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

  • *Corresponding author: Sergio Polidoro

    *Corresponding author: Sergio Polidoro

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

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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  • 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.

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


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