Calculus of Variations and Geometric Measure Theory

S. Dipierro - A. Pinamonti - E. Valdinoci

Rigidity results in Diffusion Markov Triples

created by pinamonti on 03 Jan 2018
modified on 12 Jun 2018


Accepted Paper

Inserted: 3 jan 2018
Last Updated: 12 jun 2018

Journal: J. Funct. Anal.
Year: 2018


We consider stable solutions of semilinear equations in a very general setting. The equation is set on a Polish topological space endowed with a measure and the linear operator is induced by a carré du champs (equivalently, the equation is set in a diffusion Markov triple).

Under suitable curvature dimension conditions, we establish that stable solutions with integrable carré du champs are necessarily constant (weaker conditions characterize the structure of the carré du champs and carré du champ itéré).

The proofs are based on a geometric Poincar\'e formula in this setting. {F}rom the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.