Calculus of Variations and Geometric Measure Theory

M. Manfredini - M. Piccinini - S. Polidoro

The Dirichlet problem for a family of totally degenerate differential operators

created by piccinini on 21 Jul 2022
modified on 10 Jul 2025

[BibTeX]

Published Paper

Inserted: 21 jul 2022
Last Updated: 10 jul 2025

Journal: J. Evol. Equations
Year: 2025

ArXiv: 2106.12048 PDF

Abstract:

In the framework of Potential Theory we prove existence for the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L} = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.


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