Showing papers on "Domain (mathematical analysis) published in 1995"

Approximate controllability of the semilinear heat equation

Abstract: This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in LP(Ω) for 1 ≦ p < + ∞ is proved when the nonlinearity is globally Lipschitz with a control in L∞. In the case of the interior control, we also prove approximate controllability in C0(Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

Abstract: We study the subcritical problemsP ɛ :−Δu=u p−ɛ,u>0 onΩ;u=0 on ∂Ω,ω being a smooth and bounded domain in ℝN,N−3,p+1=2N/N−2 the critical Sobolev exponent and ɛ>0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P0).

On an elliptic equation with concave and convex nonlinearities

Abstract: We study the semilinear elliptic equation -Delta u=lambdau(q-2)u+muu(p-2)u in an open bounded domain Omega subset of R(N) with Dirichlet boundary conditions; here 1 0 and mu is an element of R arbitrary there exists a sequence (upsilon(k)) of solutions with negative energy converging to 0 as k --> infinity. Moreover, for mu > 0 and lambda arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brezis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.

On the alexandroff-bakelman-pucci estimate and the reversed hölder inequality for solutions of elliptic and parabolic equations

Abstract: The constant appearing in the classical Alexandroff-Bakelman-Pucci estimate for subsolutions of second-order uniformly elliptic equations in nondivergence form was known to depend on the diameter of the domain. Using the Krylov-Safonov boundary weak Harnack inequality due to Trudinger, we show that the dependence on the diameter may be replaced by dependence on a more precise geometric quantity of the domain. As a consequence, we get dependence on the measure instead of the diameter. We also give new bounds for subsolutions in some unbounded domains, such as domains contained in cones. We apply the Fabes and Stroock reversed Holder inequality for the Green's function to improve our estimates. We also give a new proof of the reversed Holder inequality for the Green's function based on the Krylov-Safonov Harnack inequality. Finally, we find new bounds for subsolutions of uniformly parabolic equations in cylindrical and noncylindrical domains. The constant in the (Alexandroff-Bakelman-Pucci-) Krylov-Tso estimate was known to depend on the diameter of the base of the cylinder. We get dependence either on the measure of the base or on the height of the cylinder. We also give bounds for subsolutions in noncylindrical domains. ©1995 John Wiley & Sons, Inc.

Global uniqueness for a two-dimensional semilinear elliptic inverse problem

Abstract: For a general class of nonlinear Schrodinger equations -Au+a(x, u) = 0 in a bounded planar domain £2 we show that the function a(x, u) can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary dQ .

Estimates for the Stokes Operator in Lipschitz Domains

Abstract: We study the Stokes operator A in a three- dimensional Lipschitz domain Ω. Our main result asserts that the domain of A is contained in W 1,p 0 (Ω) ∩ W 3/2,2 (Ω) for some p> 3. Certain L ∞ -estimates are also established. Our results may be used to improve the regularity of strong solutions of Navier-Stokes equations in nonsmooth domains. In the ap- pendix we provide a simple proof of area integral estimates for solutions of Stokes equations.

Exponential attractors of reaction-diffusion systems in an unbounded domain

Abstract: We consider reaction-diffusion systems in unbounded domains, prove the existence of expotential attractors for such systems, and estimate their fractal dimension. The essential difference with the case of a bounded domain studied before is the continuity of the spectrum of the linear part of the equations. This difficulty is overcome by systematic use of weighted Sobolev spaces.

Characteristic-based algorithms for solving the Maxwell equations in the time domain

Abstract: Several numerical algorithms, developed in the computational-fluid-dynamics community for solving the Euler equations, are found to be equally effective for solving the Maxwell equations in the time domain. The basic approach of these numerical procedures is to achieve the Riemann approximation to the time-dependent, three-dimensional problem in each spatial direction. The three-dimensional equations are then solved by a sequence of one-dimensional problems. This approach is referred to as a characteristic-based method. The basic algorithm can be implemented for both finite-difference and finite-volume procedures, and has the potential to eliminate the spurious-wave reflections from the numerical boundaries of the computational domain. The formulation and relative merit of the finite-difference and the finite-volume approximations are presented, together with numerical results from these procedures. >

Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions

Abstract: We study symmetry properties of positive solutions to some semilinear elliptic problems with nonlinear Neumann boundary conditions. We give sufficient conditions to have symmetry around the $\e_n$-axis of positive solutions of problems on the half-space. The proofs are based on the moving plane method. Finally some symmetry results are given in the case when the domain is a ball.

Navier-Stokes Equations and Related Nonlinear Problems

Abstract: A problem of exponential decay for Navier-Stokes equations arising in the analysis of rugosity, Y. Amirat, D. Bresch, J. Lemoine and J. Simon on the existence of solutions for non-stationary second-grade fluids, D. Bresch and J. Lemoine numerical simulation for shallow lakes - first results, D. Bresch, J. Lemoine, J. Simon and R. Echevarria semi-implicit schemes for nonlinear Schrodinger type equations, R. Ciegis and O. Stikoniene on the surface diffusion flow, J. Escher, U.F. Mayer and G. Simonett on domain functionals, A. Grigelionis optimally consistent stabilization of the Inf-Sup condition and a computation of the pressure, G. Leborgne on a time periodic problem for the Navier-Stokes equations with nonstandard boundary data, G. Lukaszewicz and M. Boukrouche orlicz spaces in the global existence problem for the multidimensional compressible Navier-Stokes equations with nonlinear viscosity, A.E. Mamontov stability and uniqueness of second grade fluids in regions with permeable boundaries, R. Maritz and N. Sauer a regularity technique for non-linear Stokes-like elliptic systems, D. Maxwell a note on the existence of solutions to stationary Boussinesq equations under general outflow condition, H. Morimoto analysis of the Navier-Stokes equations for some two-layer flows in unbounded domains, K. Pileckas and J. Socolowsky compressible Stokes flow driven by capillarity on a free surface, P.I. Plotnikov weighted dirichlet type problem for the elliptic system strongly degenerate at inner point, S. Rutkauskas the finite difference method for the equation of the sessile drop, M. Sapagovas stability properties of the Boussinesq equations, B. Scarpellini the open boundary problem for inviscid compressible fluids, P. Secchi existence, uniqueness and asymptotic behaviour of viscoelastic fluids in R3 and in R3+, A. Sequeira and J.H. Videman on the decay estimate of the Stokes semigroup in a two-dimensional exterior domain, Y. Shibata Hardy's inequality for the Stokes problem, P.E. Sobolevskii artificial boundary conditions for two-dimensional exterior Stokes problems, M. Specovius-Neugebauer global analysis of 1-D Navier-Stokes equations with density dependent viscosity, I. Straskraba finite difference method for one-dimensional equations of symmetrical motion of viscous magnetic heat-conducting gas, A, Stikonas quiet flows for the steady Navier-Stokes problem in domains with quasicylindrical outlets, G. Thater.

Linear analysis of quadrature domains

Abstract: The positive definiteness of the exponential transform of a planar domain is proved by elementary means. This direct approach avoids the heavy machinery of the theory of hyponormal operators and leads to a better understanding of the linear data associated in previous works to a quadrature domain.

Transboundary extremal length

Abstract: We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short proof is given to Koebe's conjecture in the countable case: every planar domain with countably many boundary components is conformally equivalent to a circle domain. This theorem is further generalized in two direction. We show that the same statement is true for a wide class of domains with uncountably many boundary components, in particular for domains bounded byK-quasicircles and points. Moreover, these domains admit more general uniformizations. For example, every circle domain is conformally equivalent to a domain whose complementary components are heart-shapes and points.

Complementary operators: a method to annihilate artificial reflections arising from the truncation of the computational domain in the solution of partial differential equations

Abstract: The ease and simplicity with which the finite difference time domain (FDTD) or the finite elements (FE) techniques can handle complex radiation or scattering problems have lead to a remarkable surge in the use of these methods. While execution time is becoming less of an impediment when solving large problems, the biggest constraint remains the memory needed to run the FDTD or the FE methods. It is precisely this limitation that the article addresses. A boundary operation is developed to minimize the artificial reflections that arise when truncating the computational domain of an open region scattering or radiation problem. The method is based on the use of two boundary operators that are complementary in their action. By solving the problem with each of the two operators and then averaging the two solutions, the first-order reflections that arise from the artificial boundary can be completely eliminated. Numerical results are presented to show that this new technique gives significant reduction in the error when compared to other widely used boundary conditions. >

Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains

Abstract: In this work we introduce a new parameter,s≥1, in the well known Sobolev-Gagliardo-Nirenberg (abbreviated SGN) inequalities and show their validity (with an appropriates) for any compact subanalytic domain. The classical form of these SGN inequalities (s=1 in our formulation) fails for domains with outward pointing cusps. Our parameters measures the degree of cuspidality of the domain. For regular domainss=1. We also introduce an extension, depending on a parameter σ≥1, to several variables of a local form of the classical Markov inequality on the derivatives of a polynomial in terms of its own values, and show the equivalence of Markov and SGN inequalities with the same value of parameters, σ=s. Our extension of Markov's inequality admits, in the case of supremum norms, a geometric characterization. We also establish several other characterizations: the existence of a bounded (linear) extension ofC∞ functions with a homogeneous loss of differentiability, and the validity of a global Markov inequality. Our methods may broadly be classified as follows: 1. Desingularization and anLp-version of Glaeser-type estimates. In fact we obtain a bounds<-2d+1, whered is the maximal order of vanishing of the jacobian of the desingularization map of the domain. 2. Interpolation type inequalities for norms of functions and Bernstein-Markov type inequalities for multivariate polynomials (classical analysis). 3. Geometric criteria for the validity of local Markov inequalities (local analysis of the singularities of domains). 4. Multivariate Approximation Theory.

A trust-region strategy for minimization on arbitrary domains

Abstract: We present a trust-region method for minimizing a general differentiable function restricted to an arbitrary closed set. We prove a global convergence theorem. The trust-region method defines difficult subproblems that are solvable in some particular cases. We analyze in detail the case where the domain is a Euclidean ball. For this case we present numerical experiments where we consider different Hessian approximations.

On Solvability Of Nonlocal Problems For A Second-Order Elliptic Equation

Abstract: This article is an investigation of the solvability of nonlocal problems for an elliptic equation, in which the values of the solution on the boundary of the domain under consideration are expressed in terms of its values at interior points and other points of the boundary.A new concept of solution (in the space of (n?1)-dimensionally continuous functions) is introduced, broader than concepts considered previously, and sufficient conditions are established for the problem to be Fredholm with index zero. The connection between solvability of the problem in this formulation and in the classical formulation is studied. In particular, there is a class of nonlocal problems (including some problems studied previously) that are Fredholm with index zero in the formulation introduced but not in the classical formulation (sometimes not even Fredholm). For a certain class of problems a theorem on unique solvability is proved.Bibliography: 33 titles.

Holomorphic embeddings of planar domains into C 2

Abstract: In the case p = 1 the question remains open: Given an open Riemann surface E does there exist a proper holomorphic embedding of E into C2? Trying to answer this question it is natural to begin with planar domains, i.e. open connected subsets of C. Very few results are known and even in the simplest cases the construction of such an embedding is not easy. Kasahara and Nishino [KN, St] used a technique involving the Fatou-Bieberbach map from C 2 to C 2 to prove that the unit disc can be properly holomorphically embedded into C 2. Their method can be used to prove that for each M E ,W" there is an M-connected domain in C which can be properly holomorphically embedded into C 2. Laufer [La] showed that every annulus can be properly holomorphically embedded into C 2. Alexander [AI] used elliptic modular functions to construct such an embedding of the punctured disc into C 2 which, after a slight modification, becomes a proper holomorphic embedding of the unit disc into C 2.

Strong solutions of the Navier-Stokes system in Lipschitz bounded domains

Abstract: Let Ω be a bounded domain in ℝ3 with a connected Lipschitz boundary ∂Ω. Consider the Navier-Stokes system.

The diameter of the first nodal line of a convex domains

Abstract: The main goal of this paper is to prove that the first nodal line for the Dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain. More precisely, we locate the nodal line, to within a distance comparable to the inradius, near the zero of an ordinary differential equation, which is associated to the domain in a natural way. We also derive estimates for the first and second eigenvalues in terms of the corresponding eigenvalues of the ordinary differential equation and construct an approximate first eigenfunction. Two examples, a rectangle and a circular sector, illustrate the two extreme possibilities for the location of the nodal line. For the rectangle R = { (x, y) o 1, we have the second eigenfunction

Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain

Abstract: In a domain the elliptic system is considered with a Neumann boundary condition. denotes the set of solutions of this system defined for , equal to for , and bounded in uniformly for . In the space of initial data there arises the semigroup , , wherein to the point there is assigned the set , i.e., is a multivalued mapping. In the paper it is proved that has a global attractor . A theorem is proved that where is the set of solutions of the elliptic system, defined and bounded for .

High-energy and multi-peaked solutions for a nonlinear neumann problem with critical exponents

Abstract: We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem −Δu + λu = up in antipodal invariant domains including ball domains with Neumann boundary conditions. Here p is the critical Sobolev exponent (N + 2)/(N − 2). The shape of the solutions and the location of the peaks are also studied.

Positive solutions of semilinear elliptic boundary value problems

Abstract: The paper is concerned with existence questions for positive solutions (ground states) of boundary value problems for semilinear elliptic partial differential equations. Global continuation and bifurcation results are used to obtain the existence of unbounded solution continua whenever the nonlinear terms depend upon a real parameter. Results are presented for various classes of nonlinear terms which are classified depending on their asymptotic growth, such as linear, superlinear, subcritical, and supercritical growth. Results describing the influence of the geometry, topology and dimension of the domain on the solution structure are also discussed.

Multilevel preconditioning — Appending boundary conditions by Lagrange multipliers

Abstract: For saddle point problems stemming from appending essential boundary conditions in connection with Galerkin methods for elliptic boundary value problems, a class of multilevel preconditioners is developed. The estimates are based on the characterization of Sobolev spaces on the underlying domain and its boundary in terms of weighted sequence norms relative to corresponding multilevel expansions. The results indicate how the various ingredients of a typical multilevel framework affect the growth rate of the condition numbers. In particular, it is shown how to realize even condition numbers that are uniformly bounded independently of the discretization. These investigations are motivated by the idea of employing nested refinable shift-invariant spaces as trial spaces covering various types of wavelets that are of advantage for the solution of boundary value problems from other points of view. Instead of incorporating the boundary conditions into the approximation spaces in the Galerkin formulation, they are appended by means of Lagrange multipliers leading to a saddle point problem.

Conditional functional equations and orthogonal additivity

Abstract: Some examples of classes of conditional equations coming from information theory, geometry and from the social and behavioral sciences are presented. Then the classical case of the Cauchy equation on a restricted domain Ω is extensively discussed. Some results concerning the extension of local homomorphisms and the implication “Ω-additivity implies global additivity” are illustrated. Problems concerning the equations

On the Existence of Periodic Weak Solutions on the Navier-Stokes Equations in Exterior Regions with Periodically Moving Boundaries

Abstract: In this paper we consider the existence of periodic weak solutions for the Navier-Stokes equations in an exterior domain with periodically moving boundaries. To be more precise we consider a domain $$ {{\Omega }_{T}} = \mathop{ \cup }\limits_{{0 \leqslant t \leqslant T}} \Omega (t) \times \{ t\} $$ where each Ω(t) is the exterior of a bounded connected domain Ω c (t) in R 3, T is a finite number, and Ω(0) = Ω(T).