Abstract
Implicit Ordinary or Partial Differential Equations have been widely studied in recent times, essentially from the existence of solutions point of view. One of the main issues is to select a meaningful solution among the infinitely many ones. The most celebrated principle is the viscosity method. This selection principle is well adapted to convex Hamiltonians, but it is not always applicable to the non-convex setting.
Highlights
Let f be a function from [a, b] × R × R to R
We would like to conclude this introduction pointing out that we look at Problem (1) essentially as a differential inclusion
If we suppose that f+i, f−i do not assume the value zero, i.e. f−i < 0 < f+i on [a, b] × R, from the proof of Theorem 14, it follows that the solutions to (1) with minimal number of discontinuity points have derivative with exactly one singularity
Summary
Where solutions u are searched in W01,∞(a, b). As it is well known, under suitable compatibility conditions between the data and the function f , this problem admits a priori infinitely many solutions [7]. The purpose of the present work is to determine criteria to select, among these infinitely many solutions, some of them. At the moment there is no general criterion that permits to select a unique solution in all contexts. Are some of the most natural criteria. 1) The most widely used criterion is the viscosity method introduced in a systematic way by Crandall-Lions [5, 10]. 2) Another natural criterion is the one selecting the maximal solution.
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