Optimal regularity for quasilinear equations in stratified nilpotent Lie groups and applications

Abstract

We announce the optimal C1+α regularity of the gradient of weak solutions to a class of quasilinear degenerate elliptic equations in nilpotent stratified Lie groups of step two. As a consequence we also prove a Liouville type theorem for 1-quasiconformal mappings between domains of the Heisenberg group Hn. Statement of the results Consider the quasilinear elliptic equation n ∑ i=1 ∂xiAi(x,∇u) = 0, (1) where Ai(x, ξ) : R → R, i = 1, . . . , n, are differentiable functions satisfying λ|η| ≤ ∑n i,j=1 ∂ξjAi(x, ξ)ηiηj ≤ λ−1|η|2, and ∑n i,j=1 ∂xjAi(x, ξ) ≤ C(1 + |ξ|), for every η ∈ R, and almost every x, ξ ∈ R. The sharp C regularity of weak solutions to (1) is one of the pillars on which the modern theory of quasilinear partial differential equations rests. The ideas on which its proof is based form a recurring theme in nonlinear analysis: first use difference quotients to prove that the weak solutions admit second (weak) derivatives, then differentiate the equations and observe that the derivatives of the solution are themselves solutions to some linear partial differential equations, whose coefficients are not very regular. At this point the regularity theory for linear equations with nonsmooth coefficients provides the final step in the proof of the Holder continuity of the gradient of weak solutions to (1). The observation that one could reduce the study of quasilinear equations to studying linear equations with “bad” coefficients goes back to the pioneering work of Morrey, and has been developed by many mathematicians in the last thirty years. Although (1) is a relatively simple elliptic equation, the regularity theorem has far-reaching applications in calculus of variations (see, for instance, [Gi]) and in the theory of quasiconformal mappings in space [G1]. Two natural and inter-related questions arise: Can the ellipticity hypothesis in (1) be somewhat weakened, and still expect regularity of the gradient of the solutions? Is this “new” problem of some geometric relevance? In this announcement we provide positive answers to both questions. Received by the editors March 15, 1996. 1991 Mathematics Subject Classification. Primary 35H05. Alfred P. Sloan Doctoral Dissertation Fellow. c ©1996 American Mathematical Society 60 OPTIMAL REGULARITY 61 In the proof of his famous rigidity theorem [Mo], Mostow introduced quasiconformal mappings in the setting of stratified nilpotent Lie groups [F]. With this name one refers to the class of simply connected Lie groups G endowed with a stratification of the Lie algebra g = V 1 ⊕ · · · ⊕ V , with r ≥ 1 (the step of the group), such that [V , V j ] = V , j = 1, . . . , r − 1, and [V , V ] = 0. The simplest example of a stratified Lie group is the Euclidean space R, with r = 1. A less trivial, and genuinely non-Euclidean example is provided by the Heisenberg group H, n ≥ 1, whose Lie algebra is h = R ⊕ R. The central role played by the Heisenberg group in many problems of complex geometry, representation theory and partial differential equations makes H the prototype par excellence of stratified nilpotent Lie groups. The theory of quasiconformal mappings between domains of the Heisenberg group has been developed recently in a series of papers by Koranyi and Reimann [KR1]–[KR3] and by Pansu [P]. As in the Euclidean case (see [G1] and [R]), this development led to various questions concerning the regularity of weak solutions to a class of quasilinear equations similar to (1). The notion of quasiconformality is a metric one, and in this setting it is related to the Carnot-Caratheodory metric associated to a basis X 1 , . . . , X 1 m, m = m =dim(V ) of V 1 (with our notation we do not distinguish between elements of g and left invariant vector fields). Since this metric is nonisotropic, it is natural to expect some nonisotropic structure in the relevant equations. In order to be more precise we need to recall that the exponential mapping exp : g → G is a diffeomorphism, and so we can use exponential coordinates p = (p1, . . . , p 1 m, p 2 1, . . . , p 2 dim(V 2), . . . ) on G. Let Xu = (X 1u, . . . ,X 1 mu) denote the horizontal gradient of the function u. Consider the equation