Regularity for a Nonlinear Discontinuous Subelliptic System with Drift on the Heisenberg Group

Junli Zhang,Jialin Wang,Pietro D Avenia

doi:10.1155/2022/7853139

Abstract

In this paper, we prove the partial Hölder regularity of weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions to a nonlinear discontinuous subelliptic system with drift on the Heisenberg group by theA-harmonic approximation, where the coefficients in the nonlinear subelliptic system are discontinuous and satisfy the VMO condition forx, ellipticity and growth condition with the growth index1<p<2for the Heisenberg gradient variable, and the nonhomogeneous terms satisfy the controllable growth condition and the natural growth condition, respectively.

Highlights

Kohn in [1] proved L2 estimates for the operator k Lu = 〠 X 2 j u +X0u cu ð1Þ j=1 constructed by Hörmander’s vector fields fX1, X2,⋯,Xq, X0g based on the energy estimate and a subelliptic estimate
Some authors inspected the regularity of solutions to linear degenerate elliptic equations with drift term by establishing singular integral estimates
I, j=1 constructed by Hörmander’s vector fields, Bramanti and Zhu in [4] established Lp estimates with aijðxÞ and a0ðxÞ belonging to VMO spaces related to fX1, X2,⋯,Xq, X0g and Schauder estimates with aijðxÞ and a0ðxÞ being in Hölder spaces for strong solutions

Summary

Introduction

X0u cu ð1Þ j=1 constructed by Hörmander’s vector fields fX1, X2,⋯,Xq, X0g (see [2]) based on the energy estimate and a subelliptic estimate. Ð3Þ i, j=1 constructed by Hörmander’s vector fields, Bramanti and Zhu in [4] established Lp estimates with aijðxÞ and a0ðxÞ belonging to VMO spaces related to fX1, X2,⋯,Xq, X0g and Schauder estimates with aijðxÞ and a0ðxÞ being in Hölder spaces for strong solutions It is important in [4] that the difference between equations without X0 and with X0 was pointed out. We consider the regularity to the weak solution of discontinuous subelliptic systems with drift term Tu on Hn. ð5Þ where Ω is the bounded domain in Hn, Aki belongs to the vanishing mean oscillation space (which is abbreviated as VMO) and satisfies the ellipticity on R2n×N and polynomial growth conditions with the growth index 1 < p < 2 for ∇Hu, and Aki is continuous for u and differentiable for ∇Hu with continuous derivatives,.

Preliminaries

Some Lemmas

Proof of Theorem 1