Rigidity results in diffusion Markov triples

S. Dipierro, A. Pinamonti, E. Valdinoci

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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 Poincaré formula in this setting. From the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.
Original languageEnglish
Pages (from-to)785-814
JournalJournal of Functional Analysis
Volume276
Issue number3
DOIs
Publication statusPublished - Feb 2019

Fingerprint

Stable Solution
Rigidity
Semilinear Equations
Topological space
Linear Operator
Curvature
Theorem

Cite this

@article{db5f4323b6944faab6d4fc526a286b3b,
title = "Rigidity results in diffusion Markov triples",
abstract = "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 carr{\'e} du champs (equivalently, the equation is set in a diffusion Markov triple). Under suitable curvature dimension conditions, we establish that stable solutions with integrable carr{\'e} du champs are necessarily constant (weaker conditions characterize the structure of the carr{\'e} du champs and carr{\'e} du champ it{\'e}r{\'e}). The proofs are based on a geometric Poincar{\'e} formula in this setting. From the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.",
author = "S. Dipierro and A. Pinamonti and E. Valdinoci",
year = "2019",
month = "2",
doi = "10.1016/j.jfa.2018.06.005",
language = "English",
volume = "276",
pages = "785--814",
journal = "Journal of Functional Analysis",
issn = "0022-1236",
publisher = "Academic Press",
number = "3",

}

Rigidity results in diffusion Markov triples. / Dipierro, S.; Pinamonti, A.; Valdinoci, E.

In: Journal of Functional Analysis, Vol. 276, No. 3, 02.2019, p. 785-814.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Rigidity results in diffusion Markov triples

AU - Dipierro, S.

AU - Pinamonti, A.

AU - Valdinoci, E.

PY - 2019/2

Y1 - 2019/2

N2 - 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 Poincaré formula in this setting. From the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.

AB - 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 Poincaré formula in this setting. From the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.

UR - http://www.scopus.com/inward/record.url?scp=85048710910&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2018.06.005

DO - 10.1016/j.jfa.2018.06.005

M3 - Article

VL - 276

SP - 785

EP - 814

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -