We prove by giving an example that when the asymptotic behavior of functionals is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.
La primitiva della funzione non è esprimibile in termini elementari: si tratta di un fatto ben noto, dimostrato per la prima volta da Liouville nel diciannovesimo secolo. Ciononostante la dimostrazione è poco conosciuta. In questa nota mi propongo di dame un resoconto completo, ponendo l’accento sulle idee più importanti, ma includendo anche una trattazione il più elementare possibile di tutti i dettagli tecnici. Questo lavoro è l’elaborazione di una conferenza tenuta dall’autore il 29 settembre...
In a recent joint paper with L. Székelyhidi we have proposed a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in with . We give a reformulation of the Euler equations as a differential inclusion, and in this way we obtain transparent proofs of several celebrated results of V. Scheffer and A. Shnirelman concerning the non-uniqueness of weak solutions and the existence of energy-decreasing solutions. Our results are stronger...
In this article we survey some recent results in the regularity theory of admissible solutions to hyperbolic conservation laws and Hamilton-Jacobi equations.
``Lo scopo di questa nota è illustrare, nel modo più elementare possibile, la classificazione dei politopi regolari in ogni dimensione, un risultato raggiunto nella metaÁ dell'ottocento dal grande matematico svizzero Ludwig Schlaefli''.
We prove by giving an example that when ≥ 3 the asymptotic behavior of functionals is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some -norm of the gradient with is controlled...
Building upon the techniques introduced in [15], for any we construct periodic weak solutions of the incompressible Euler equations which dissipate the total kinetic energy and are Hölder-continuous with exponent . A famous conjecture of Onsager states the existence of such dissipative solutions with any Hölder exponent . Our theorem is the first result in this direction.
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