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Side Boundary potentials for a Kolmogorov-type PDE

Richard Sowers

Abstract

We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak Hörmander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel discontinuity at the boundary. We introduce some polynomial corrections to this approximate boundary potential and then construct a boundary-domain Volterra equation to solve the original partial differential equation. This Volterra integral equation is iteratively solved, and the bounds contain a periodic behavior resulting from the boundary effects.

Side Boundary potentials for a Kolmogorov-type PDE

Abstract

We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak Hörmander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel discontinuity at the boundary. We introduce some polynomial corrections to this approximate boundary potential and then construct a boundary-domain Volterra equation to solve the original partial differential equation. This Volterra integral equation is iteratively solved, and the bounds contain a periodic behavior resulting from the boundary effects.
Paper Structure (16 sections, 27 theorems, 339 equations, 3 figures)

This paper contains 16 sections, 27 theorems, 339 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Solutions of Parabolic Equations
  3. Hypoellipticity
  4. Assumptions and Notation
  5. Spaces and standard notation
  6. Regularity
  7. Bounds
  8. Partial Differential Operator
  9. Main Parts of Parametrix and Approximate Boundary Potential
  10. Parametrix Calculations in the Interior
  11. Sides
  12. Corrections
  13. Setup
  14. Solution and Error Bounds
  15. Integrability and Continuity
  16. ...and 1 more sections

Key Result

Proposition 3.2

Fix $(x_\circ,y_\circ)\in \mathbb{R}^2$. Then $\mathscr{P}^L_{x_\circ,y_\circ}\mathcal{K}_{x_\circ,y_\circ}=0$ on $\mathcal{U}$.

Figures (3)

  • Figure 1: Geometry of Problem
  • Figure 2: Cases in proof of Proposition \ref{['P:integrablekernel']}
  • Figure 3: Integrand of \ref{['E:coreI']}

Theorems & Definitions (66)

  • Remark 1.1
  • Remark 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Proposition 4.1
  • ...and 56 more