![]() What is a good response to this question? Imagine a student asks the question why it is worth it to study continuity. They have to be shown the differences between continuity and the lack of it. This silly example shows me every year that my students do not understand how some functions can act differently from others. This assumption is correct, but when I ask them they cannot even imagine any discontinuous possibility such as a Star Trek beam transporter. Students assume that the distance from my home is a continuous function, so I must at some point be exactly ten miles from my home (school is 22 miles away). This student will get into trouble when this is not the case.Įxample story: when I begin to teach the Intermediate Value Theorem for continuous functions, I give the example of me driving to school from my home. Catastrophe theory, developed in the 1970's, shows there are a variety of ways to get discontinuity, with a variety of applications.Īny student who does not directly think about continuous and discontinuous functions early in his/her math training may assume that all functions are continuous.The study of Chaos, resulting from weather and other such topics, gives results that are practically discontinuous, even if continuous in theory.Schrodinger's equation assumes continuity and differentiability, but in the Copenhagen interpretation of quantum theory any measurement causes a collapse of the waveform and things become discontinuous: the spin is either up or down, etc. Quantum theory is a mixture of the continuous and the discontinuous.Many computer routines produce discontinuous output, even if the data is near-continuous. Computers and their digitization of data.However, more and more discontinuous functions are appearing in the various sciences. Zbl0692.Most functions that are studied by physicists and other scientists are continuous. Ziemer, “Weakly Differentiable Functions”, Springer-Verlag, Berlin, 1989. Stampacchia, On some regular multiple integral problems in the calculus of variations, Comm. Morrey, “Multiple Integrals in the Calculus of Variations”, Springer-Verlag, New York, 1966. Miranda, Un teorema di esistenza e unicità per il problema dell’area minima in n variabili, Ann. Treu, Gradient maximum principle for minima, J. Treu, Existence and Lipschitz regularity for minima, Proc. Marcellini, Regularity for some scalar variational problems under general growth conditions, J. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. Hartman, On the bounded slope condition, Pacific J. Giusti, “Direct Methods in the Calculus of Variations” World Scientific, Singapore, 2003. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer-Verlag, Berlin, 1998. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems”, Princeton University Press, Princeton, N.J., 1983. Gariepy, “Measure Theorey and Fine Properties of Functions”, CRC Press, Boca Raton, FL, 1992. ![]() De Arcangelis, Some remarks on the identity between a variational integral and its relaxed functional, Ann. Wolenski, “Nonsmooth Analysis and Control Theory”, Graduate Texts in Mathematics, vol. ![]() Sinestrari, “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Birkhäuser, Boston, 2004. Belloni, A survey on old and recent results about the gap phenomenon, In: “Recent Developments in Well-Posed Variational Problems”, R. Clarke, Local Lipschitz continuity of solutions to a basic problem in the calculus of variations, to appear. Bousquet, On the lower bounded slope condition, to appear. In certain cases, as when Γ is a polyhedron or else of class C 1, 1, we obtain in addition a global Hölder condition on Ω ¯. We prove in particular that the solution is locally Lipschitz in Ω. ![]() This condition, which is less restrictive than the familiar bounded slope condition of Hartman, Nirenberg and Stampacchia, allows us to extend the classical Hilbert-Haar regularity theory to the case of semiconvex (or semiconcave) boundary data (instead of C 2). A new type of hypothesis on the boundary function φ is introduced: thelower (or upper) bounded slope condition. The lagrangian F and the domain Ω are assumed convex. We study the problem of minimizing ∫ Ω F ( D u ( x ) ) d x over the functions u ∈ W 1, 1 ( Ω ) that assume given boundary values φ on Γ : = ∂ Ω. Institut universitaire de France Université Claude Bernard Lyon 1, FranceĪnnali della Scuola Normale Superiore di Pisa - Classe di Scienze. ![]() Continuity of solutions to a basic problem in the calculus of variations ![]()
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