Erratum to: ``Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator'' [Ann. Inst. H. Poincaré Analyse Non Linéaire 8 (5) (1991) 443-457]
Erratum to: ``Classical solvability in dimension two of the second boundary value problem associated with the Monge-Ampère operator'' [Ann. Inst. H. Poincaré Analyse Non Linéaire 8 (5) (1991) 443-457]
复制标题
勘误：“与 Monge-Ampère 算子相关的第二个边值问题的经典可解性”[Ann. H. Poincaré Analysis Non Linéaire 8 (5) (1991) 443-457]
DOI：
10.1016/j.anihpc.2007.03.001
复制
发表时间：
2007
影响因子：
1.9
通讯作者：
Philippe Delanoë
中科院分区：
文献类型：
--
作者：
Philippe Delanoë
最近，西蒙·布伦德尔（Simon Brendle）（我想感谢他）向我指出，在[1，p.449，14行自上而下]中的断言ut= tu 1+（1-t）u 0 ∈ S（D，D）是不正确的（除非u1-u 0是常数）。所以我们必须修正它所进入的唯一性证明。由于在随后的几篇文章中，在考虑相同的非线性边界条件的情况下（见例如[3-7]），唯一性已经被断言而没有证明，我们将提供一个对所有人都有效的相当普遍的证明。我们需要一个引理，无处说明的一般性，虽然它的证明（在这里给出的完整性）已成为标准[6，页。870-871]，[7，p. 65]：引理1（严格的一致性）.设D（resp. D）是Rn的有界域（分别为（Rn）n）与C2（分别为C1）边界。边界条件du（D）= D，考虑在D上严格凸的真实的函数u∈ C2（D）上，这个条件是严格斜的。
Recently, Simon Brendle (whom I would like to thank) pointed out to me that the assertion ut= tu1+(1− t) u0∈ S (D, D∗) made in [1, p. 449, 14 lines from top] is incorrect (unless u1− u0 is constant). So we must fix the uniqueness proof in which it enters. Since uniqueness has been asserted without proof in several subsequent articles where the same nonlinear boundary condition is considered (see eg [3–7]), we will provide a fairly general proof valid for all. We require a lemma, nowhere stated in that generality although its proof (given here for completeness) has become standard [6, pp. 870–871],[7, p. 65]:Lemma 1 (strict obliqueness). Let D (resp. D∗) be a bounded domain of Rn (resp. of (Rn)∗) with C2 (resp. C1) boundary. The boundary condition du (D)= D∗, considered on real functions u∈ C2 (D) which are strictly convex (meaning they have a positive definite Hessian matrix at each point) on D, this condition, is strictly oblique.