Regularity to a Kohn-Laplace Equation with Bounded Coefficients on the Heisenberg Group

Junli Zhang,Pengcheng Niuand Xiuxiu Wang

Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710129,China.

Abstract.In this paper,we concern the divergence Kohn-Laplace equation

Key words:Heisenberg group,Kohn-Laplace equation,local maximum principle,H¨older regularity,weak Harnack inequality.

1 Introduction

Under the assumption that the coefficients in the divergence elliptic equation

are bounded measurable,De Giorgi([8])proved that the bounded weak solution must be locally H¨older continuous,and deduced the priori estimate of the corresponding H¨older modulus.Nash([29])obtained independently this type of estimates of the solutions to the parabolic equation,but the method used is different.Afterwords,Moser([28])discovered a new iteration method(later called the Moser iteration)and established the above results for both elliptic equation and parabolic equation with this method.

In recent years,more attention has been paid to the regularity of degenerate partial differential equations(including elliptic equations,parabolic equations and many specific degenerate elliptic equations).Extension of classical regularity theory to degenerate partial differential equations composed of vector fields is an important issue and obtains the substantial development(see[2,3,6,9,12,20]).TheC1,αregularity of solutions to thep-Laplace equation in the Heisenberg group H1was considered by Ricciotti[30],Domokos[10],and Domokos and Manfredi[11].Dong and Niu studided nondiagonal quasilinear degenerate elliptic systems and gained regularity for weak solutions in[13].Du,Han and Niu obtained interior Morrey estimates and H¨older continuity for weak solutions to degenerate equations with drift on homogenous groups in[14].Hou,Feng and Cui proved global H¨older estimates for hypoelliptic operators with drift on homogeneous groups in[22].Bramanti and Zhu gainedLpand Schauder estimates for nonvariational operators structured on H¨ormander vector fields with drift in[4].Austin and Tyson work with the smoothness of solutions to the operator

(cis a real number)in[1].The aim of this paper is to establish the H¨older regularity of weak solutions to the divergence Kohn-Laplace equation with bounded coefficients on the Heisenberg group.

More concretely,we consider the divergence Kohn-Laplace equation

This paper is organized as follows.In Section 2,we introduce the knowledge of Heisenberg group,known Sobolev embedding theorem and Poincar′e inequality on the Heisenberg group.The H¨older space,Capanato space,and the doubling condition in the Heisenberg group are described.Two new properties(Theorems 2.2 and 2.3 below)is given.In Section 3,we show that the weak solution of the equation(1.1)belongs to the De Giorgi classDG(Ω;M,γ0,γ1,η,δ)and prove the Theorem 1.1.In Section 4,oscillation properties(Lemmas 4.1-4.5)of weak solutions to(1.1)are derived.In Section 5,we use the previous results to prove Theorems 1.2 and 1.3.

2 Preliminaries

The Heisenberg group Hnis a two-step nilpotent Lie group on R2n+1with the law

The homogeneous dimension on HnisQ=2n+2.Denote an open ball of radiusrand centerξbyBHn(ξ,r)={ζ∈Hn:d(ξ,ζ)＜r}.

Now we introduce the Sobolev embedding theorem and Poincar′e inequality to be used in later proofs.

3 Local maximum principle

4 Oscillation properties of functions of the De Giorgi class

5 Proof of Theorems 1.2 and 1.3

Acknowledgments

This work was supported by National Natural Science Foundation of China(Grant No.11771354 and 11701454).