The aim of this paper is to study some classes of second-order divergence-form partial differential operators L of sub-Riemannian type. Our main assumption is the C^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(x, y) for L. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the L-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to M_r; the role of M_r in solving the homogeneous Dirichlet problem related to L in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Omega_r(x), superlevel sets of Gamma(x, ·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for L using M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).

B. Abbondanza, A. Bonfiglioli (2013). The Dirichlet problem and the inverse mean value theorem for a class of divergence form operators. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 87, 321-346 [10.1112/jlms/jds050].

The Dirichlet problem and the inverse mean value theorem for a class of divergence form operators

ABBONDANZA, BEATRICE;BONFIGLIOLI, ANDREA
2013

Abstract

The aim of this paper is to study some classes of second-order divergence-form partial differential operators L of sub-Riemannian type. Our main assumption is the C^infinity-hypoellipticity of L, together with the existence of a well-behaved fundamental solution Gamma(x, y) for L. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the L-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to M_r; the role of M_r in solving the homogeneous Dirichlet problem related to L in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Omega_r(x), superlevel sets of Gamma(x, ·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for L using M_r, a Poisson–Jensen formula for the L-subharmonic functions and several results concerning the geometry of the sets Omega_r(x).
2013
B. Abbondanza, A. Bonfiglioli (2013). The Dirichlet problem and the inverse mean value theorem for a class of divergence form operators. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 87, 321-346 [10.1112/jlms/jds050].
B. Abbondanza; A. Bonfiglioli
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/120835
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