Internal Schauder estimates for Hörmander type equations with Dini continuous source

2023/06/05 by Giovanna Citti, Bianca Stroffolini, Citti, Giovanna +1
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2306.02799

Abstract

We study the regularity properties of a general second order Hörmander operator with Dini continous coefficients a_ij. Precisely if X₀, X₁,⋯ X_m are smooth self adjoint vector fields satisfying the Hörmander condition, we consider the linear operator in ℝ^N, with N>m+1: L u := ∑_{i, j= 1}^m a_ij X_iX_j u - X₀ u. The vector field X₀ plays a role similar to the time derivative in a parabolic problem so that it is a vector of degree two. We prove that, if f is a Dini continuous function, then the second order derivatives of the solution u to the equation L u = f are Dini continuous functions as well. A key step in our proof is a Taylor formula in this anisotropic setting, that we establish under minimal regularity assumptions.

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