Showing papers on "Domain (mathematical analysis) published in 2007"

The ODE/IM correspondence

Abstract: This paper reviews a recently discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

Controllability and Observability of Partial Differential Equations: Some Results and Open Problems

Abstract: In this chapter we present some of the recent progresses done on the problem of controllability of partial differential equations (PDE). Control problems for PDE arise in many different contexts and ways. A prototypical problem is that of controllability. Roughly speaking it consists in analyzing whether the solution of the PDE can be driven to a given final target by means of a control applied on the boundary or on a subdomain of the domain in which the equation evolves. In an appropriate functional setting this problem is equivalent to that of observability which concerns the possibility of recovering full estimates on the solutions of the uncontrolled adjoint system in terms of partial measurements done on the control region. Observability/controllability properties depend in a very sensitive way on the class of PDE under consideration. In particular, heat and wave equations behave in a significantly different way, because of their different behavior with respect to time reversal. In this paper we first recall the known basic controllability properties of the wave and heat equations emphasizing how their different nature affects their main controllability properties. We also recall the main tools to analyze these problems: the so-called Hilbert uniqueness method (HUM), multipliers, microlocal analysis and Carleman inequalities. We then discuss some more recent developments concerning equations with low regularity coefficients, equations with potentials, bang-bang controls, etc. We also analyze the way control and observability properties depend on the norm and regularity of these coefficients, a problem which is also relevant when addressing nonlinear models. We then present some recent results on coupled models of wave–heat equations arising in fluid–structure interaction. We also present some open problems and future directions of research.

Limit shapes and the complex Burgers equation

Abstract: In this paper we study surfaces in R 3 that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a functional of the form ∫σ(∇h) dx dy among all Lipschitz functions h taking given values on the boundary of the domain. The surface tension σ has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes. We find a change of variables that reduces the Euler–Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.

An open string analogue of Viterbo functoriality

Abstract: Liouville domains are a special type of symplectic manifolds with boundary (they have an everywhere defined Liouville flow, pointing outwards along the boundary). Symplectic cohomology for Liouville domains was introduced by Cieliebak-Floer-Hofer-Wysocki and Vitero. The latter constructed a restriction (or transfer) map associated to an embedding of one Liouville domain into another. In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called "wrapped Floer cohomology". We construct an A_\infty-structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A_\infty-homomorphism realizing the restriction to a Liouville subdomain. The construction of the A_\infty-structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations.

Fractional Laplacian in bounded domains.

Abstract: The fractional Laplacian operator -(-delta)(alpha/2) appears in a wide class of physical systems, including Levy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.

On uniqueness in the inverse conductivity problem with local data

Abstract: We show that the Dirichlet-to-Neumann map given on an arbitrary part of the boundary of a three-dimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on unaccessible part of the boundary. Proofs use some modification of the method of complex geometrical solutions due to Calderon-Sylvester-Uhlmann.

Finite elements for symmetric tensors in three dimension

Abstract: We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

hp-Version a priori Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

Abstract: We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in $$\mathbb{R}^d, d \geqslant 2$$ . For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L2 norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson---Kirchhoff thin plate theory are presented.

Distributions, Sobolev Spaces, Elliptic Equations

Abstract: The Laplace-Poisson equation -- Distributions -- Sobolev spaces on R-n and R-n+ -- Sobolev spaces on domains -- Elliptic operators in L-2 -- Spectral theory in Hilbert spaces and Banach spaces -- Compact embeddings, spectral theory of elliptic operators -- Domains, basic spaces, and integral formulae -- Orthonormal bases of trigonometric functions -- Operator theory -- Some integral inequalities -- Function spaces.

Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains

Abstract: In the first part of this article we give intrinsic characterizations of the classes of Lipschitz and C1 domains. Under some mild, necessary, background hypotheses (of topological and geometric measure theoretic nature), we show that a domain is Lipschitz if and only if it has a continuous transversal vector field. We also show that if the geometric measure theoretic unit normal of the domain is continuous, then the domain in question is of class C1. In the second part of the article, we study the invariance of various classes of domains of locally finite perimeter under bi-Lipschitz and C1 diffeomorphisms of the Euclidean space. In particular, we prove that the class of bounded regular SKT domains (previously called chord-arc domains with vanishing constant, in the literature) is stable under C1 diffeomorphisms. A number of other applications are also presented.

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Abstract: We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

Abstract: We study uniformly elliptic fully nonlinear equations $$ F(D^2u, Du, u, x)=0, $$ and prove results of Gidas--Ni--Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

Null controllability of degenerate parabolic operators with drift

Abstract: We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.

Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions

Abstract: A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

A note on confined diffusion

Abstract: The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion operator. Because the latter depend on space dimensionality and on the particular shape of the domain, an analytical expression is in most circumstances not available. In this article, it is shown that the series may in some circumstances sum up exactly. Explicit calculations are performed for 2D diffusion restricted to a circular domain and 3D diffusion inside a sphere. In both cases, the short-time behaviour of the mean square displacement is obtained.

Analysis of a Spectral-Galerkin Approximation to the Helmholtz Equation in Exterior Domains

Abstract: An error analysis is presented for the spectral-Galerkin method to the Helmholtz equation in 2- and 3-dimensional exterior domains. The problem in unbounded domains is first reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator, and then a spectral-Galerkin method is employed to approximate the reduced problem. The error analysis is based on exploring delicate asymptotic behaviors of the Hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems. Explicit error bounds with respect to the wave number are derived, and some illustrative numerical examples are also presented.

Boundary behaviour for p-harmonic functions in Lipschitz and starlike Lipschitz ring domains

Abstract: In this paper we prove new results for p harmonic functions, p ≠ 2 , 1 p ∞ , in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p , n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Holder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p ≠ 2 , 1 p ∞ , of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p = 2 ).

Quantum Algorithms for Learning and Testing Juntas

Abstract: In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.

Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold

Abstract: For any bounded regular domain $\Omega$ of a real analytic Riemannian manifold $M$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $\Omega$. In this paper, we consider $\lambda_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $\lambda_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.

On a generalized resolvent estimate for the Stokes system with Robin boundary condition

Abstract: We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L q framework ( 1 ν = h on the boundary of the domain with α , β ≥ 0 and α + β = 1 , where u denotes a velocity vector, p a pressure, T ( u , p ) the stress tensor for the Stokes flow, and ν the unit outer normal to the boundary of the domain It presents the slip condition when β = 1 and the non-slip one when α = 1 , respectively

Jump problem and removable singularities for monogenic functions

Abstract: In this article the jump problem for monogenic functions (Clifford holomorphicity) on the boundary of a Jordan domain in Euclidean spaces is investigated. We shall establish some criteria that imply the uniqueness of the solution in terms of a natural analogue of removable singularities in the plane to ℝn+1 (n ≥ 2). Sufficient conditions to extend monogenically continuous Clifford algebra valued functions across a hypersurface are proved.

A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems

Abstract: We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N 2m$. We prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if $u$ is a classical, bounded, non-negative solution of $(-\Delta)^m u = u^q$ in a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically.