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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 1399 (16 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 435 (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 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 211 (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....
The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature.
 Comm. Pure Appli. Math.
, 1980
"... The purpose of this paper is to study minimal surfaces in threedimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian threemanifold N, then the condition that M be stable is expressed analytically by the requirement that o n a ..."
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Cited by 192 (1 self)
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The purpose of this paper is to study minimal surfaces in threedimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian threemanifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+(AI* be positive. Here Ric (v) is the Ricci curvature of N in the normal direction to M and (A)' is the square of the length of the second fundamental form of M. In the case that N is the flat R3, we prove that any complcte stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true provided the image of the Gauss map of M omits an open set on the sphere. The relationship of the stable regions on M to the area of their Gaussian image has been studied by Barbosa and do Carmo [l] (cf. Remark 5 ) . The methods of SchoenSimonYau [ 113 give a proof of this result provided the area growth of a geodesic ball of radius r in M is not larger than r6. An interesting feature of our theorem is that it does not assume that M is of finite type topologically, or that the area growth of M is suitably small. The theorem for R3 is a special case of a classification theorem which we prove for stable surfaces in threedimensional manifolds N having scalar curvature SZO. We use an observation of SchoenYau [8] to rearrange the stability operator so that S comes into play (see formula (12)). Using this, and the study of certain differential operators on the disc (Theorem 2), we are
Variational theory for the total scalar curvature functional for Riemannian metrics and related topics
 in Topics in Calculus of Variations (Montecatini
, 1987
"... The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We ..."
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Cited by 177 (2 self)
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The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss variational characterizations of constant curvalure m trics, and give a proof of 0bata's uniqueness theorem. Much of the material in this section can be found in papers of Berger Ebin [3], FischerMarsden [8], N. Koiso [14], and also in the recent book by A. Besse [4] where the reader will find additional references. In §2 we give a general discussion of the Yamabe problem and its resolution. We also give a detailed analysis of the solutions of the Yamabe equation for the product conformal structure on SI(T) x S~1(1), a circle of radius T crossed with a sphere of radius one. These exhibit interesting variational fea,tures uch a.s symmetry breaking and the existence of solutions with high Morse index. Since the time of the summer school in Montecatini, the beautiful survey paper of J. Lee and T. Parker [15] has appeared. This gives a detailed discussion of the
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 145 (9 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.
Regularity of the obstacle problem for a fractional power of the Laplace operator
 Comm. Pure Appl. Math
"... Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α & ..."
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Cited by 135 (4 self)
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Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in Rn, • (−)su ≥ 0 in Rn, • (−)su(x) = 0 for those x such that u(x)> ϕ(x), • limx →+ ∞ u(x) = 0. We show that when ϕ is C1,s or smoother, the solution u is in the space C1,α for every α < s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C1,s. When ϕ is only C1,β for a β < s, we prove
Proof of the Riemannian Penrose inequality using the positive mass theorem
 MR MR1908823 (2004j:53046) MATHEMATICAL GENERAL RELATIVITY 73
"... We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature which contain minimal sphe ..."
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Cited by 119 (14 self)
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We prove the Riemannian Penrose Conjecture, an important case of a conjecture [41] made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3manifold as a totally geodesic submanifold of a spacetime in the context of general relativity, then outermost minimal spheres with total area A correspond to apparent horizons of black holes contributing a mass A/16π, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass (also called the ADM mass). The Riemannian Penrose Conjecture then states that the total mass of an asymptotically flat 3manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3metric to a Schwarzschild 3metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the Positive Mass Theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black hole in a Schwarzschild metric, the Riemannian Penrose Conjecture follows. We also refer the reader to the beautiful work of Huisken and Ilmanen [30], who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole. In Sections 1 and 2, we motivate the problem, discuss important quantities like total mass and horizons of black holes, and state the