Schauder estimates for degenerate elliptic and parabolic equations in \(\mathbb R^N\) with Lipschitz drift. (English) Zbl 1212.35257

Saintier, Nicolas

Differ. Integral Equ. 20, No. 4, 397-428 (2007).

Summary: We prove existence, uniqueness and Schauder estimates for the degenerate elliptic and parabolic equations \(\lambda -\mathcal Ku=f\) in \(\mathbb R^{2n}\) and \(\partial _tu=\mathcal Ku+g\) in \((0,+\infty )\times \mathbb R^{2n}\), \(u(0,\cdot )=f\) in \(\mathbb R^{2n}\), associated to the degenerate Kolmogorov operator \(\mathcal K\), \(\mathcal K u:=\frac 12\Delta _xu+F(x,y)D_xu+xD_yu\), where \(u\) is a smooth function on \(\mathbb R^n\times \mathbb R^n\) and \(F:\mathbb R^{2n}\to \mathbb R^{2n}\) is of class \(C^3\) with bounded derivatives up to the third order.

MSC:

35K65	Degenerate parabolic equations
35J70	Degenerate elliptic equations
35B65	Smoothness and regularity of solutions to PDEs
35K15	Initial value problems for second-order parabolic equations
47D06	One-parameter semigroups and linear evolution equations

Keywords:

existence; uniqueness; Schauder estimates; degenerate elliptic equation; degenerate parabolic equation; Kolmogorov operator