The role of fundamental solution in potential and regularity theory for subelliptic PDE. (English) Zbl 1338.35010
Citti, Giovanna (ed.) et al., Geometric methods in PDE’s. Cham: Springer (ISBN 978-3-319-02665-7/hbk; 978-3-319-02666-4/ebook). Springer INdAM Series 13, 341-373 (2015).
Summary: In this survey we consider a general Hörmander 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 formula on the level set of the fundamental solution, which allow 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: 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.
For the entire collection see [Zbl 1342.35007].
For the entire collection see [Zbl 1342.35007].
MSC:
| 35A08 | Fundamental solutions to PDEs |
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