论文标题
非线性电位理论与完全非线性椭圆PDE之间的相互作用
Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs
论文作者
论文摘要
我们讨论了众多主题之一，这些主题说明了布莱恩·劳森（Blaine Lawson）的深刻几何和分析见解的相互作用。第一作者非常感谢与布莱恩（Blaine）在许多愉快的岁月中与布莱恩（Blaine）合作的乐趣。要讨论的话题涉及非线性潜在理论之间富有成果的相互作用。也就是说，对2束束中设定的一般约束以及非线性（退化）椭圆PDE的亚物种和超溶液的研究。主要结果包括（但不限于）比较原理的有效性以及对域上相关迪里奇问题的解决方案的存在和唯一性，这些域上适当地“伪共酮”。所采用的方法是几何和灵活的，并且在潜在理论方面非常笼统，这本身就是有趣的。此外，在许多重要的几何环境中，可能没有出现自然算子。另一方面，潜在的理论方法可以在给定差分运算符上的非标准结构条件下在PDE方面产生结果。
We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between nonlinear potential theory; that is, the study of subharmonics with respect to a general constraint set in the 2-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably "pseudoconvex". The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natutral operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structual conditions on a given differential operator.
