[Paper Review] Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients

Malte Litsgård, Kaj Nyström|arXiv (Cornell University)|Dec 7, 2020

Numerical methods in inverse problems30 references4 citations

TL;DR

This paper establishes a robust potential theory for strongly degenerate parabolic Kolmogorov-type operators with rough, bounded, measurable coefficients in unbounded Lipschitz-type domains. Using scale- and translation-invariant estimates, it proves boundary Harnack inequalities, doubling properties of parabolic measures, and comparison principles, with all constants depending only on ellipticity and Lipschitz constants, extending classical Fabes-Safonov results to degenerate settings.

ABSTRACT

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:= abla_X\cdot(A(X,Y,t) abla_X)+X\cdot abla_{Y}-\partial_t, \] in unbounded domains of the form \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1} imes\mathbb R imes\mathbb R^{m-1} imes\mathbb R imes\mathbb R\mid x_m>ψ(x,y,y_m,t)\}, \] where $ψ$ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we assume that $A$ is bounded, measurable, symmetric and uniformly elliptic (as a matrix in $\mathbb R^{m}$). 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.

Motivation & Objective

• To extend classical boundary potential theory—previously known for uniformly parabolic equations—to a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients.
• To establish solvability of the Dirichlet problem in unbounded Lipschitz-type domains with geometry adapted to the underlying non-Euclidean Lie group structure.
• To derive scale- and translation-invariant boundary comparison principles, boundary Harnack inequalities, and doubling properties for parabolic measures associated with the operator.
• To generalize results of Fabes and Safonov to degenerate operators by proving quantitative estimates with explicit dependence only on ellipticity and Lipschitz constants of the domain.

Proposed method

• The analysis is conducted on unbounded domains defined by a Lipschitz boundary function $\psi$ adapted to the dilation structure of the Kolmogorov operator's underlying Lie group.
• The operator $\mathcal{L} = \nabla_X \cdot (A(X,Y,t)\nabla_X) + X\cdot\nabla_Y - \partial_t$ is studied in divergence form with $A$ bounded, measurable, symmetric, and uniformly elliptic in $\mathbb{R}^m$.
• Key tools include the use of fundamental solutions and Green's functions $G$ associated with the operator, particularly in parabolic cylinders $A_{r,\Lambda}^+(Z_0,t_0)$, to analyze harmonic measures.
• Boundary Harnack inequalities are derived via comparison of Green's functions and parabolic measures $\omega$, using estimates from Lemma 8.1 and Lemma 8.4 to control ratios of measures.
• The proof relies on the Harnack inequality for the adjoint operator and scaling arguments to derive uniform estimates independent of location or scale.
• The theory is built on the structure of the Kolmogorov operator as a prototype of hypoelliptic, degenerate parabolic equations with non-isotropic scaling.

Experimental results

Research questions

• RQ1 Can a boundary potential theory be developed for strongly degenerate parabolic operators of Kolmogorov type when the coefficient matrix $A$ is only bounded and measurable, not smooth?
• RQ2 Do boundary Harnack inequalities and doubling properties of parabolic measures hold in unbounded Lipschitz-type domains under such rough coefficient assumptions?
• RQ3 Can the constants in boundary estimates be made independent of spatial and temporal location, depending only on ellipticity and Lipschitz constants of the domain?
• RQ4 Is there a scale- and translation-invariant comparison principle for non-negative solutions vanishing on the boundary in such degenerate settings?
• RQ5 To what extent do classical results for uniformly parabolic equations extend to this class of degenerate Kolmogorov-type operators?

Key findings

• The paper establishes a boundary Harnack inequality with constants depending only on the ellipticity constants of $A$ and the Lipschitz constant of $\psi$, independent of location or scale.
• Parabolic measures associated with the operator are shown to be doubling, with doubling constants depending only on the ellipticity and Lipschitz constants.
• A scale- and translation-invariant boundary comparison principle is proven, allowing comparison of non-negative solutions vanishing on the boundary.
• The ratio of parabolic measures $\omega(A_{cr,\Lambda}^+(Z_0,t_0), E)/\omega(A_{cr,\Lambda}^+(Z_0,t_0), \Delta_{\tilde{r}}(\tilde{Z}_0,\tilde{t}_0))$ is shown to be comparable to $\omega(A_{c\tilde{r},\Lambda}^+(\tilde{Z}_0,\tilde{t}_0), E)$ for subsets $E \subset \Delta_{\tilde{r}}(\tilde{Z}_0,\tilde{t}_0)$.
• The Green's function ratio $G(A_{c\tilde{r},\Lambda}^+(\tilde{Z}_0,\tilde{t}_0), A_{\tilde{r}/\tilde{c},\Lambda}(\tilde{Z}_0,\tilde{t}_0)) / G(A_{cr,\Lambda}^+(Z_0,t_0), A_{\tilde{r}/\tilde{c},\Lambda}(\tilde{Z}_0,\tilde{t}_0))$ is uniformly bounded from below and above.
• All estimates are translation- and scale-invariant, with constants depending only on $m$, $\kappa$, $M$, and the ellipticity and Lipschitz constants, not on the specific geometry of the domain beyond these.