2022/12/04 by Timur Akhunov, Akhunov, Timur, Lyudmila Korobenko +1
Computer Science · Mathematics · #35A18 #35B65 #35G05 #35H10 #35H20 #35S05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2212.01727
openalex publication_date 2022/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form L1(x)+g(x)L2(y), where L1(x) is one-dimensional and g(x) may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our results do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.