Space W1 Research Articles

Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C1,α boundary to ℝ3 which belongs to the critical Sobolev space W2,2 is C1 up to the boundary. More generally C1 regularity up to the boundary holds for all scalar functions V ∈ W2,2(S) which satisfy det ∇2V=0. If S has only Lipschitz boundary we show such V can be approximated in W2,2 by functions Vk ∈ W1,∞∩W2,2 with det ∇2Vk=0.

Nonlinear riemann problem for nonlinear elliptic systems in sobolev space W1,p (D)

The nonlinear Riemann problems were converted into nonlinear singular integral equations and the existence of the solution for the problem was proved by means of contract principle.

Weak solutions to a nonlinear variational wave equation with general data

We establish the existence of global weak solutions to the initial value problem for a nonlinear variational wave equation utt−c(u)(c(u)ux)x=0 with general initial data (u(0),ut(0))=(u0,u1)∈W1,2×L2 under the assumptions that the wave speed c(u) satisfies c′(⋅)⩾0 and c′(u0(⋅))>0. Moreover, we obtain high regularity for the spatial derivative ∂xu of the wave amplitude u away from where c′(u)=0. This equation arises from studies in nematic liquid crystals, long waves on a dipole chain, and a few other fields. We use Young measure method in the setting of Lp spaces and method of renormalization to overcome the difficulty that oscillations in a sequence of approximations get amplified by the quadratic growth term of the equation. We use a high space-time estimate for ∂xu to handle possible concentrations. This result improves our earlier existence result for initial data in the space W1,∞×L∞ to the natural space W1,2×L2.

Multiplication operators on Sobolev disk algebra

In this paper, we study the algebra consisting of analytic functions in the Sobolev space W2,2(D) (D is the unit disk), called the Sobolev disk algebra, explore the properties of the multiplication operators Mf on it and give the characterization of the commutant algebra A′(Mf) of Mf. We show that A′(Mf) is commutative if and only if M*f is a Cowen-Douglas operator of index 1.

On an estimate related to the Hessian and application to an oblique derivative problem

We prove an estimate on the L2(Ω) -norm of the Hessian of a function u ∈ W2,q(Ω) , satisfying an oblique derivative type condition on the boundary, allowing the oblique axis to be tangential at a finite number of points of ∂Ω . Using this inequality, the solvability in Sobolev spaces W2,q(Ω) , with q close to 2 , follows for a class of nonlinear differential equations in the plane with quadratic growth. Mathematics subject classification (2000): 35J65, 35R05.

Fast Diffusion to Self-Similarity: Complete Spectrum, Long-Time Asymptotics, and Numerology

The complete spectrum is determined for the operator Open image in new window on the Sobolev space W1,2ρ(Rn) formed by closing the smooth functions of compact support with respect to the norm Open image in new window Here the Barenblatt profile ρ is the stationary attractor of the rescaled diffusion equation Open image in new window in the fast, supercritical regime mOpen image in new window the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d2. Formally, the operator H=HessρE is the Hessian of this entropy at its minimum ρ, so the spectral gap H≧α:=2−n(1−m) found below suggests the sharp rate of asymptotic convergence: Open image in new window from any centered initial data 0≦u(0,x) ∈ L1(Rn) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)−nρ(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rn and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to ρ with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of ρ.

ON ISOMETRIC IMMERSIONS OF A RIEMANNIAN SPACE WITH LITTLE REGULARITY

We consider a Riemannian metric in an open subset of the d-dimensional Euclidean space and assume that its Riemann curvature tensor vanishes. If the metric is of class C2, a classical theorem in differential geometry asserts that the Riemannian space is locally isometrically immersed in the d-dimensional Euclidean space. We establish that if the metric belongs to the Sobolev space W1,∞ and its Riemann curvature tensor vanishes in the space of distributions, then the Riemannian space is still locally isometrically immersed in the d-dimensional Euclidean space.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

On the definition of the Monge-Amp�re operator in ?2

We show that if u is a plurisubharmonic function defined on an open subset Ω of ℂ2 then the Monge-Ampere measure (dd cu )2 can be well defined if and only if u belongs to the Sobolev space W 1,2 loc (Ω).

Weak density of smooth maps for the Dirichlet energy between manifolds

We prove that given a simply connected compact manifold M and a closed manifold N, any map in the Sobolev space W 1,2(M,N) can be approximated weakly by smooth maps between M and N.

Sobolev capacity on the space W1, p(⋅)(ℝn)

We define Sobolev capacity on the generalized Sobolev space W1, p(⋅)(ℝn). It is a Choquet capacity provided that the variable exponent p : ℝn → [1, ∞) is bounded away from 1 and ∞. We discuss the relation between the Hausdorff dimension and the Sobolev capacity. As another application we study quasicontinuous representatives in the space W1, p(⋅)(ℝn).

Certain imbeddings of weighted Sobolev spaces

We characterize weight functions for which the weighted Sobolev space W1,p(Ω, dβ M) [and also W1,p(Ω, s0(dM))] is imbedded continuously or compactly into the weighted Lebesgue space Lq(Ω, dα M) [and also L q(Ω, s1(dM))] where 1 q < p < ∞ and M ⊂ ∂Ω . Some of the imbeddings are also extended to the higher order weighted Sobolev spaces. Mathematics subject classification (2000): 46E35.

Relaxation and attainment results for an integral functional with unbounded energy well

Consider the functional I(u) = ∫Ω ‖Du|n − L det Du| dx, whose energy well consists of matrices satisfying |ξ|n = L det ξ. We show that the relaxations of this functional in various Sobolev spaces are significantly different. We also make several remarks concerning various p-growth semiconvex hulls of the energy-well set and prove an attainment result for a special Hamilton-Jacobi equation, |Du|n = L det Du, in the so-called grand Sobolev space W1,q)(Ω; Rn), with q = nL/(L + 1).

Formula omitted]-weighted imbedding inequalities for [formula omitted]-harmonic tensors

We prove new versions of Arλ(Ω)-weighted imbedding inequalities for A-harmonic tensors locally and globally. The results are used to estimate the integrals of a homotopy operator T from the Banach space Ls(Ω,∧l) to the Sobolev space W1,s(Ω,∧l−1), l=1,2,…,n, and Sobolev–Poincaré imbedding inequality with Arλ(Ω) weight.

A MEAN CONTINUITY TYPE RESULT FOR CERTAIN SOBOLEV SPACES WITH VARIABLE EXPONENT

For functions f in Sobolev spaces W1,p(x)(Ω) with exponent lower semicontinuous, bounded away from 1 and ∞ and with the property of the density of smooth functions, it is shown that for each open set ω ⊂⊂ Ω, for each h ∈ RN such that ω+th ⊂ Ω ∀ t∈ [0,1], the following inequality holds [Formula: see text] where min p(x, x + h) denotes the minimum of p along the segment whose endpoints are x, x + h. As a consequence, if p(x) is also continuous, for mollifiers (ρn){n ∈ N the liminf and the limsup of [Formula: see text] are respectively minorized and majorized by expressions equivalent to ‖|∇f|‖Lp(x)(Ω).

Estimates of Jacobians by subdeterminants

Let ƒ: Ω → ℝn be a mapping in the Sobolev space W1,n−1(Ω,ℝn), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D#ƒ satisfies\(|D^\sharp f|^{\frac{n}{{n - 1}}} \in L^P (\Omega )\), where Lp(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in Lloc1 (Ω). Estimates above and below Lloc1 (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.

A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthdayJ. Kinnunen proved that of P &gt; 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

Multiwavelets on the Interval

Smooth orthogonal and biorthogonal multiwavelets on the real line with their scaling function vectors being supported on [−1,1] are of interest in constructing wavelet bases on the interval [0,1] due to their simple structure. In this paper, we shall present a symmetric C2 orthogonal multiwavelet with multiplicity 4 such that its orthogonal scaling function vector is supported on [−1,1], has accuracy order 4, and belongs to the Sobolev space W2.56288. Biorthogonal multiwavelets with multiplicity 4 and vanishing moments of order 4 are also constructed such that the primal scaling function vector is supported on [−1,1], has the Hermite interpolation properties, and belongs to W3.63298, while the dual scaling function vector is supported on [−1,1] and belongs to W1.75833. A continuous dual scaling function vector of the cardinal Hermite interpolant with multiplicity 4 and support [−1,1] is also given. All the wavelet filters constructed in this paper have closed form expressions. Based on the above constructed orthogonal and biorthogonal multiwavelets on the real line, both orthogonal and biorthogonal multiwavelet bases on the interval [0,1] are presented. Such multiwavelet bases on the interval [0,1] have symmetry, small support, high vanishing moments, good smoothness, and simple structures. Furthermore, the sequence norms for the coefficients based on such orthogonal and biorthogonal multiwavelet expansions characterize Sobolev norms ‖·‖Ws([0,1]) for s∈(−2.56288,2.56288) and for s∈(−1.75833,3.63298), respectively.

A Singular Boundary Value Problem for Uniformly Elliptic Operators

Strong solvability in the Sobolev space W2,p(Ω) is proved for the oblique derivative problem∑j,i=1naij(x)Diju=f(x)almost every where in Ω,∂u/∂ℓ+σ(x)u=ϕ(x)in the trace sense on ∂Ωin the case when the vector field ℓ(x) has a contact of infinite order with ∂Ω at the points of some non-empty subset E⊂∂Ω.

On Radially Symmetric Minima of Nonconvex Functionals

We study a minimization problem in the space W1,10(BR) where BR is the ball of radius R with center at the origin; the functional considered is not necessarily convex. Under suitable assumptions, we prove the existence of a radially symmetric (decreasing) solution. By strengthening the assumptions we obtain uniqueness results. Finally, we study under which assumptions and in which sense the solutions found solve the corresponding Euler equation. The proofs are very direct and simple: they only make use of the functions T±n introduced by the author [Arch. Rational Mech. Anal., 1999].