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Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications

Emily Shores

TL;DR

The paper studies hypoellipticity and regularity for constant-coefficient linear degenerate elliptic systems on Carnot groups, showing that a weak solution $u$ is actually smooth when there is strong ellipticity along the horizontal direction and with no restriction on the Carnot step. The authors develop a layered-differentiation framework across Carnot layers, using commutator analysis and energy (Caccioppoli) inequalities to control higher-order horizontal derivatives, and they rely on Sobolev embeddings and prior linear estimates to deduce smoothness. Building on this linear theory, they perform a blow-up (Giusti–Miranda) analysis to obtain partial Hölder regularity for weak solutions of certain nonlinear systems, and then apply appcor-type arguments to promote regularity. The results provide a robust method for hypoellipticity and regularity in constant-coefficient degenerate elliptic systems on Carnot groups, applicable without restriction on the group step, and they enable partial to full smoothness via a quantitative regularity framework.

Abstract

We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to develop blow-up analysis to prove a partial regularity result for weak solutions of certain non-linear systems.

Hypoellipticity for linear degenerate elliptic systems in Carnot groups and applications

TL;DR

The paper studies hypoellipticity and regularity for constant-coefficient linear degenerate elliptic systems on Carnot groups, showing that a weak solution is actually smooth when there is strong ellipticity along the horizontal direction and with no restriction on the Carnot step. The authors develop a layered-differentiation framework across Carnot layers, using commutator analysis and energy (Caccioppoli) inequalities to control higher-order horizontal derivatives, and they rely on Sobolev embeddings and prior linear estimates to deduce smoothness. Building on this linear theory, they perform a blow-up (Giusti–Miranda) analysis to obtain partial Hölder regularity for weak solutions of certain nonlinear systems, and then apply appcor-type arguments to promote regularity. The results provide a robust method for hypoellipticity and regularity in constant-coefficient degenerate elliptic systems on Carnot groups, applicable without restriction on the group step, and they enable partial to full smoothness via a quantitative regularity framework.

Abstract

We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to develop blow-up analysis to prove a partial regularity result for weak solutions of certain non-linear systems.

Paper Structure

This paper contains 8 sections, 16 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main Theorem
  4. Notation
  5. Results
  6. Auxillary Lemmas
  7. Proof of Theorems 9 and 10
  8. Sketch of Proof of Corollary $\ref{['Gi']}$

Key Result

Theorem 1

(Main Theorem) Let G be a Carnot group of step r and \Omega \subset G an open, bounded set. If u \in S^{1,2}_{loc}(\Omega) is a weak solution to $1.1$ and f^\alpha, f_i^\alpha are smooth functions, then for any ball B such that 2B \subset \Omega the following inequality holds: Here, X^{I_1} \cdots X^{I_r} represent differentiation of indefinite order in each of the layers, 1, \ldots, r. Also, we

Theorems & Definitions (27)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • proof : Sketch of proof
  • Theorem 8: Peetre
  • Theorem 9: Hörmander
  • ...and 17 more