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An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. (English) Zbl 0802.35024

From the introduction: Let X 1 ,,X m be C vector fields in n satisfying Hörmander’s condition for hypoellipticity: rank Lie [X 1 ,,X m ]=n at every point x n . Denote by X j * the formal adjoint of X j . The linear operator = j=1 m X j * X j is the subelliptic Laplacian associated to the vector fields X 1 ,,X m .

Given an open set U n , and a function uC 1 (U), denote by D u=(X 1 u,,X m u) the subelliptic gradient of u. For 1<p< we consider the functional

J p (u)= U |D u| p dx= U j=1 m (X j u) 2 p/2 dx,

and define S 1,p (U) to be the completion of C 0 1 (U) in the norm generated by J p . The Euler equation of J p is

j=1 m X j * | D u| p-2 X j u=0·(1)

We call the operator in (1) the subelliptic p-Laplacian. Critical points of J p are (weak) solutions of (1), and vice-versa.

In this paper, we propose to study a general class of nonlinear subelliptic equations, whose prototype is constituted by (1) above. Our objectives are: a) To establish an optimal embedding result of Sobolev type for the subelliptic spaces S 1,p ; b) To prove a Harnack type inequality for nonnegative solutions. From b) the Hölder continuity of solutions with respect to the (X 1 ,,X m )-control distance will follow.

MSC:
65H10Systems of nonlinear equations (numerical methods)
35B45A priori estimates for solutions of PDE
31C45Nonlinear potential theory, etc.