The Neumann problem for Sub-Laplacians on Carnot groups and the extension theorem for Sobolev spaces

Abstract

As witnessed by papers [1], [25], [14], the recent results [4], [5] and several other works, boundary value problems on Carnot groups (and beyond) are both challenging and difficult. The purpose of this paper is to prove the existence (and uniqueness) of a variational solution to the Neumann boundary problem for Sub-Laplacians on Carnot groups. According to the classical approach to this problem, the extension and trace theorems for Sobolev spaces are indispensable tools. Therefore, we first develop an extension theorem for the Folland–Stein nonisotropic Sobolev spaces Lk,p on Carnot groups and apply it to prove the existence of a variational solution to the Neumann boundary problem while the trace theorem has been developed in a separate work [13]. Aside from having important applications in pdes, in view of the recent fundamental lifting theorem of Rothschild and Stein [38], the study of the extension theorem on Carnot groups is an important step in trying to understand the analogous problem for systems of smooth vector fields satisfying Hormander’s finite rank condition [24]. This general open question is another motivation behind the present paper. The application of the Sobolev extension theorem and (approximation by smooth functions up to the boundary for) the study of boundary value problems goes beyond the scope of the present paper, see for instance (in the classical setting) [29], [15]. The first basic step in the study of the Neumann problem for Sub-Laplacians is to establish the existence of a variational solution. Following the classical development, to accomplish this goal one needs to develop extension and