The Dirichlet problem for a family of totally degenerate differential operators

2021/06/22 by Maria Manfredini, Mirco Piccinini, Manfredini, Maria +3
Computer Science · Mathematics · #31B20 #31B25 #31D05 #35B65 #35K20 #35K70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2106.12048

openalex publication_date 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the framework of Potential Theory we prove existence or 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 L = t²Δ_x + ⟨ x, ∇_y ⟩ -∂_t, for (x,y,t) ∈ ℝ^N × ℝ^N × ℝ.

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