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259
Infinitedimensional linear systems with unbounded control and observation: A functional analytic approach
 Transactions of the American Mathematical Society
, 1987
"... ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and ..."
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Cited by 59 (1 self)
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ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. 1. Introduction. For
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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A leastsquares finite element method for the NavierStokes equations
 Appl. Math. Lett
, 1993
"... Abstract. In this paper we study finite element methods of leastsquares type for the stationary, incompressible NavierStokes equations in 2 and 3 dimensions. We consider methods based on velocityvorticitypressure form of the NavierStokes equations augmented with several nonstandard boundary con ..."
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Cited by 46 (16 self)
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Abstract. In this paper we study finite element methods of leastsquares type for the stationary, incompressible NavierStokes equations in 2 and 3 dimensions. We consider methods based on velocityvorticitypressure form of the NavierStokes equations augmented with several nonstandard boundary conditions. Leastsquares minimization principles for these boundary value problems are developed with the aid of AgmonDouglisNirenberg elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations, and analysis of some nonstandard boundary conditions. Results of several computational experiments with leastsquares methods which illustrate, among other things, the optimal convergence rates are also reported. Key words. NavierStokes equations, leastsquares principle, finite element methods, velocityvorticitypressure equations. AMS subject classifications. 76D05, 76D07, 65F10, 65F30 1. Introduction. In
Fredholm operators and Einstein metrics on conformally compact manifolds
"... Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, t ..."
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Cited by 42 (2 self)
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Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
, 1996
"... this paper we will derive a 3G type theorem as in (1) but with G 1;n replaced by the Green function G m;n for the mpolyharmonic operator with Dirichlet boundary conditions and with ..."
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Cited by 36 (11 self)
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this paper we will derive a 3G type theorem as in (1) but with G 1;n replaced by the Green function G m;n for the mpolyharmonic operator with Dirichlet boundary conditions and with
Finite element methods of leastsquares type
, 1998
"... We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticit ..."
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Cited by 31 (4 self)
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We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convectiondiffusion. For many of these problems, leastsquares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of leastsquares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the infsup condition arising in mixed methods for the Stokes and NavierStokes equations. As a result, application of leastsquares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.
Asymptotically flat initial data with prescribed regularity at infinity
 Comm. Math. Phys
, 2001
"... We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1 ..."
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Cited by 29 (8 self)
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We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1
Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients
 Arch. Rational Mech. Anal
, 2000
"... In this paper we derive global W 1, ∞ and piecewise C 1,α estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the ..."
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Cited by 28 (4 self)
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In this paper we derive global W 1, ∞ and piecewise C 1,α estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces. 1.
Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions
 MATHEMATISCHE NACHRICHTEN
, 1996
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Solutions of the Einstein constraint equations with apparent horizon boundaries Comm
 Math. Phys
, 2005
"... Abstract. We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the ..."
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Cited by 24 (2 self)
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Abstract. We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method introduced by Isenberg for compact manifolds without boundary,suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary. 1.