On the existence and uniqueness of minima and maxima on spheres of the integral functional of the calculus of variations

[摘要] Given a bounded domain Omega subset of R-n, we prove that if f : Rn+1 R is a C-1 function whose gradient is Lipschitzian in Rn+1 and non-zero at 0, then, for each r > 0 small enough, the restriction of the integral functional u -> integral(Omega) f (u(x), del u (x)) dx to the sphere {u is an element of H-1 (Omega): integral(Omega) (vertical bar del u(x)vertical bar(2) + vertical bar u(x)vertical bar(2)) dx = r } has a unique global minimum and a unique global maximum. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-15

[关键词] Sobolev space;integral functional;minimum;maximum;sphere;existence;uniqueness