GÃ¥rding Polynomials and its applications in fully nonlinear PDEs
GÃ¥rding Polynomials and its applications in fully nonlinear PDEs
In this talk we report some recent joint work with Biao Ma.
We introduce a new type of Monge-Ampere type PDEs in complex geometry and relate their solvability to corresponding paired slope stability conditions through the bridge of analytical condition of sub-solution existence. This work generalizes previous works on J-equations and deformed Hermitian-Yang-Mills equations by G. Chen, Datar-Pingali, Song and many others.
In order to find more examples, we also explore a generalization of ÌýtheÌý1959 works of GÃ¥rding on hyperbolic polynomials, which hasÌýbeen widely used to study geometric PDEs under Ìýthe framework of Caffarelli-Nirenberg-Spruck theory.Ìý
Generalizing earlir works of C.-M. Lin and some of our previous results, we introduce a class of polynomials that is named afterÌýGÃ¥rding. GÃ¥rding polynomials are characterized by their associated cones and defined in a similar fashion of Lorentzian polynomial ofÌýBranden-Huh.Ìý GÃ¥rdingÌý polynomials share similar algebraic properties of stable and Lorentzian polynomials and have potential applications in other fields of mathematics.