Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

Diego Pallara; Sergio Polidoro

doi:10.3934/dcdss.2022144

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2022. Doi: 10.3934/dcdss.2022144

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Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

Diego Pallara^1, and
Sergio Polidoro^2, ,

1.
Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, and INFN, Sezione di Lecce, Ex Collegio Fiorini - Via per Arnesano - Lecce, Italy
2.
Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41125 Modena, Italy

^*Corresponding author: Sergio Polidoro
^*Corresponding author: Sergio Polidoro

We dedicate this paper to Jerry Goldstein on the occasion of his 80th birthday.

Received: April 2022

Revised: June 2022

Early access: August 2022

The first author is supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by INFN, Seione di Lecce. The second author is supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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Abstract

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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