Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations
Partial Regularity of Strong Local Minimizers in the Multi-Dimensional Calculus of Variations
复制标题
多维变分演算中强局部极小化的部分正则性
DOI：
10.1007/s00205-003-0275-4
复制
发表时间：
2003
影响因子：
2.5
通讯作者：
A. Taheri
中科院分区：
文献类型：
--
作者：
Jan Kristensen;A. Taheri
设Ω⊂ℝn是有界域，F：？→ℝ是给定的C2类的强拟凸被积，且对某些c>；0和2≤p<；∞满足增长条件$${{|F(\xi)|\le c(1+|\xi|^p)}}$$.考虑重整数${{i[u]=\int_{{\omega}}\！F(\nabla u)}}$$其中uW1，p(Ω，ℝN)。本文的主要结果是证明了i[·]的任何强局部极小元对全n维度量开集上的任意α(0，1)都是C1类的α(0，1)。在弱局部极小点的情况下，我们在额外的假设下建立了相同的结果，即极小点梯度的振荡不是太大。在没有这样的假设的情况下，弱局部极小化不必是部分正则的，如我们通过一类例子所示。我们还简要地讨论了I[·]的强局部极小点的存在性问题，以及我们的结果与把魏尔斯特拉斯的充分性定理推广到多维环境之间的联系。
Abstract.Let Ω⊂ℝn be a bounded domain and F:?→ℝ a given strongly quasiconvex integrand of class C2 satisfying the growth condition $${{ |F(\xi)| \le c (1 + |\xi|^p)}}$$ for some c>0 and 2≤p<∞. Consider the multiple integral $${{ I[u] = \int_{{\Omega}} \! F(\nabla u) }}$$ where uW1,p(Ω, ℝN). The main result of the paper is the proof that any strong local minimizer of I[·] is of class C1,αloc for any α(0,1) on an open set of full n-dimensional measure. In the case of weak local minimizers we establish the same result under the extra assumption that the oscillations in the gradient of the minimizer are not too large. Without such an assumption weak local minimizers need not be partially regular as we show by a class of examples. We also briefly discuss the question of existence of strong local minimizers for I[·] and connections of our results to extensions of Weierstrass’ sufficiency theorem to the multi-dimensional setting.