In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.
The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE / Bonfiglioli, Andrea; Citti, Giovanna; Cupini, Giovanni; Manfredini, Maria; Montanari, Annamaria; Morbidelli, Daniele; Pascucci, Andrea; Polidoro, Sergio; Uguzzoni, Francesco. - STAMPA. - 13:(2015), pp. 341-373. (Intervento presentato al convegno Indam Meeting on Geometric Methods in PDEs on the Occasion of the 70th birthday of Ermanno Lanconelli tenutosi a Cortona, ITALY nel MAY 27-31, 2013) [10.1007/978-3-319-02666-4_18].
The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE
Manfredini, MariaMembro del Collaboration Group
;POLIDORO, SergioMembro del Collaboration Group
;
2015
Abstract
In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.File | Dimensione | Formato | |
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