Determination of a Domain in Complex Space by its Automorphism Group

We consider results, both in one complex variable and several, which show that the algebraic or geometric structure of the automorphism group of a domain z can determine that domain. The domains considered include Ω = B, the unit ball in C ⁿ , and Ω = C ⁿ . Various illustrative examples are provided.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.     

An integral operator related to the Stokes system in exterior domains

Consider a bounded domain Ω in ?³ with C²-boundary ?Ω. In [1] the Stokes problem in the exterior domain ?³/Ω , with resolvent parameter [λ??\] ? [∞,0], is solved by using the method of integral equations. However, for estimating the corresponding solutions in L^p norms, it turns out that a certain operator defined on the spaces L^r(?Ω)³, for r ?]1, ∞[, has to be evaluated in the norm of L^r(?Ω)³. This estimate is proved in the present paper.     

Regularity of minimizers of semilinear elliptic problems up to dimension 4

We consider the class of semistable solutions to semilinear equations ?Δu = f(u) in a bounded smooth domain Ω of \input amssym $\Bbb R^n$ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an a priori L^∞‐bound that holds for every positive semistable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n = 4 and Ω is convex. This result was previously known only in dimensions n ≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω = B_R by a result of A. Capella and the author. © 2010 Wiley Periodicals, Inc.     

A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

     

Nonlinear Elliptic Variational Inequalities

We consider the problem of finding a solution to a class of nonlinear elliptic variational inequalities. These inequalities may be defined on bounded or unbounded domains Ω, and the nonlinearity can depend on gradient terms. Appropriate definitions of sub-and supersolutions relative to the constraint sets are given. By using a mixture of maximal monotone operator theory and compactness arguments we prove the existence of a H²(Ω) solution lying between a given subsolution φ₁ and a given supersolution φ₂≧φ₁, when Ω is bounded, and a H¹(Ω) solution when Ω is unbounded.     

A Unified Constructive Approach to the Topological Degree in Rn

The paper gives an approach to the topological degree in Rⁿ which takes into account numerical requirements and permits derivation of the known degree computation formulas in a simple way. The new approach subsumes several earlier approaches and represents a general principle of construction of degree computation formulas. The basic idea consists of computing the degree of a continuous function relative to a bounded open subset Ω of Rⁿ by means of an auxiliary function which is defined on a polyhedron approximating Ω and maps into a known fixed convex polyhedron containing the origin of Rⁿ. It is further shown that the topological degree of a continuous function relative to an n-dimensional polyhedron P can be computed alone by means of a subset of the boundary of P .     

An error estimate for a finite volume scheme for a diffusion–convection problem on a triangular mesh

We study here a finite volume scheme for a diffusion-convection equation on an open bounded set Ω of ?², using a triangular mesh for the discretization of Ω. The 4-point numerical scheme is presented along with the geometrical assumptions on the mesh. An error estimate of order h on the discrete L² norm is obtained, where h denotes the “size” of the mesh. The proof uses an estimate of order h of the consistency error on the fluxes and an estimate of the number of edges of the mesh between one given triangle and the boundary Ω. © 1995 John Wiley & Sons, Inc.     

On the Uniform Poincaré Inequality

We give a proof of the Poincaré inequality in W ^{1, p} (Ω) with a constant that is independent of Ω ? , where  is a set of uniformly bounded and uniformly Lipschitz domains in ? ⁿ . As a byproduct, we obtain the following: The first non vanishing eigenvalues λ₂(Ω) of the standard Neumann (variational) boundary value problem on Ω for the Laplace operator are bounded below by a positive constant if the domains Ω vary and remain uniformly bounded and uniformly Lipschitz regular.     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Polar factorization and monotone rearrangement of vector-valued functions

Given a probability space (X, μ) and a bounded domain Ω in ?^d equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ L^p (X, μ; ?^d) there is a unique “polar factorization” u = ?Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u^?1(E)) = 0 for each Lebesgue negligible subset E of ?^d. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

Über das Verhalten der Lösungen der Maxwellschen Randwertaufgabe in Gebieten mit Kegelspitzen

We study the behavior of the solution E of the Maxwell's boundary value problem ? × ? × E + λE = F, n × E|r = 0 in domains Ω which have conical boundary points. In a neighbourhood K(R) = B(a,R) ∩ Ω of a singular boundary point a the field E is expanded using a theorem of N. Weck. It e.g. turns out that the solution lies in H₁⁽³⁾(K(R)) if K(R) is convex.     

On sparse reconstruction from Fourier and Gaussian measurements

This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the sparse approximation theory is to relax this highly nonconvex problem to a convex problem and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every r‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size A random set Ω satisfies this with high probability. This estimate is optimal within the log logn and log³r factors. We also give a relatively short argument for a similar problem with k(r, n) ≈ r[12 + 8 log(n/r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short. © 2007 Wiley Periodicals, Inc.     

Coincidence of Two Classes of Universal Laurent Series

There exist functions, called U.L.S. (Universal Laurent Series), holomorphic on finitely connected domains Ω in C, whose Laurent-type partial sums approximate everything we can hope for, on compact subsets outside Ω ∪ {a ₁,…,a}, for certain prescribed points a ₁,…,a _k. In this paper we prove that, under additional assumptions, for every U.L.S. there exists a subsequence of its Laurent-type partial sums, which converges to the function itself in the whole of Ω and which approximates everything we can hope for outside Ω ∪ {a ₁,…,a, _k}.     

Schauder Estimates for the Single Layer Potential in Hydrodynamics

Let Ω be an open set in ?^N(N ? 3), with compact boundary ?Ω of type C^1,α(?(0,1)). We show that the single layer potential Ef, related to the stationary Stokes system on Ω, belongs to C^1,α(?Ω)^N, provided the source density f belongs to C^α(?Ω)^N. In addition, we prove a related estimate of the function E(f)_?Ω and its tangential derivatives.