Publications (107)89.06 Total impact
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ABSTRACT: In this article it is shown that the three conditions on the norm · of a Banach space called “geometric convexity”, “balanced” and “doubling” in an earlier work by the authors related to eikonal equations are in fact all equivalent. Moreover, each of them is equivalent to a condition called “Property Γ” by Ganichev and Kalton. A fifth condition, that the second derivative of the function t↦x+ty is a doubling measure on [2,2] for suitable x,y∈X, is also equivalent to the various other properties, and this formulation occupies a central place in the analysis.  [Show abstract] [Hide abstract]
ABSTRACT: We show that an absolutely minimizing function with respect to a convex Hamiltonian \({H : \mathbb{R}^{n} \rightarrow \mathbb{R}}\) is uniquely determined by its boundary values under minimal assumptions on H. Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in L ∞.  [Show abstract] [Hide abstract]
ABSTRACT: Consider a function u defined on Rn, except, perhaps, on a closed set of potential singularities S. Suppose that u solves the eikonal equation {double pipe}Du{double pipe} = 1 in the pointwise sense on Rn\S, where Du denotes the gradient of u and {double pipe}·{double pipe} is a norm on Rn with the dual norm {double pipe}·{double pipe}*. For a class of norms which includes the standard pnorms on Rn, 1 < p < ∞, we show that if S has Hausdorff 1measure zero and n ≥ 2, then u is either affine or a "cone function," that is, a function of the form u(x) = a ± {double pipe}x  z{double pipe}*. 
Article: The Problem of Two Sticks
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ABSTRACT: Let $ l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\ l_1\bar l_0\ \ge \ l_1l_0\, \ \bar l_1l_0\ \ge \ \bar l_1\bar l_0\.$ Here $\ \cdot\ $ is a norm on ${\mathbb R}^n.$ We explore the manner in which $l_1\bar l_1$ is then constrained when assumptions are made about "intermediate points" $l_* \in l$, $\bar l_* \in \bar l.$ Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to $\ l_* \bar l_*\ $ such that $l_1\bar l_1$ must lie between these planes, provided that $\ \cdot\ $ is "geometrically convex" and "balanced", as defined herein. The standard $p$norms are shown to be geometrically convex and balanced. Other results estimate $\ l_1\bar l_1 \$ in a Lipschitz or H\"older manner by $\ l_* \bar l_* \ $. All these results have implications in the theory of eikonal equations, from which this "problem of two sticks" arose. Comment: AMSLaTeX, 34 pages 
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ABSTRACT: It is proved herein that any absolute minimizer u for a suitable Hamiltonian H 2 C1(Rn ◊ R ◊ U) is a viscosity solution of the Aronsson equation: 
Chapter: A Visit with the ∞Laplace Equation
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ABSTRACT: In these notes we present an outline of the theory of the archetypal L∞ variational problem in the calculus of variations. Namely, given an open U ⊂ IRn and b ∈ C(∂U), find u ∈ C (Ū) which agrees with the boundary function b on ∂U and minimizes  [Show abstract] [Hide abstract]
ABSTRACT: David Hilbert used to say every real progress walks hand in hand with the discovery of more and more rigorous tools and simpler methods which meanwhile make easier the understanding of previous theories. Nevertheless Augustus De Morgan used to say: The mental attitude which stimulate the mathematical invention is not only a sharp reasoning but rather a deep imagination. The “progress” Hilbert was talking about is based – in mathematics more than in other scientific fields – upon teaching and collaboration and the “imagination” De Morgan was referring to, must be stimulated through a progressive and gradual learning. Both history and everybody personal experience show that mathematical learning and its improvement is not just a matter of studying books and original articles, but rather that of a continuous and effective relationships with our own teacher(s) and collegues, rising new questions and discussing together their possible answers. In the Fifties a group of outstanding Italian mathematicians, all member of the Scientific Committee of UMI (the Union of Italian Mathematicians) under the presidency of Enrico Bompiani, decided that it was the moment to rise the mathematical research in Italy to the level it was before the Second Worldly War and that it should be done through the organisation of high level courses. They realized the importance of providing the young researchers with the possibility of learning the new theories, subjects and themes which were appearing in those years and of mastering the new techniques and tools. It was right in those years that the CIME was founded and the first course was held in Varenna (a charming small city on the Como lake) in 1954. The subject was on Functional Analysis, which can be considered at that time a new subject. More precisely:  [Show abstract] [Hide abstract]
ABSTRACT: Symmetrization is one of the most powerful mathematical tools with several applications both in Analysis and Geometry. Probably the most remarkable application of Steiner symmetrization of sets is the De Giorgi proof (see [14], [25]) of the isoperimetric property of the sphere, while the spherical symmetrization of functions has several applications to PDEs and Calculus of Variations and to integral inequalities of Poincaré and Sobolev type (see for instance [23], [24], [19], [20])  [Show abstract] [Hide abstract]
ABSTRACT: Comparison results are obtained between infinity subharmonic and infinity superharmonic functions defined on unbounded domains. The primary new tool employed is an approximation of infinity subharmonic functions that allows one to assume that gradients are bounded away from zero. This approximation also demystifies the proof in the case of a bounded domain. A second, quite different, topic is also taken up. This is the uniqueness of absolutely minimizing functions with respect to other norms besides the Euclidean, norms that correspond to comparison results for partial differential equations which are quite discontinuous. 


Chapter: Viscosity solutions: A primer

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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 apriori 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.  [Show abstract] [Hide abstract]
ABSTRACT: For 1<p<∞, the equation which characterizes minima of the functional u↦∫ U Du p ,dx subject to fixed values of u on ∂U is −Δ p u=0. Here −Δ p is the wellknown ``pLaplacian''. When p=∞ the corresponding functional is u↦ Du2 L∞(U) . A new feature arises in that minima are no longer unique unless U is allowed to vary, leading to the idea of ``absolute minimizers''. Aronsson showed that then the appropriate equation is −Δ∞ u=0, that is, u is ``infinity harmonic'' as explained below. Jensen showed that infinity harmonic functions, understood in the viscosity sense, are precisely the absolute minimizers. Here we advance results of Barron, Jensen and Wang concerning more general functionals u↦f(x,u,Du) L∞(U) by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses. 
Article: Another way to say harmonic
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ABSTRACT: It is known that solutions of Δ<sub>∞</sub> u=Σ<sup>n</sup><sub>i,j=1</sub>u<sub>x<sub>i</sub></sub>u<sub>x<sub>j</sub></sub>u<sub>x<sub>i</sub> x<sub>j</sub></sub>= 0 that is, the ∞ harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions G(x) = ax. We establish a more difficult linear result: a function in R<sup>n</sup> is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions G(x) = ax<sup>2n</sup> for $n \not= 2$ and G(x) = bln(x) for n = 2. An attempt to generalize these results for  Δ<sub>∞</sub> u = 0 (p = ∞) and  Δ u = 0 (p = 2) to the general pLaplacian leads to the fascinating discovery that certain sums of translates of radial psuperharmonic functions are again psuperharmonic. Mystery remains: the class of psuperharmonic functions so constructed for p ∉ {2,∞} does not suffice to characterize psubharmonic functions.  [Show abstract] [Hide abstract]
ABSTRACT: We reconsider in this paper boundary value problems for the socalled "infinity Laplacian" PDE and the relationships with optimal Lipschitz extensions of the boundary data. We provide some fairly elegant new proofs, which clarify and simplify previous work, and in particular draw attention to the fact that solutions may be characterized by a comparison principle with appropriate cones. We in particular show how comparison with cones directly implies the variational principle associated with the equation. In addition, we establish a Liouville theorem for subsolutions bounded above by planes.  [Show abstract] [Hide abstract]
ABSTRACT: A realvalued function $u$ is said to be {it infinity harmonic} if it solves the nonlinear degenerate elliptic equation $sum_{i,j=1}^nu_{x_1}u_{x_j}u_{x_ix_j}=0$ in the viscosity sense. This is equivalent to the requirement that $u$ enjoys comparison with cones, an elementary notion explained below. Perhaps the primary open problem concerning infinity harmonic functions is to determine whether or not they are continuously differentiable. Results in this note reduce the problem of whether or not a function $u$ which enjoys comparison with cones has a derivative at a point $x_0$ in its domain to determining whether or not maximum points of $u$ relative to spheres centered at $x_0$ have a limiting direction as the radius shrinks to zero.  [Show abstract] [Hide abstract]
ABSTRACT: This paper provides a numberofworking tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of #strong" solutions of extremal equations by L n+1 norms of the forcing term over the #contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient estimate in L p for p##n+ 1##n + 2# for solutions of extremal equations with forcing terms in L n+1 ,the use of this estimate in improving the range of p for which the maximum principle #rst alluded to holds #obtaining some p#n+ 1  but without the contact set#, a proof of the strong solvability of Dirichlet problems for extremal equations with forcing terms in L p for some p#n+1,and the twice parabolic di#erentiability a.e. of W 2;1;p functions for #n +2#=2#p. 0. Introduction. In this work we provide a number of tools for the discussion of nonlinear parabolic equations under appropriate structure conditions. In particular, we...
Publication Stats
12k  Citations  
89.06  Total Impact Points  
Top Journals
Institutions

19912014

University of California, Santa Barbara
 Department of Mathematics
Santa Barbara, California, United States


19692006

University of California, Los Angeles
 Department of Mathematics
Los Angeles, California, United States


19861989

Brown University
 Department of Applied Mathematics
Providence, RI, United States


19751989

University of Wisconsin, Madison
 • Department of Mathematics
 • Center for Climatic Research
Mississippi, United States


19851986

Paris Dauphine University
Lutetia Parisorum, ÎledeFrance, France


1984

Loyola University Maryland
Baltimore, Maryland, United States


1980

Université ParisSud 11
Orsay, ÎledeFrance, France


1972

Hebrew University of Jerusalem
Yerushalayim, Jerusalem, Israel


1971

City University Los Angeles
Los Ángeles, California, United States


1970

Mathematical Sciences Research Institute
Berkeley, California, United States


19681969

Stanford University
 Department of Mathematics
Palo Alto, California, United States
