, not necessarily H\"older continuous Sobolev function with bounded derivative. Thirdly, if the varifold additionally has bounded mean curvature and finite measure, the present Sobolev spaces are isomorphic to those previously available for finite Radon measures yielding many new results for those classes as well. Suitable versions of the embedding results obtained for Sobolev functions hold in the larger class of generalised weakly differentiable functions.

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

In this paper we study the local and global regularity properties of the cubic nonlinear Schrödinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS [16] and the cubic NLS on the line [12]. We also prove that in the defocusing case the norm of the solution grows at most polynomially-in-time while in the focusing case it grows exponentially-in-time. As a byproduct of our analysis we provide a different proof of an almost sharp local wellposedness in Hs(R+). Sharp L2 local wellposedness was obtained in [19] and [2]. Our methods simplify some ideas in the wellposedness theory of initial and boundary value problems that were developed in [11,19,20,2].

A Class of High Order Nonlocal Operators

We study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators. We present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of Sobolev inequalities such as the nonlocal Poincare and nonlocal Gagliardo–Nirenberg inequalities. Nonlocal characterizations of high order Sobolev spaces in the spirit of Bourgain–Brezis–Mironescu are provided. Applications of nonlocal calculus of variations to the well-posedness of linear nonlocal models of elastic beams and plates are also considered.

On supercritical Sobolev type inequalities and related elliptic equations

Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved: let $$B \subset \mathbb R^N, N \ge 3$$ , be the unit ball, and let $$H_{0,\mathrm rad}^1(B)$$ denote the first order Sobolev space of radial functions, and $$2^* = 2N/(N-2)$$ the corresponding critical Sobolev embeddding exponent. Let $$r = |x|$$ , and $$p(r)=2^*+r^{\alpha }$$ , with $$\alpha > 0$$ ; then 0.1 $$\begin{aligned} \sup \left\{ \int _{B} |u|^{p(r)} \; dx \big | u \in H^1_{0,\mathrm{rad}}(B), \Vert \nabla u \Vert _2=1 \right\} < +\infty . \end{aligned}$$ We point out that the growth of p(r) is strictly larger than $$2^*$$ , except in the origin. Furthermore, we show that for $$p(r) = 2^* + r^\alpha $$ , with $$0< \alpha < \min \{\frac{N}{2};N-2\}$$ , the supremum in (0.1) is attained. Finally, we prove that associated elliptic equations admit nontrivial radial solutions. This is somewhat surprising since the nonlinearities have strictly supercritical growth except in the origin.

Breaking spaces and forms for the DPG method and applications including Maxwell equations

Discontinuous Petrov–Galerkin (DPG) methods are made easily implementable using “broken” test spaces, i.e., spaces of functions with no continuity constraints across mesh element interfaces. Broken spaces derivable from a standard exact sequence of first order (unbroken) Sobolev spaces are of particular interest. A characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided. Stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces. This technique is used to provide a complete error analysis of DPG methods for Maxwell equations with perfect electric boundary conditions. The technique also permits considerable simplifications of previous analyses of DPG methods for other equations. Reliability and efficiency estimates for an error indicator also follow. Finally, the equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and various forms in between.

Lower order regularity for the generalized Camassa–Holm equation

This paper deals with the Cauchy problem for the generalized Camassa–Holm equation. The existence of weak solutions for the generalized Camassa–Holm equation in lower order Sobolev space with is obtained.

A new approach to Sobolev spaces in metric measure spaces

Let (X,dX,μ) be a metric measure space where X is locally compact and separable and μ is a Borel regular measure such that 0<μ(B(x,r))<∞ for every ball B(x,r) with center x∈X and radius r>0. We define X to be the set of all positive, finite non-zero regular Borel measures with compact support in X which are dominated by μ, and M=X∪{0}. By introducing a kind of mass transport metric dM on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F:X→R, and then for functions f:X→[−∞,∞] by identifying them with the unique element Ff:X→R defined by the mean-value integral:Ff(η)=1‖η‖∫fdη. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space Rn with Lebesgue measure.

On the regularity of the global attractor for a damped Rosenau equation on R

In this paper, we consider the asymptotic behaviour of the solution for a damped Rosenau equation on . We apply a variant of Riesz–Rellich criteria, which involves the Littlewood–Paley projection operators, to prove that the damped Rosenau equation possesses a global attractor in for any . Moreover, the global attractor is contained in , if the time-independent source term is in and the initial data are in . Our results establish the regularity of the global attractor for the damped Rosenau equation in fractional order Sobolev space, which is a new ingredient in this paper.

A bilinear estimate and its application to a quadratic nonlinear Klein–Gordon equation in two space dimensions

We study the existence of the wave operators for the nonlinear Klein–Gordon equation with quadratic nonlinearity in two space dimensions ( ∂ t 2 − Δ + 1 ) u = λ u 2 , ( t , x ) ∈ R × R 2 . We prove existence of wave operators in lower order Sobolev spaces by using estimates of bilinear operators associated with Klein–Gordon equation.

Extra-strong uncertainty principles in relation to phase derivative for signals in Euclidean spaces

This paper devotes to studying uncertainty principles of Heisenberg type for signals defined on Rn taking values in a Clifford algebra. For real-para-vector-valued signals possessing all first-order partial derivatives we obtain two uncertainty principles of which both correspond to the strongest form of the Heisenberg type uncertainty principles for the one-dimensional space. The lower-bounds of the new uncertainty principles are in terms of a scalar-valued phase derivative. Through Hardy spaces decomposition we also obtain two forms of uncertainty principles for real-valued signals of finite energy with the first order Sobolev type smoothness.

Characterizations of Sobolev spaces via averages on balls

In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on Rn, using the average Btf(x)≔1|B(x,t)|∫B(x,t)f(y)dy of a function f over the ball B(x,t)≔{y∈Rn:|y−x|<t} with x∈Rn and t∈(0,∞). These characterizations rely only on the metric and the Lebesgue measure on Rn and are simpler than those obtained recently by Alabern et al. (2012). Moreover, these results may shed new light on the theory of high order Sobolev spaces on spaces of homogeneous type.

Decay of solutions to the three-dimensional generalized Navier–Stokes equations

In this paper, temporal decay estimates for weak solutions to the three dimensional generalized Navier-Stokes equa- tions are established firstly. With these estimates at disposal, algebraic time decay for higher order Sobolev norms of small initial data solutions are obtained. The decay rates are optimal in the sense that they coincide with ones of the corresponding generalized heat equation.

Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence

We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ on the d-dimensional torus, where the approximation error is measured in the $$L_2$$ -norm. In other words, we study the approximation numbers $$a_n$$ of the Sobolev embeddings $$H^s_\mathrm{mix}(\mathbb {T}^d)\hookrightarrow L_2(\mathbb {T}^d)$$ , with particular emphasis on the dependence on the dimension d. For any fixed smoothness $$s>0$$ , we find two-sided estimates for the approximation numbers as a function in n and d. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small n). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

Sequential weak approximation for maps of finite Hessian energy

Consider the space $$W^{2,2}(\Omega ;N)$$ of second order Sobolev mappings $$\ v\ $$ from a smooth domain $$\Omega \subset \mathbb {R}^m$$ to a compact Riemannian manifold N whose Hessian energy $$\int _\Omega |\nabla ^2 v|^2\, dx$$ is finite. Here we are interested in relations between the topology of N and the $$W^{2,2}$$ strong or weak approximability of a $$W^{2,2}$$ map by a sequence of smooth maps from $$\Omega $$ to N. We treat in detail $$W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$$ where we establish the sequential weak $$W^{2,2}$$ density of $$W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\,\cap \,{\mathcal C}^\infty $$ . The strong $$W^{2,2}$$ approximability of higher order Sobolev maps has been studied in the recent paper of Bousquet et al. (J. Eur. Math. Soc. (JEMS) 17(4), 763–817, 2015). For an individual map $$v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$$ , we define a number L(v) which is approximately the total length required to connect the isolated singularities of a strong approximation u of v either to each other or to $$\partial Bx^5$$ . Then $$L(v)=0$$ if and only if v admits $$W^{2,2}$$ strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of u, is the bound $$\ L(u)\le c+c\int _{\mathbb {B}^5}|\nabla ^2 u|^2\, dx\ $$ . This allows us to construct, for the given Sobolev map $$v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$$ , the desired $$W^{2,2}$$ weakly approximating sequence of smooth maps. To find suitable connecting curves for u, one uses the twisting of a u pull-back normal framing on a suitable level surface of u.

Existence and regularity results for a class of equations with logarithmic growth

We study the existence and the regularity of the solutions of the following Dirichlet problem {−div(A(x,∇u))=finΩu=0on∂Ω where A(x,ξ) is slowly increasing at ∞ as |ξ|→+∞. More precisely, we consider the model case A(x,ξ)=M(x)ξ|ξ|logα(1+|ξ|), where M(x) is a bounded elliptic matrix and α>0 is a fixed exponent. We will show that if f∈Ln(Ω) then there exists a unique solution u∈W01,1(Ω)∩L∞(Ω), such that ∇u∈LlogαL(Ω). Moreover, we prove that there exists an exponent ε>0 such that |∇u|logα+ε(1+|∇u|)∈L1(Ω). In case the matrix M(x) is assumed to be Hölder continuous, we establish a higher differentiability result for the solution in the scale of fractional order Sobolev spaces which yields a higher integrability result in the scale of Lebesgue spaces.

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.