Classical Sobolev Spaces Research Articles

New Locking-Free Mixed Method for the Reissner--Mindlin Thin Plate Model

We are interested here in the Reissner--Mindlin model for a bending thin plate with physical boundary conditions. It is well known that this problem depends singularly upon the plate's thickness \( \varepsilon \). By decomposing the bending moment and by dualizing its symmetry, we obtain an equivalent mixed formulation of the initial problem whose unknowns now belong to classical Sobolev spaces. We then propose a low-order conforming finite element method for which we obtain optimal error estimates independently upon the small parameter \( \varepsilon . \) Thus, the discrete method is unconditionally convergent and locking-free. It directly gives an approximation of the bending moment and allows us to recover the two other variables, which are the deflection and the rotation vector.

Intractability Results for Integration and Discrepancy

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight γj moderates the behavior of functions with respect to the jth variable. For γj≡1, we obtain the classical Sobolev spaces whereas for decreasing γj's the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n function values for the unit ball of the space Fd. It is known that if the smoothness parameter of the Sobolev space is one, then the minimal error is the same as the discrepancy. Let n(ε, Fd) be the smallest n for which we reduce the initial error, i.e., the error with n=0, by a factor ε. The main problem studied in this paper is to determine whether the problem is intractable, i.e., whether n(ε, Fd) grows faster than polynomially in ε−1 or d. In particular, we prove intractability of integration (and discrepancy) if limsupd∑dj=1γj/lnd=∞. Previously, such results were known only for restricted classes of quadratures. For γj≡1, the, following bounds hold for discrepancy •with boundary conditions1.0628d(1+o(1))⩽n(ε, Fd)⩽1.5dε−2,asd→∞,•without boundary conditions1.0463d(1+o(1))⩽n(ε, Fd)⩽1.125dε−2,asd→∞. These results are obtained by analyzing arbitrary linear tensor product functionals Id defined over weighted tensor product reproducing kernel Hilbert spaces Fd of functions of d variables. We introduce the notion of a decomposable kernel. For reproducing kernels that have a decomposable part we prove intractability of all Id with non-zero subproblems with respect to the decomposable part of the kernel, as long as the weights satisfy the condition mentioned above.

Connected components of sets of finite perimeter and applications to image processing

This paper contains a systematic analysis of a natural measure theoretic notion of connectedness for sets of finite perimeter in \mathbb R^N , introduced by H. Federer in the more general framework of the theory of currents. We provide a new and simpler proof of the existence and uniqueness of the decomposition into the so-called M -connected components. Moreover, we study carefully the structure of the essential boundary of these components and give in particular a reconstruction formula of a set of finite perimeter from the family of the boundaries of its components. In the two dimensional case we show that this notion of connectedness is comparable with the topological one, modulo the choice of a suitable representative in the equivalence class. Our strong motivation for this study is a mathematical justification of all those operations in image processing that involve connectedness and boundaries. As an application, we use this weak notion of connectedness to provide a rigorous mathematical basis to a large class of denoising filters acting on connected components of level sets. We introduce a natural domain for these filters, the space \mathrm{WBV}(\Omega) of functions of weakly bounded variation in \Omega , and show that these filters are also well behaved in the classical Sobolev and BV spaces.

On the ill-posedness of some canonical dispersive equations

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

On the local and global existence of solution for a general Ginzburg-Landau like equation coupled with a Poisson equation in {$L^p({\Bbb R}^d)$}

We investigate the existence of a local or global semi-group for a complex Ginzburg-Landau like equation in $u$ coupled with a Poisson equation in $\phi$ defined on the whole space $\mathbb R^d$. At first, we consider the Cauchy problem in the classical Sobolev spaces $L_{\mathbb C}^{p}(\mathbb R ^{d})$, and later we study it in the weighted Sobolev spaces $L_{\rho\mathbb C }^{p}(\mathbb R ^{d})$, where $p\geq 3/2$, $p\geq d$, $d$ is a positive integer and the weight $\rho$ is increasing. Using the smoothing properties of the linear part, we obtain, for initial data in $L_{\mathbb C }^{p}(\mathbb R ^{d})$, a continuous strong solution in $W_{\mathbb C }^{1,p}(\mathbb R ^{d})$ with a singularity at $t=0$ behaving like $t^{-\frac{1}{2}}$. We obtain analogous results in weighted Sobolev spaces.

Low regularity solutions for a generalized Zakharov system

Using the method of Bourgain (actually we use techniques developed in the paper of Ginibre, Tsutsumi, Velo (cf. [9])) we prove well-posedness of a generalized Zakharov system, describing the spontaneous generation of magnetic field in a cold plasma, in the framework of classical Sobolev spaces. Using the conservation laws of the system we extend these solutions globally in the energy space.

Global existence and decay of solutions of a coupled system of BBM-Burgers equations

The global well-posedness of the initial-value problem associated to the coupled system of BBM-Burgers equations ut-uxxt-a3vxxt+ upux+ a1vpvx+ a2(upv)x -e1uxx= 0 b1vt- vxxt- b2a3uxxt+ vpvx+b2a2upux+ b2a1(uvp)x-e2vxx = 0 in the classical Sobolev spaces Hs(R) x Hs(R) for s > - 2 is studied. Furthermore we find decay estimates of the solutions of in the norm Lq(R) x Lq(R), 2 < - q < - for general initial data provided that a32b2<1 and p|3. Model (*) is motivated by a work due to Gear and Grimshaw [10] who considered strong interaction of weakly nonlinear long waves governed by a coupled system of KdV equations.

The Distribution of Eigenfrequencies of Anisotropic Fractal Drums

Let Γ be an anisotropic fractal. The aim of the paper is to investigate the distribution of the eigenvalues of the fractal differential operator (−Δ)−1 o trΓ acting in the classical Sobolev space W ^ 2 1 ( Ω ) where Ω is a bounded C∞ domain in the plane R2 with Γ⊂Ω. Here −Δ is the Dirichlet Laplacian in Ω and trΓ is closely related to the trace operator trΓ.

Regularity of the solutions of elliptic systems in polyhedral domains

The solution of the Dirichlet problem relative to an elliptic system in a polyhedron has a complex singular behaviour near edges and vertices. Here, we show that this solution has a global regularity in appropriate weighted Sobolev spaces. Some useful embeddings of these spaces into classical Sobolev spaces are also established. As applications, we consider the Lam e, Stokes and Navier-Stokes systems. The present results will be applied in a forthcoming work to the constructive treatment of these problems by optimal convergent nite element method.

Analyticity of Solutions of the Generalized Korteweg–de Vries Equation with Respect to Their Initial Values

The initial value problem (IVP) of the generalized Korteweg–de Vries (KdV) equation \[ \partial _t u + \partial _x (a(u)) + \partial _x^3 u = 0,\quad u(x,0) = \phi (x)\] is well posed in the classical Sobolev space $H^s (R)$ with $s > {3 / 4}$, which establishes a nonlinear map K from $H^s (R)$ to $C ( [ - T,T];H^s (R)$. It is shown that (i) if $a = a(x)$ is a $C^\infty $ function on R to R, then K is infinitely many times Frechet differentiable; (ii) if $a = a(x)$ is a polynomial, then K is analytic, i.e., for any $\phi \in H^s (R)$, K has a Taylor series expansion \[ K(\phi + h) = \sum_{n = 0}^\infty {\frac{1}{{n!}}} K^{(n)} (\phi )[h^n ].\] Each term $y_n = K^{(n)} (\phi )[h^n ]$ in the series solves a linearized KdV equation. Consequently, any “small” perturbation $K(\phi + h)$ of $K(\phi )$ can be obtained by solving a series of linear problems. The proof of these results relies on various smoothing properties of the associated linear KdV equation.

A fast algorithm for solving the generalized airfoil equation

The present paper is devoted to the quadrature method for Cauchy-type singular integral equations on [-1, 1] with an additional logarithmic singularity of the kernel. An algorithm being based on this method is given whose realization only requires a computational work of O( n ln n) operations in order to determine n unknown coefficients of the approximate solution. Furthermore, in a scale of weighted Sobolev-like spaces on [-1, 1] (having analogous properties like the classical Sobolev spaces on the unit circle) the order of convergence is optimal.

Differential equations in Hilbert spaces and applications to boundary value problems in nonsmooth domains

We give existence and uniqueness results in weighted Sobolev spaces for a solution of a first order differential equation with operator coefficients of type ∂u ∂t − Au(t) = ƒ(t) , on the line R . This equation is an abstract version of elliptic boundary value problems in infinite cylinders (see [V. A. Kondratiev, Trans. Moscow Math. Soc. 16 (1967), 227–313; V. G. Maz'ya and B. A. Plamenevskii, Trans. Moscow Math. Soc. 1 (1980), 49–97; V. G. Maz'ya and B. A. Plamenevskii, Amer. Math. Soc. Transl. Ser. 2 123 (1984), 1–56; V. G. Maz'ya and B. A. Plamenevskii. J. Soviet Math. 9 (1978), 750–764; V. G. Maz'ya and B. A. Plamenevskii, Amer. Math. Soc. Transl. Ser. 2 123 (1984), 57–88]); therefore, our results translate the classical ones to this setting. The difference between two solutions belonging to different weighted Sobolev spaces is a finite linear combination of singular functions depending on the eigenvalues of A and on the corresponding eigenspaces. We compute the coefficients of these singular functions, because we can give explicitly the Laurent series of the resolvent of A near an eigenvalue of the operator A. We also give an abstract polynomial resolution, which corresponds to the resolution of an elliptic boundary value problem in an infinite cone with polynomial right-hand sides. In practice, it allows one to pass from weighted Sobolev spaces to classical Sobolev spaces. Finally, we consider some applications of this abstract theory.

Well-posedness of the initial value problem for the Korteweg-de Vries equation

(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been considered in different contexts, namely in its relation with the inverse scattering method, in plasma physics, and in algebraic geometry (see [24], and references therein). Our purpose is to study local and global well-posedness of the IVP (1.1) in classical Sobolev spaces Hs(R) . We shall say that the IVP (1.1) is locally (resp. globally) well-posed in the function space X if it induces a dynamical system on X by generating a continuous local (resp. global) flow. It was established in the works of Bona and Smith [3], Bona and Scott [2], Saut and Temam [30], and Kato [ 1 5] that the IVP (1. 1) is locally (resp. globally) well-posed in Hs with s > 3/2 (resp. s > 2). Roughly speaking, global well-posedness in Hs depends on the available local theory and on the conservation laws satisfied by solutions of (1.1), namely:

On Singular Integro‐Differential Operators on the Halfaxis

AbstractIn the frame of classical SOBOLEV spaces the FREDHOLM property will be characterized for a class of singular integro‐differential operators on the positive halfaxis. A symbol calculus moreover permits to specify an index formula and a special FREDHOLM inverse.

A model for the buckling and the stability of thin elastoplastic plates

Two mathematical models are presented which yield some mechanical aspects of this elastoplastic plate bending. The frist model proves insufficient if the classical Sobolev space framework is kept up. With the second model, an existence result for the transverse displacement problem formulation is obtained when the load does not exceed a specific critical value. The study of the stability problem leads to differentiation of a projector on a closed convex set, which is a difficult question; nevertheless, a hypothesis of regularity of the solution of some plasticity problem is introduced, and the existence of a critical load under which there is stability in Some sense is shown.

The state density of elliptic operators with random potentials

The existence of the state density N(λ) is established for certain linear elliptic coercive selfadjoint operators subject to perturbation by a unbounded random potential and its asymptotic properties discussed as λ →∞ (and for certain Gaussian perturbations as λ→ — ∞). The estimates depend on recent worlc by Metivier on the approximation numbers of the classical Sobolev spaces.