Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs

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 natural operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structural conditions on a given differential operator.

Keywords

subequations, potential theory, fully nonlinear degenerate elliptic PDEs, comparison principles, viscosity solutions, admissibility constraints, monotonicity, duality, fiberegularity

2010 Mathematics Subject Classification

31C45, 35B51, 35D40, 35E20, 35J60, 35J70

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The second-named author was partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and the project: GNAMPA 2022 “Proprietà qualitative e quantitative per EDP non lineari: dai principi del massimo alla teoria di regolarità”.

Received 25 March 2022

Accepted 3 March 2023

Published 30 January 2024