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Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain

Luan Hoang, Akif Ibragimov

TL;DR

The paper addresses linear second-order elliptic equations in non-divergence form on a bounded planar domain that winds around a fixed circle infinitely many times and accumulates on it. By combining the Maximum Principle with an explicit Landis-type Growth Lemma, the authors derive global and asymptotic estimates for both homogeneous and inhomogeneous problems, including a discrete dichotomy for cross-section maxima and oscillation bounds, and they prove uniqueness and continuous dependence on boundary data and forcing. The analysis covers both bounded and unbounded drifts (under growth constraints) and yields precise exponential decay or growth rates with respect to the angular or arc-distance coordinates, providing a rigorous framework for diffusion-transport models in porous media with highly winding geometries. These results lay groundwork for extensions to nonlinear problems and more general limiting sets, highlighting the role of geometric growth control in elliptic theory and offering tools for stability and asymptotics in complex domains.

Abstract

We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function.

Linear non-divergence elliptic equations in a bounded, infinitely winding planar domain

TL;DR

The paper addresses linear second-order elliptic equations in non-divergence form on a bounded planar domain that winds around a fixed circle infinitely many times and accumulates on it. By combining the Maximum Principle with an explicit Landis-type Growth Lemma, the authors derive global and asymptotic estimates for both homogeneous and inhomogeneous problems, including a discrete dichotomy for cross-section maxima and oscillation bounds, and they prove uniqueness and continuous dependence on boundary data and forcing. The analysis covers both bounded and unbounded drifts (under growth constraints) and yields precise exponential decay or growth rates with respect to the angular or arc-distance coordinates, providing a rigorous framework for diffusion-transport models in porous media with highly winding geometries. These results lay groundwork for extensions to nonlinear problems and more general limiting sets, highlighting the role of geometric growth control in elliptic theory and offering tools for stability and asymptotics in complex domains.

Abstract

We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function.

Paper Structure

This paper contains 12 sections, 16 theorems, 167 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Description of the domain
  3. Growth lemmas
  4. A general Growth Lemma
  5. Planar domains inside an annulus
  6. The homogeneous problems
  7. Bounded drifts
  8. Unbounded drifts
  9. Inhomogeneous problems
  10. Homogeneous equation with bounded boundary data
  11. Inhomogeneous equation with a bounded forcing function
  12. Uniqueness and continuous dependence

Key Result

Lemma 2.3

Under Assumption bassum2, one has the following estimates with $x=X_\lambda(\theta)\in \widetilde{U}$.

Figures (2)

  • Figure 1: The domain $U$.
  • Figure 2: Construction for the Growth Lemma.

Theorems & Definitions (38)

  • Definition 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 3.1
  • Lemma 3.4: A general Growth Lemma
  • proof
  • Lemma 3.5
  • proof
  • Lemma 4.1: Discrete dichotomy for solutions
  • ...and 28 more