Sobolev Spaces Of Order Research Articles

Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces

We exhibit sharp embedding constants for Sobolev spaces of any order into Zygmund spaces, obtained as the product of sharp embedding constants for second order Sobolev space into Lorentz spaces. As a consequence, we derive a new proof of Adams’ inequality, which holds in the larger hypotheses of homogenoeous Navier boundary contidions.

Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schrödinger Equation

We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schrödinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.

An inverse problem for a wave equation with arbitrary initial values and a finite time of observations

We consider a solution u(p, g, a, b) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T) and ∂itu(p, g, a, b)(x, T0), x ∊ Ω, i = 0, 1, with T0 < T. We prove that if T − T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, …, gN, so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj, bj}1 ⩽ j ⩽ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized H01 simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to O(plogp), required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with O(plogp) unknowns. Several numerical examples support the theoretical estimates.

Well-posedness of the Einstein–Euler system in asymptotically flat spacetimes: The constraint equations

This paper deals with the construction of initial data for the coupled Einstein–Euler system. We consider the condition where the energy density might vanish or tend to zero at infinity, and where the pressure is a fractional power of the energy density. In order to achieve our goals we use a type of weighted Sobolev space of fractional order. The common Lichnerowicz–York scaling method (Choquet-Bruhat and York, 1980 [9]; Cantor, 1979 [7]) for solving the constraint equations cannot be applied here directly. The basic problem is that the matter sources are scaled conformally and the fluid variables have to be recovered from the conformally transformed matter sources. This problem has been addressed, although in a different context, by Dain and Nagy (2002) [11]. We show that if the matter variables are restricted to a certain region, then the Einstein constraint equations have a unique solution in the weighted Sobolev spaces of fractional order. The regularity depends upon the fractional power of the equation of state.

The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa‐Holm Equation

Using the Kato theorem for abstract differential equations, the local well‐posedness of the solution for a nonlinear dissipative Camassa‐Holm equation is established in space C([0, T), Hs(R))∩C1([0, T), Hs−1(R)) with s &gt; 3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1 ≤ s ≤ 3/2 is developed.

Real interpolation of spaces of differential forms

In this paper, we study interpolation of Hilbert spaces of differential forms using the real method of interpolation. We show that the scale of fractional order Sobolev spaces of differential l-forms in H s with exterior derivative in H s can be obtained by real interpolation. Our proof heavily relies on the recent discovery of smoothed Poincare lifting for differential forms [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincare integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265(2): 297–320, 2010]. They enable the construction of universal extension operators for Sobolev spaces of differential forms, which, in turns, pave the way for a Fourier transform based proof of equivalences of K-functionals.

Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces

Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$loc$(0,1) : A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.

The existence of weak solutions for a generalized Camassa-Holm equation

A Camassa-Holm type equation containing nonlinear dissipative effect is investigated. A sufficient condition which guarantees the existence of weak solutions of the equation in lower order Sobolev space $H^s$ with $1 \leq s \leq \frac{3}{2}$ is established by using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself.

Complexity of numerical integration over spherical caps in a Sobolev space setting

Let r ≥ 2 , let S r be the unit sphere in R r + 1 , and let C ( z ; γ ) : = { x ∈ S r : x ⋅ z ≥ cos γ } be the spherical cap with center z ∈ S r and radius γ ∈ ( 0 , π ] . Let H s ( S r ) be the Sobolev (Hilbert) space of order s of functions on the sphere S r , and let Q m be a rule for numerical integration over C ( z ; γ ) with m nodes in C ( z ; γ ) . Then the worst-case error of the rule Q m in H s ( S r ) , with s > r / 2 , is bounded below by c r , s , γ m − s / r . The worst-case error in H s ( S r ) of any rule Q m ( n ) that has m ( n ) nodes in C ( z ; γ ) , positive weights, and is exact for all spherical polynomials of degree ≤ n is bounded above by c ̃ r , s , γ n − s . If positive weight rules Q m ( n ) with m ( n ) nodes in C ( z ; γ ) and polynomial degree of exactness n have m ( n ) ∼ n r nodes, then the worst-case error is bounded above by c ˆ r , s , γ ( m ( n ) ) − s / r , giving the same order m − s / r as in the lower bound. Thus the complexity in H s ( S r ) of numerical integration over C ( z ; γ ) with m nodes is of the order m − s / r . The constants c r , s , γ and c ˆ r , s , γ in the lower and upper bounds do not depend in the same way on the area | C ( z ; γ ) | ∼ γ r of the cap. A possible explanation for this discrepancy in the behavior of the constants is given. We also explain how the lower and upper bounds on the worst-case error in a Sobolev space setting can be extended to numerical integration over a general non-empty closed and connected measurable subset Ω of S r that is the closure of an open set.

A model containing both the Camassa–Holm and Degasperis–Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa–Holm and Degasperis–Procesi equations as special cases, is investigated. Although the H 1 -norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space H s with 1 < s ⩽ 3 2 is established under the assumptions u 0 ∈ H s and ‖ u 0 x ‖ L ∞ < ∞ . The local well-posedness of solutions for the equation in the Sobolev space H s ( R ) with s > 3 2 is also developed.

Geometric optics and instability for NLS and Davey–Stewartson models

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schr¨odinger equations with gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.

Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality

The authors prove Carleman estimates for the Schrodinger equation in Sobolev spaces of negative orders, and use these estimates to prove the uniqueness in the inverse problem of determining Lp-potentials. An L2-level observability inequality and unique continuation results for the Schrodinger equation are also obtained.

Boundary Regularity in Variational Problems

We prove that, if $${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}$$ is a solution to the Dirichlet variational problem $$\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,$$ involving a regular boundary datum (u 0, ∂Ω) and a regular integrand F(x, w, Dw) strongly convex in Dw and satisfying suitable growth conditions, then $${{\mathcal H}^{n-1}}$$ -almost every boundary point is regular for u in the sense that Du is Holder continuous in a relative neighborhood of the point. The existence of even one such regular boundary point was previously not known except for some very special cases treated by Jost & Meier (Math Ann 262:549–561, 1983). Our results are consequences of new up-to-the-boundary higher differentiability results that we establish for minima of the functionals in question. The methods also allow us to improve the known boundary regularity results for solutions to non-linear elliptic systems, and, in some cases, to improve the known interior singular sets estimates for minimizers. Moreover, our approach allows for a treatment of systems and functionals with “rough” coefficients belonging to suitable Sobolev spaces of fractional order.

Toeplitz and Hankel operators with [formula omitted] symbols on Dirichlet space

We show that Toeplitz or small Hankel operators with symbols in L ∞ , 1 is a generalization of the case with the harmonic symbol in C ⊕ D ⊕ D ¯ , where D is the Dirichlet space on D . We also give a decomposition theorem for the Sobolev space of first order on D . Using this result, some characterizations for algebraic properties of Toeplitz or small Hankel operators with symbols in L ∞ , 1 are given.

The existence and uniqueness of the local solution for a Camassa–Holm type equation

A shallow water equation of Camassa–Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H s ( R ) with s > 3 2 . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H s with 1 < s ⩽ 3 2 are presented.

Existence of weak solutions in lower order Sobolev space for a Camassa–Holm-type equation

A generalized Camassa–Holm equation containing a nonlinear dissipative effect is investigated. The existence of the weak solution of the equation in lower order Sobolev space Hs with is established by using the techniques of the pseudoparabolic regularization and some a priori estimates derived from the equation itself.

The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation

A generalization of the Camassa–Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space H s ( R ) with s > 3 2 is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space H s with 1 < s ⩽ 3 2 is developed.

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω⊂Rn, we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L1(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.

Coercive energy estimates for differential forms in semi-convex domains

In this paper, we prove a $H^1$-coercive estimate for differentialforms of arbitrary degrees in semi-convex domains of the Euclidean space.The key result is an integral identity involving a boundary term in whichthe Weingarten matrix of the boundary intervenes, establishedfor any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whoseoutward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$,the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.