Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is p...Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is proved in [2]. Wieser extended the notion of Q-minima to the parabolic case, and obtained the Hlder continuity. In this note, under those conditions, by which the Hlder continuity of the parabolic Q-minima was obtained in [ 3], we prove that the Harnack inequalities hold for Q-minima.展开更多
In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of E...In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of Euclidean N-space （N ≥ 3）, u：Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition ｜Du｜^p＊ ｜u｜^p＊-a（x） ≤ f（x,u,Du） ≤ L（｜Du｜^p＋｜u｜^p＊ ＋ a（x））, （2） where L≥1, 1pN,p^＊ = Np/N-p , and a（x） is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of （1）. Our method is to compare the ω-minima of （1） with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is proved in [2]. Wieser extended the notion of Q-minima to the parabolic case, and obtained the Hlder continuity. In this note, under those conditions, by which the Hlder continuity of the parabolic Q-minima was obtained in [ 3], we prove that the Harnack inequalities hold for Q-minima.
基金Supported by the Program of Fujian Province-HongKong
文摘In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of Euclidean N-space （N ≥ 3）, u：Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition ｜Du｜^p＊ ｜u｜^p＊-a（x） ≤ f（x,u,Du） ≤ L（｜Du｜^p＋｜u｜^p＊ ＋ a（x））, （2） where L≥1, 1pN,p^＊ = Np/N-p , and a（x） is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of （1）. Our method is to compare the ω-minima of （1） with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.
