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USER’S GUIDE TO VISCOSITY SOLUTIONS OF SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS
, 1992
"... The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking argume ..."
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Cited by 808 (10 self)
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The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a selfcontained exposition of the basic theory of viscosity solutions.
Motion of level sets by mean curvature
 II, Trans. Amer. Math. Soc
"... We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set Γ o a unique generalized motion by mean curvature, existing for all time. We investigate the various ..."
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Cited by 322 (6 self)
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We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set Γ o a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution. 1.
The gradient theory of phase transitions and the minimal interface criterion
, 1987
"... In this paper I prove some conjectures of GURTIN [15] concerning the Van der WaalsCahnHilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed func ..."
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Cited by 171 (0 self)
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In this paper I prove some conjectures of GURTIN [15] concerning the Van der WaalsCahnHilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded container 12 Q R', whose Gibbs free energy, per unit volume, is a prescribed func
The Partition of Unity Method
 International Journal of Numerical Methods in Engineering
, 1996
"... A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partitionofu ..."
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Cited by 108 (2 self)
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A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partitionofunity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for aposteriori error estimation for this new method are also proved. Key words: Finite element method, meshless finite element method, finite element methods for highly oscillatory solutions TICAM, The University of Texas at Austin, Austin, TX 78712. Research was partially supported by US Office of Naval Research under grant N0001490J1030 y Seminar for Applied Mathematics, ETH Zurich, CH8092 Zurich, Switzerland....
A tour of the theory of absolutely minimizing functions
"... Abstract. These notes are intended to be a rather complete and selfcontained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding ..."
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Cited by 80 (6 self)
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Abstract. These notes are intended to be a rather complete and selfcontained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely selfcontained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron’s method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.
FirstOrder System Least Squares For SecondOrder Partial Differential Equations: Part I
, 1994
"... . This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to esta ..."
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Cited by 77 (22 self)
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. This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers of the discrete systems. Key words. leastsquares discretization, secondorder elliptic problems, RayleighRitz, finite elements 1. Introduction. The purpose of this paper is to analyse the leastsquares finite element method for secondorder convectiondiffusion equations written as a firstorder system. In general, the standard Galerkin finite element methods applied to nonselfadjoint elliptic equations with significant convection terms exhibit a variety of deficiencies, including oscillations or nonmonotonocity of the solution and poor approximation of its derivatives. A variety of stabilization techniques, such as upwin...
Multiple Boundary Peak Solutions For Some Singularly Perturbed Neumann Problems
"... We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses bound ..."
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Cited by 67 (51 self)
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We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as " approaches zero, at a critical point of the mean curvature function H(P ); P 2 @ It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P ) or multiple local maximum points of H(P ). In this paper, we prove that for any fixed positive integer K there exist boundary K \Gamma peak solutions at a local minimum point of H(P ). This implies that for any smooth and bounded domain there always exist boundary K \Gamma peak solutions. We first use the LiapunovSchmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes. 1.
A variational principle for domino tilings
"... Abstract. We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entrop ..."
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Cited by 66 (11 self)
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Abstract. We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within ε (for an appropriate metric) of the unique entropymaximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges. The effect of boundary conditions is, however, not entirely trivial and will be discussed in more detail in a subsequent paper. P. W. Kasteleyn, 1961 1.
Analysis of the heterogeneous multiscale method for ordinary differential equations
 Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 56 (7 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
 MATH. COMP
, 2002
"... The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly h ..."
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Cited by 56 (10 self)
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The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an oversampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random lognormal relative permeability to demonstrate the efficiency and accuracy of the proposed method.