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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 629 (9 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 262 (5 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.
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 65 (4 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.
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 62 (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.
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 61 (14 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...
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 54 (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....
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 boundary spike ..."
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Cited by 50 (38 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.
Mixed finite element methods on nonmatching multiblock grids
 SIAM J. Numer. Anal
, 2000
"... Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux ..."
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Cited by 48 (25 self)
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Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.
A Finite Element Method for Surface Restoration with Smooth Boundary Conditions
, 2004
"... In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundar ..."
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Cited by 46 (5 self)
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In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as system of two coupled second order equations. As it is wellknown, fourth order problems require an implicit time discretization.
Twisted connected sums and special Riemannian holonomy
 J. Reine Angew. Math
"... Abstract. We give a new, connected sum construction of Riemannian metrics with special holonomy G2 on compact 7manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian ma ..."
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Cited by 46 (5 self)
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Abstract. We give a new, connected sum construction of Riemannian metrics with special holonomy G2 on compact 7manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7manifolds with holonomy G2 are constructed using Fano 3folds. The purpose of this paper is to give a new construction of compact 7dimensional Riemannian manifolds with holonomy group G2. The holonomy group of a Riemannian manifold is the group of isometries of a tangent space generated by parallel transport using the Levi–Civita connection over closed paths based at a point. For an oriented ndimensional manifold the holonomy group may be identified as a subgroup of SO(n). If there is a structure on a manifold defined by a tensor field and parallel with respect to the Levi–Civita connection then the holonomy may be a proper subgroup of SO(n) (it is just the subgroup leaving invariant the corresponding tensor on R n). There is essentially just one possibility for such holonomy reduction in odd dimensions, as follows from the wellknown Berger