Comparison principles for general potential theories and fully nonlinear PDEs with directionality
speaker: Kevin Payne (Università degli Studi di Milano Statale)abstract: We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets F ⊂ J 2 (X) of the 2-jets
on X ⊂ R n open) and subsolutions of degenerate elliptic PDEs of the form F(x, u, Du, D2u) = 0. We will describe the monotonicity-duality method begin by Harvey and Lawson Comm. Pure
Appl. Math, 2009 for proving comparison principles for potential theories where F has suffi- cient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all
differential operators F which are compatible with F in a precise sense. Particular attention will
be given to gradient dependent examples with the requisite sufficient monotonicity of proper el- lipticity and directionality. Examples operators we will discuss include those of optimal transport
in which the target density is strictly increasing in some directions as well as operators which are parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of G ̊arding, produce additional examples in which the comparison principles holds, but standard viscosity structural conditions fail to hold.
timetable:
Tue 9 May, 14:00 - 14:30, Aula Dini
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