Showing papers on "Domain (mathematical analysis) published in 1997"

Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain

Abstract: The exact boundary controllability of linear and nonlinear Korteweg-de Vries equation on bounded domains with various boundary conditions is studied. When boundary conditions bear on spatial derivatives up to order 2, the exact controllability result by Russell-Zhang is directly proved by means of Hilbert Uniqueness Method. When only the first spatial derivative at the right endpoint is assumed to be controlled, a quite different analysis shows that exact controllability holds too. From this last result we derive the exact boundary controllability for nonlinear KdV equation on bounded domains, for sufficiently small initial and final states.

Mathematical theory of medial axis transform

Abstract: The medial axis of a plane domain is dened to be the set of the centers of the maximal inscribed disks. It is essentially the cut loci of the inward unit normal bundle of the boundary. We prove that if a plane domain has nite number of boundary curves each of which consists of nite number of real analytic pieces, then the medial axis is a connected geometric graph in R 2 with nitely many vertices and edges. And each edge is a real analytic curve which can be extended in the C 1 manner at the end vertices. We clarify the relation between the vertex degree and the local geometry of the domain. We also analyze various continuity and regularity results in detail, and show that the medial axis is a strong deformation retract of the domain which means in the practical sense that it retains all the topological informations of the domain. We also obtain parallel results for the medial axis transform.

Fine Properties of Functions with Bounded Deformation

Abstract: The paper is concerned with the fine properties of functions in , the space of functions with bounded deformation. We analyse the set of Lebesgue points and the set where these functions have one-sided approximate limits. Moreover, following the analogy with , we decompose the symmetric distributional derivative into an absolutely continuous part , a jump part , and a Cantor part . The main result of the paper is a structure theorem for functions, showing that these parts of the derivative can be recovered from the corresponding ones of the one-dimensional sections. Moreover, we prove that functions are approximately differentiable in almost every point of their domain.

Positive solutions for the p-Laplacian: application of the fibrering method

Abstract: Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.

A Strange Term Coming from Nowhere

Abstract: Let Ω be a bounded open set in ℝ N and let us perforate it by holes: we obtain an open set Ωe. Consider the Dirichlet problem in the domain Ωe. The general questions with which we are concerned are the following. Do the solutions u e converge to a limit u when the parameter e tends to zero? If this limit exists, can it be characterized?

The complex Ginzburg - Landau equation on large and unbounded domains: sharper bounds and attractors

Abstract: Using weighted -norms we derive new bounds on the long-time behaviour of the solutions improving on the known results of the polynomial growth with respect to the instability parameter. These estimates are valid for quite arbitrary, possibly unbounded domains. We establish precise estimates on the maximal influence of the boundaries on the dynamics in the interior. For instance, the attractor for the domain with periodic boundary conditions is upper semicontinuous to .

C1,? domains and unique continuation at the boundary

Abstract: It is shown that the square of a nonconstant harmonic function u that either vanishes continuously on an open subset V contained in the boundary of a Dini domain or whose normal derivative vanishes on an open subset V in the boundary of a C1,1 domain in ℝd satisfies the doubling property with respect to balls centered at points Q ∈ V. Under any of the above conditions, the module of the gradient of u is a B2(dσ)-weight when restricted to V, and the Hausdorff dimension of the set of points {Q ∈ V : ∇u(Q) = 0} is less than or equal to d−2. These results are generalized to solutions to elliptic operators with Lipschitz second-order coefficients and bounded coefficients in the lower-order terms. © 1997 John Wiley & Sons, Inc.

Holomorphic functional calculi of operators, quadratic estimates and interpolation

Abstract: We develop some connections between interpolation theory and the theory of bounded holomorphic functional calculi of operators in Hilbert spaces, via quadratic estimates. In particular we show that an operator T of type ! has a bounded holomorphic functional calculus if and only if the Hilbert space is the complex interpolation space midway between the completion of its domain and of its range. We also characterise the complex interpolation spaces of the domains of all the fractional powers of T , whether or not T has a bounded functional calculus. This treatment extends earlier ones for self{adjoint and maximal accretive operators. This work is motivated by the study of rst order elliptic systems which are related to the square root problem for non{degenerate second order operators under boundary conditions on an interval. See our subsequent paper [AMcN].

First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains

Abstract: Given a complete Riemannian manifoldM (or a regionU inR N ) and two second-order elliptic operators L1, L2 in M (resp.U, conditions, mainly in terms of proximity near infinity (resp. near ∂U) between these operators, are found which imply that their Green’s functions are equivalent in size. For the case of a complete manifold with a given reference pointO the conditions are as follows:L 1 andL 2 are weakly coercive and locally well-behaved, there is an integrable and nonincreasing positive function Ф on [0, ∞[ such that the “distance” (to be defined) betweenL 1 andL 2 in each ballB(x, 1 ) ⊂M is less than Ф(d(x, O)). At the same time a continuity property of the bottom of the spectrum of such elliptic operators is proved. Generalizations are discussed. Applications to the domain case lead to Dini-type criteria for Lipschitz domains (or, more generally, Holder-type domains).

The Differentiability of the Drag with Respect to the Variations of a Lipschitz Domain in a Navier--Stokes Flow

Abstract: This paper is concerned with the computation of the drag $T$ associated with a body traveling at uniform velocity in a fluid governed by the stationary Navier--Stokes equations. It is assumed that the fluid fills a domain of the form $\Omega+u$, where $\Omega\subset\reels^3$ is a reference domain and $u$ is a displacement field. We assume only that $\Omega$ is a Lipschitz domain and that $u$ is Lipschitz-continuous. We prove that, at least when the velocity of the body is sufficiently small, $u\mapsto T(\Omega+u)$ is a $C^{\infty}$ mapping (in a ball centered at $0$). We also compute the derivative at $0$.

The Bergman kernel function: explicit formulas and zeroes

Abstract: We show how to compute the Bergman kernel functions of some special domains in a simple way. As an application of the explicit formulas, we show that the Bergman kernel functions of some convex domains, for instance the domain in C^3 defined by the inequality |z_1|+|z_2|+|z_3|<1, have zeroes.

A fast and accurate imaging algorithm in optical/diffusion tomography

Abstract: An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.

Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation

Abstract: In this paper the linear gravimetric boundary-value problem is discussed in the sense of the so-called weak solution. For this purpose a Sobolev weight space was constructed for an unbounded domain representing the exterior of the Earth and quantitative estimates were deduced for the trace theorem and equivalent norms. In the generalized formulation of the problem a special decomposition of the Laplace operator was used to express the oblique derivative in the boundary condition which has to be met by the solution. The relation to the classical formulation was also shown. The main result concerns the coerciveness (ellipticity) of a bilinear form associated with the problem under consideration. The Lax-Milgram theorem was used to decide about the existence, uniqueness and stability of the weak solution of the problem. Finally, a clear geometrical interpretation was found for a constant in the coerciveness inequality, and the convergence of approximation solutions constructed by means of the Galerkin method was proved.

Harmonic mappings of multiply connected domains

Abstract: In this paper the theorem of Rad o-Kneser-Choquet is extended in two dierent ways to multiply connected domains. One is a direct continuation of Kneser’s idea and has nothing to do with convexity; while the other asserts that a nitely connected domain can be mapped harmonically with prescribed outer boundary correspondence onto a given convex domain with suitable punctures. It is also shown that a domain containing innity admits a unique harmonic mapping, with standard normalization at innity, onto a punctured plane. For domains of connectivity n the dilatation of the canonical mapping covers the unit disk exactly 2n times. Furthermore, no other normalized harmonic mapping has the same dilatation. In 1926, T. Rad o[ 22] posed the problem to show that for any homeomorphism of the unit circle onto the boundary @ of a bounded convex domain ; the harmonic extension f maps the unit disk D univalently onto : In response, H. Kneser [14] supplied an elegant proof. Some 20 years later G. Choquet [4], apparently unaware that the theorem was known, rediscovered it and gave another proof. Fortunately, the two proofs are dierent and even for simply connected domains they have dierent generalizations. The dichotomy between the two approaches of Kneser and Choquet comes into sharper focus as the theorem is generalized to multiply connected domains. In presenting these generalizations, it will be expedient to distinguish between \Kneser’s theorem" and \Choquet’s theorem". Kneser’s proof has little to do with convexity, while Choquet’s proof uses convexity in a more essential way. Indeed, Kneser’s proof applies (as he indicates in [14]) when is not convex, under the additional hypothesis that f(D) : We shall see that the main idea of his proof carries over to multiply connected domains. On other hand, by methods more akin to Choquet’s proof we will show that a nitely connected domain D can be mapped harmonically, with prescribed boundary values, onto a given convex domain with punctures at suitable points. Another result is that D can be mapped harmonically onto a punctured plane, and such a mapping is unique up to a normalization. Our proofs adapt an idea of Clunie and Sheil-Small [5], which gives yet another proof of the Rad o-Kneser-Choquet theorem.

Asymptotic analysis of the Navier-Stokes equations in the domains

Abstract: We are interested in this article with the Navier–Stokes equations of viscous incompressible fluids in three dimensional thin domains Let Ωe be the thin domain Ωe = ω × (0, e), where ω is a suitable domain in R and 0 < e < 1 Our aim is to derive an asymptotic expansion of the strong solution u of the Navier–Stokes equations in the thin domain Ωe when e is small, which is valid uniformly in time This study should give a better understanding of the global existence results in thin domains obtained previously; see [15]–[17] and [23], [22] We consider in this work two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition For the first type of boundary condition we derive an asymptotic expansion of the solution u in terms of the solution of the associated Stokes problem More precisely, we prove that the solution can be written, for e small, as

An adaptive collocation method based on interpolating wavelets

Abstract: A wavelet collocation method for the adaptive solution of second order elliptic partial differential equations in dimension $d$ is presented. The method is based of the use of the Deslaurier-Dubuc interpolating functions. The method is tested on an advection dominated advection diffusion problem, and on a Laplace problem posed on a non rectangular domain. EMAIL:: ian@microian.ian.pv.cnr.it

The Thermal Equilibrium Solution of a Generic Bipolar Quantum Hydrodynamic Model

Abstract: The thermal equilibrium state of a bipolar, isothermic quantum fluid confined to a bounded domain ,d = 1,2 or d = 3 is entirely described by the particle densities n, p, minimizing the energy where G 1,2 are strictly convex real valued functions, . It is shown that this variational problem has a unique minimizer in and some regularity results are proven. The semi-classical limit is carried out recovering the minimizer of the limiting functional. The subsequent zero space charge limit leads to extensions of the classical boundary conditions. Due to the lack of regularity the asymptotics can not be settled on Sobolev embedding arguments. The limit is carried out by means of a compactness-by-convexity principle.

Smooth approximation in weighted Sobolev spaces

Abstract: We give necessary and sufficient conditions for the equality $H=W$ in weighted Sobolev spaces. We also establish a Rellich-Kondrachov compactness theorem as well as a Lusin type approximation by Lipschitz functions in weighted Sobolev spaces.

Abstract Evolution Equations

Abstract: In this chapter we reconsider partial differential equations in which there is a distinguished variable t, usually time in physical problems We might think of such equations as ordinary differential equations in Banach spaces Consider, for example, the heat equation in a bounded domain Ω If we let A = Δ, then the equation can be written $$ \frac{{du}}{{dt}} = Au, $$ (E) where u(t) is a vector-valued function (Chapter 8) As we will see, if the temperature u vanishes on the boundary of Ω, one reasonable choice of the “state space” is \(H_0^1\)(Ω)

The resolution of the Gibbs phenomenon for "spliced" functions in one and two dimensions

Abstract: In this paper we study approximation methods for analytic functions that have been “spliced” into nonintersecting subdomains. We assume that we are given the first 2N + 1 Fourier coefficients for the functions in each subdomain. The objective is to approximate the “spliced” function in each subdomain and then to “glue” the approximations together in order to recover the original function in the full domain. The Fourier partial sum approximation in each subdomain yields poor results, as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain. Thus once we “glue” the subdomain approximations back together, the approximation for the function in the full domain will exhibit oscillations throughout the entire domain. Recently methods have been developed that successfully eliminate the Gibbs phenomenon for analytic but nonperiodic functions in one dimension. These methods are based on the knowledge of the first 2N + 1 Fourier coefficients and use either the Gegenbauer polynomials (Gottlieb et al.) or the Bernoulli polynomials (Abarbanel, Gottlieb, Cai et al., and Eckhoff). We propose a way to accurately reconstruct a “spliced” function in a full domain by extending the current methods to eliminate the Gibbs phenomenon in each nonintersecting subdomain and then “gluing” the approximations back together. We solve this problem in both one and two dimensions. In the one-dimensional case we provide two alternative options, the Bernoulli method and the Gegenbauer method, as well as a new hybrid method, the Gegenbauer-Bernoulli method. In the two-dimensional case we prove, for the very first time, exponential convergence of the Gegenbauer method, and then we apply it to solve the “spliced” function problem.

Weakly Smooth Nonselfadjoint Spectral Elliptic Boundary Problems

Abstract: The paper is devoted to general elliptic boundary problems (A− λ)u = f in G, Bju = 0 (j = 1, . . . ,m) on Γ = ∂G, (1) generally nonselfadjoint, where G is a bounded domain in R. The main goal is to minimize, to some extent, the smoothness assumptions under which the known spectral results are true. The main results concern the asymptotics of the trace of R(λ) with q > n/2m, where R(λ) is the resolvent, in an angle of ellipticity with parameter. For example, for the Dirichlet problem these asymptotics are obtained in the case of bounded and measurable coefficients in A and continuous coefficients in the principal part of A, while the boundary is assumed to belong to C2m−1,1. The asymptotics of the moduli of the eigenvalues are investigated. The last section is devoted to indefinite spectral problems, with a real-valued multiplier ω(x) before λ changing the sign. The references contain 44 items. 1991 Mathematics Subject Classification. 35Pxx, 35J40, 47F05, 58G03.

Towards Computing Distances Between Programs via Scott Domains

Abstract: This paper introduces an approach to defining and computing distances between programs via continuous generalized distance functions ρ: A×A→D, where A and D are directed complete partial orders with the induced Scott topology, A is a semantic domain, and D is a domain representing distances (usually, some version of interval numbers). A continuous distance function ρ can define a To topology on a nontrivial domain A only if the axiom ∃0 e D.∀x e A.ρ(x,x)=0 does not hold. Hence, the notion of relaxed metric is introduced for domains — the axiom ρ(x,x)=0 is eliminated, but the axiom ρ(x,y)=ρ(y,x) and a version of the triangle inequality tailored for the domain D remain.