A remark on the regularity of solutions of Maxwell's equations on Lipschitz domains

Let u be a vector field on a bounded Lipschitz domain in ?³, and let u together with its divergence and curl be square integrable. If either the normal or the tangential component of u is square integrable over the boundary, then u belongs to the Sobolev space H^1/2 on the domain. This result gives a simple explanation for known results on the compact embedding of the space of solutions of Maxwell's equations on Lipschitz domains into L².     

Spectral Tau approximation of the two-dimensional stokes problem

Summary We analyse the Spectral Tau method for the approximation of the Stokes system on a square with Dirichlet boundary conditions. We provide an error estimate, in the norm of the Sobolev spaceH ^s, for the approximation of a divergence free vector field with polynomial divergence free vector fields. We apply this result to prove some convergence estimates for the solution of the discrete Stokes problem.This work has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C 0035     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Piecewise-polynomial approximation of functions fromH ℓ((0, 1) d ), 2ℓ=d,and applications to the spectral theory of the Schrödinger operator

For the selfadjoint Schrödinger operator ?Δ?αV on ?² the number of negative eigenvalues is estimated. The estimates obtained are based upon a new result on the weightedL ₂-approximation of functions from the Sobolev spaces in the cases corresponding to the critical exponent in the embedding theorem.     

The Gradient Flow of the L 2 Curvature Functional with Small Initial Energy

We investigate the low-energy behavior of the gradient flow of the L ² norm of the Riemannian curvature on a four-manifold. In particular we show that if the initial energy is chosen small enough with respect to the initial Sobolev constant and the H ¹ norm of the gradient vector then the flow exists for all time and converges to a flat metric. We also improve the regularity requirement for the flow proved in Streets (J. Geom. Anal. 18:249, 2008) in the case of four-manifolds.     

Estimates for L1-vector fields

A simple proof of an integral inequality involving L¹-vector fields is provided. This gives a short proof of estimates of Bourgain and Brezis for elliptic and div–curl systems. To cite this article: J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Domain perturbations and estimates for the solutions of second order elliptic equations

We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal L² and energy estimates for the difference of two solutions in two open sets in terms of the “distance” between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the L² family of Sobolev–Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.     

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

Given an open domain (possibly unbounded) Ω?R ⁿ , we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L ¹(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.     

On a parabolic logarithmic Sobolev inequality

In order to extend the blow-up criterion of solutions to the Euler equations, Kozono and Taniuchi [H. Kozono, Y. Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Comm. Math. Phys. 214 (2000) 191-200] have proved a logarithmic Sobolev inequality by means of isotropic (elliptic) BMO norm. In this paper, we show a parabolic version of the Kozono-Taniuchi inequality by means of anisotropic (parabolic) BMO norm. More precisely we give an upper bound for the L^∞ norm of a function in terms of its parabolic BMO norm, up to a logarithmic correction involving its norm in some Sobolev space. As an application, we also explain how to apply this inequality in order to establish a long-time existence result for a class of nonlinear parabolic problems.     

On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions

In this paper we extend the DiPerna-Lions theory of flows associated to Sobolev vector fields to the case of Cameron-Martin-valued vector fields in Wiener spaces E having a Sobolev regularity. The proof is based on the analysis of the continuity equation in E, and on uniform (Gaussian) commutator estimates in finite-dimensional spaces.     

Carleman Estimates for the Schrdinger Equation and Applications to an Inverse Problem and an Observability Inequality

The authors prove Carleman estimates for the Schrdinger 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 Schrdinger equation are also obtained.     

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">P</Emphasis></Superscript>

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L² spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L ^p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L ^p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L ^p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L ^p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L² extends to L ^p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L² proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.     

Sobolev embeddings involving symmetry

Given a bounded regular domain with cylindrical symmetry, then functions having such symmetry and belonging to the first Sobolev space can be embedded compactly into some weighted Lp spaces, with p superior to the critical Sobolev exponent. A simple application to elliptic boundary value problem is also considered.     

Boundary estimates for elliptic systems with L<Superscript>1</Superscript>–data

We obtain boundary estimates for the gradient of solutions to elliptic systems with Dirichlet or Neumann boundary conditions and L ¹–data, under some condition on the divergence of the data. Similar boundary estimates are obtained for div–curl and Hodge systems.     

Embedding of Sobolev spaces and properties of the domain

We establish the embedding of the Sobolev space W _p ^s (G) ? L _q (G) for an irregular domain G in the case of a limit exponent under new relations between the parameters depending on the geometric properties of the domain G.     

On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L² tangential fields and then the attention is focused on some particular Sobolev spaces of order $‐{1\over 2}$\nopagenumbers\end . In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H ( curl , Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.     

The Cauchy problem for critical and subcritical semilinear parabolic equations in Lr (II), initial data in critical Sobolev spaces Ḣ−s,rs

By presenting some time–space L^p−L^r estimates, we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the Cauchy data in critical Sobolev spaces of negative indices. Our results contain the complex (derivative) Ginzburg–Landau equation and the Cahn–Hilliard equation as special cases.     

Divergence L 2-Coercivity Inequalities

This paper derives sharp L ²-coercivity inequalities for the divergence operator on bounded Lipschitz regions in ? ⁿ . They hold for fields in H(div,Ω) that are orthogonal to N(div). The optimal constants in the inequality are defined by a variational principle and are identified as the least eigenvalue of a nonstandard boundary value problem for a linear biharmonic type operator. The dependence of the optimal constant under dilations of the region is described and a generalization that involves weighted surface integrals is also proved. When n = 2, this also yields a similar coercivity result for the curl operator.     

Local stability of mappings with bounded distortion on Heisenberg groups

This is the second of the author’s three papers on stability in the Liouville theorem on the Heisenberg group. The aim is to prove that each mapping with bounded distortion of a John domain on the Heisenberg group is close to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\).In this paper we prove a local variant of the desired result: each mapping on a ball with bounded distortion and distortion coefficient K near to 1 is close on a smaller ball to a conformal mapping with order of closeness \(\sqrt {K - 1} \) in the uniform norm and order of closeness K ? 1 in the Sobolev norm L _p ¹ for all \(p < \tfrac{C}{{K - 1}}\). We construct an example that demonstrates the asymptotic sharpness of the order of closeness of a mapping with bounded distortion to a conformal mapping in the Sobolev norm.     

An LP estimate for Maxwell's equations with source term

In this Note we establish an interior L^p-type estimate for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a $\text{C}{}^2$ interface. Due to the singularity of the dielectric permittivity, the usual elliptic estimates cannot be applied directly. A special curl–div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with discontinuous coefficients. The potential theory analysis and the jump condition lead to the L^p estimates which are superior to the straightforward Nash–Moser estimates. The reduction procedure is expected to be useful for numerical simulation. Such an estimate is crucial for solving nonlinear Maxwell's equations that arise for example in the modeling of nonlinear optics. To cite this article: G. Bao et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).