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## Solution U (solution + u)
## Selected Abstracts## A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2009M. PicassoAbstract We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u (the superscript n stands for the CG iteration number, the subscript h for the typical mesh size) and on heuristics to relate the error between uh and u to the residual vector. Numerical experiments with anisotropic adaptive meshes show that the total number of CG iterations can be divided by 10 without significant discrepancy in the computed results. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Estimates of hyperbolic equations in Hardy spaces MATHEMATISCHE NACHRICHTEN, Issue 1 2003Chen ChangAbstract The aim of this paper is to study estimates of hyperbolic equations in Hardy classes. Consider the Cauchy problem P(Dt,Dx)u(t, x) = 0 for x , ,d and t > 0 with the initial conditions Djtu(0, x) = gj (x), j = 0, 1, ,, m , 1. We assume that the symbol ,,(,, ,) of P(Dt,Dx) can be factorized as ,,(,, ,) = (,,,j(,)) where ,j (,) = , j = 1, ,, m. We assume further that gj , Hpk (,d) for j = 1, ,, m , 1. Then the solution u of the problem (3.13) is in Hp(,d) provided k , (d, 1) and < p < ,. Here n = max{n1, ,, nm}. In particular, P(Dt, Dx)u = , ,u = 0 with u(0, x) = f(x) and (0, x) = g(x), then the solution u of the wave equation is in Hp(,d) provided k , (d , 1) and 0 < p < ,. [source] ## Comparison results for nonlinear elliptic equations with lower,order terms MATHEMATISCHE NACHRICHTEN, Issue 1 2003Vincenzo FeroneAbstract We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray,Lions operator defined on and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric. [source] ## Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2008Tobias H. ColdingWe show that for a Schrödinger operator with bounded potential on a manifold with cylindrical ends, the space of solutions that grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently, for a surface for a fixed potential and a dense set of metrics), the constant function 0 is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution u to a Schrödinger equation on a product N × [0, T], where N is a closed manifold with a certain spectral gap. Examples of such N's are all (round) spheres ,,n for n , 1 and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrödinger operators.© 2007 Wiley Periodicals, Inc. [source] ## Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2008Elena KosyginaWe consider a family {u, (t, x, ,)}, , < 0, of solutions to the equation ,u,/,t + ,,u,/2 + H (t/,, x/,, ,u,, ,) = 0 with the terminal data u,(T, x, ,) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ,) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u,(t, x, ,) as , , 0 to the solution u(t, x) of a deterministic averaged equation ,u/,t + H,(,u) = 0, u(T, x) = U(x). The "effective" Hamiltonian H, is given by a variational formula. © 2007 Wiley Periodicals, Inc. [source] ## Regularity of the obstacle problem for a fractional power of the laplace operator COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2007Luis SilvestreGiven a function , and s , (0, 1), we will study the solutions of the following obstacle problem: u , , in ,n, (,,)su , 0 in ,n, (,,)su(x) = 0 for those x such that u(x) > ,(x), lim|x| , + ,u(x) = 0. We show that when , is C1, s or smoother, the solution u is in the space C1, , for every , < s. In the case where the contact set {u = ,} is convex, we prove the optimal regularity result u , C1, s. When , is only C1, , for a , < s, we prove that our solution u is C1, , for every , < ,. © 2006 Wiley Periodicals, Inc. [source] ## Concentration on curves for nonlinear Schrödinger Equations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2007Manuel Del PinoWe consider the problem where p > 1, , > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let , be a closed curve, nondegenerate geodesic relative to the weighted arc length ,,V,, where , = (p + 1)/(p , 1) , 1/2. We prove the existence of a solution u, concentrating along the whole of ,, exponentially small in , at any positive distance from it, provided that , is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two-dimensional case. © 2006 Wiley Periodicals, Inc. [source] ## On the convergence rate of vanishing viscosity approximations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2004Alberto BressanGiven a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ,u(t, ·) , u,(t, ·), = O(1)(1 + t) · |ln ,| on the distance between an exact BV solution u and a viscous approximation u,, letting the viscosity coefficient , , 0. In the proof, starting from u we construct an approximation of the viscous solution u, by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ,. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc. [source] ## Continuous dependence on the geometry and on the initial time for a class of parabolic problems I MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2007L. E. PayneAbstract In this paper, we investigate the continuous dependence on the geometry and the initial time for solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Resonance phenomena in compound cylindrical waveguides MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2006Günter HeinzelmannAbstract We study the large time asymptotics of the solutions u(x,t) of the Dirichlet and the Neumann initial boundary value problem for the wave equation with time-harmonic right-hand side in domains , which are composed of a finite number of disjoint half-cylinders ,1,,,,r with cross-sections ,,1,,,,,r and a bounded part (,compound cylindrical waveguides'). We show that resonances of orders t and t1/2 may occur at a finite or countable discrete set of frequencies ,, while u(x,t) is bounded as t,, for the remaining frequencies. A resonance of order t occurs at , if and only if ,2 is an eigenvalue of the Laplacian ,, in , with regard to the given boundary condition u=0 or ,u/,n=0, respectively. A resonance of order t1/2 occurs at , if and only if (i) ,2 is an eigenvalue of at least one of the Laplacians for the cross-sections ,,1,,,,r, with regard to the respective boundary condition and (ii) the respective homogeneous boundary value problem for the reduced wave equation ,U+,2U=0 in , has non-trivial solutions with suitable asymptotic properties as | x | ,, (,standing waves'). Copyright © 2006 John Wiley & Sons, Ltd. [source] ## Time asymptotics for the polyharmonic wave equation in waveguides MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2003P. H. LeskyAbstract Let , denote an unbounded domain in ,n having the form ,=,l×D with bounded cross-section D,,n,l, and let m,, be fixed. This article considers solutions u to the scalar wave equation ,u(t,x) +(,,)mu(t,x) = f(x)e,i,t satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t,, is investigated. Depending on the choice of f ,, and ,, two cases occur: Either u shows resonance, which means that ,u(t,x),,, as t,, for almost every x , ,, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## On the regularity of flows with Ladyzhenskaya Shear-dependent viscosity and slip or non-slip boundary conditions, COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 4 2005H. Beirão da VeigaNavier-Stokes equations with shear dependent viscosity under the classical non-slip boundary condition have been introduced and studied, in the sixties, by O. A. Ladyzhenskaya and, in the case of gradient dependent viscosity, by J.-L. Lions. A particular case is the well known Smagorinsky turbulence model. This is nowadays a central subject of investigation. On the other hand, boundary conditions of slip type seems to be more realistic in some situations, in particular in numerical applications. They are a main research subject. The existence of weak solutions u to the above problems, with slip (or non-slip) type boundary conditions, is well known in many cases. However, regularity up to the boundary still presents many open questions. In what follows we present some regularity results, in the stationary case, for weak solutions to this kind of problems; see Theorems 3.1 and 3.2. The evolution problem is studied in the forthcoming paper [6]; see the remark at the end of the introduction. © 2004 Wiley Periodicals, Inc. [source] |