2020/12/07 by Malte Litsgård, Litsgård, M., Kaj Nyström +1 · 1 citation
Computer Science · Mathematics · #35H20 #35K65 #35K70 #35R03 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2012.03654
openalex publication_date 2020/12/07 · openalex created_date 2020/12/21 · openalex updated_date 2026/07/28
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L:=∇X⋅(A(X,Y,t)∇X)+X⋅∇Y-∂t, in unbounded domains of the form Ω=\(X,Y,t)=(x,xm,y,ym,t)∈\mathbb Rm-1×\mathbb R×\mathbb Rm-1×\mathbb R×\mathbb R| xmgt;ψ(x,y,ym,t)\, where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L. Concerning A=A(X,Y,t) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in \mathbb Rm). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, any several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.