کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4611118 1338603 2010 13 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Relaxation and regularity in the calculus of variations
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Relaxation and regularity in the calculus of variations
چکیده انگلیسی

In this work we prove that, if L(t,u,ξ)L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ  , then any absolutely continuous solution u¯ to the variational problemmin{∫abL(t,u(t),u˙(t))dt:u∈W01,1(a,b)} is quasi-regular in the sense of Tonelli, i.e. u¯ is locally Lipschitz on an open set of full measure of [a,b][a,b], under the further assumption that either L is Lipschitz continuous in u, locally uniformly in ξ, but not necessarily in t, or L   is invariant under a group of C1C1 transformations (as in the Noether's theorem). Without one of those further assumptions the solution could be not regular as shown by a recent example in Gratwick and Preiss (2010) [13]; our result is then optimal in this sense. Moreover, we improve the standard hypothesis used so far in Buttazzo et al. (1998) [1], Clarke and Vinter (1985) [5] and [6], Csörnyei et al. (2008) [7], Tonelli (1915) [15] which have been the Lipschitz continuity of L in u, locally uniform in ξ and t, and some growth condition in ξ.We also show that the relaxed and the original problem have the same solutions (without assuming any of the two further assumptions above). This extends a result in Mariconda and Treu (2004) [14] to the non-autonomous case.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Differential Equations - Volume 249, Issue 10, 15 November 2010, Pages 2548–2560
نویسندگان
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