Michael G. Crandall

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

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Publications (78)61.24 Total impact

  • Luis A. Caffarelli, Michael G. Crandall
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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.
    Proceedings of the American Mathematical Society 01/2014; 142(7). · 0.61 Impact Factor
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    ABSTRACT: We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their 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 $L^\infty.$ Comment: 34 pages
    Archive for Rational Mechanics and Analysis 03/2010; · 2.29 Impact Factor
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    Luis A. Caffarelli, Michael G. Crandall
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    ABSTRACT: Consider a function u defined on n , except, perhaps, on a closed set of potential singularities . Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on n \, where Du denotes the gradient of u and ‖·‖ is a norm on n with the dual norm ‖·‖. For a class of norms which includes the standard p-norms on n , 1 < p < ∞, we show that if has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x − z‖.
    Communications in Partial Differential Equations 03/2010; 35(3):391-414. · 1.03 Impact Factor
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    Luis A. Caffarelli, Michael G. Crandall
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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": $\| l_1-\bar l_0\| \ge \| l_1-l_0\|, \| \bar l_1-l_0\| \ge \| \bar l_1-\bar l_0\|.$ 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
    Expositiones Mathematicae 01/2010; · 0.78 Impact Factor
  • Lecture Notes in Mathematics -Springer-verlag- 08/2008; 1927. · 0.55 Impact Factor
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    Michael G. Crandall, Changyou Wang, Yifeng Yu
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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:
    Transactions of the American Mathematical Society 01/2008; 361(01):103-124. · 1.02 Impact Factor
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    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:
    12/2007: pages 183-189;
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    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])
    12/2007: pages 155-181;
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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
    12/2007: pages 75-122;
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    Michael G Crandall, Gunnar Gunnarsson, Peiyong Wang
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    ABSTRACT: Comparison results are obtained between infinity subharmonic and infinity superharmonic func-tions defined on unbounded domains. The primary new tool employed is an approximation of infinity subhar-monic 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.
    Communications in Partial Differential Equations 01/2007; 32(10). · 1.03 Impact Factor
  • 11/2006: pages 1-43;
  • 11/2006: pages 134-185;
  • 11/2006: pages 186-242;
  • H. Brezis, M. G. Crandall, A. Pazy
    Communications on Pure and Applied Mathematics 10/2006; 23(1):123 - 144. · 3.34 Impact Factor
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    Gunnar Aronsson, Michael G Crandall, Petri Juutinen
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    ABSTRACT: These notes are intended to be a rather complete and self-con-tained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowl-edge of the subject. In particular, we improve known results regarding exis-tence 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 self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geo-metric 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 bur-densome 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 ques-tions, 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.
    Bulletin of The American Mathematical Society - BULL AMER MATH SOC. 01/2004; 41(04).
  • Michael G. Crandall
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    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 well-known ``p-Laplacian''. When p=∞ the corresponding functional is u↦|| |Du|2|| 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.
    Archive for Rational Mechanics and Analysis 01/2003; 167(4):271-279. · 2.29 Impact Factor
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    M.G. Crandall, L.C. Evans, R.F. Gariepy
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    ABSTRACT: We reconsider in this paper boundary value problems for “infinity Laplacian” PDE and the relationships with optimal Lipschitz extensions of the boundary data. fairly elegant new proofs, which clarify and simplify previous work, and may be characterized by a comparison principle with appropriate cones. We in comparison with cones directly implies the variational principle associated Liouville theorem for subsolutions bounded above by planes.
    Calculus of Variations 08/2001; 13(2):123-139. · 1.24 Impact Factor
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    Michael G. Crandall, L. C. Evans
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    ABSTRACT: A real-valued 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.
    Electronic Journal of Differential Equations 01/2001; · 0.43 Impact Factor
  • M. G. Crandall, K. Fok, M. Kocan
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    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...
    07/2000;
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    M. g. Crandall, M. Kocan, A. Świech
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    ABSTRACT: this paper so that a reader might scan the resultsit contains without reference to other works, it otherwise depends stronglyon #9# for some perspective on the literature and for preliminary results. Resultsfrom #9# are recounted in Section 2 after a discussion of preliminariesin Section 1. In addition,the issue of existence of L
    Communications in Partial Differential Equations - COMMUN PART DIFF EQUAT. 01/2000; 25:1997-2053.

Publication Stats

5k Citations
61.24 Total Impact Points

Institutions

  • 1994–2010
    • University of California, Santa Barbara
      • Department of Mathematics
      Santa Barbara, California, United States
  • 1969–2006
    • University of California, Los Angeles
      • Department of Mathematics
      Los Angeles, California, United States
  • 1985–1991
    • Paris Dauphine University
      Lutetia Parisorum, Île-de-France, France
  • 1972–1989
    • University of Wisconsin, Madison
      • • Department of Mathematics
      • • Center for Climatic Research
      Mississippi, United States
    • Hebrew University of Jerusalem
      Yerushalayim, Jerusalem District, Israel
  • 1968–1969
    • Stanford University
      • Department of Mathematics
      Palo Alto, CA, United States