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Zbl 0802.35024
Capogna, Luca; Danielli, Donatella; Garofalo, Nicola
An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. (English)
[J] Commun. Partial Differ. Equations 18, No.9-10, 1765-1794 (1993). ISSN 0360-5302; ISSN 1532-4133/e

From the introduction: Let $X\sb 1, \dots, X\sb m$ be $C\sp \infty$ vector fields in $\bbfR\sp n$ satisfying Hörmander's condition for hypoellipticity: rank Lie $[X\sb 1, \dots, X\sb m] = n$ at every point $x \in\bbfR\sp n$. Denote by $X\sp*\sb j$ the formal adjoint of $X\sb j$. The linear operator ${\cal L} = \sum\sp m\sb{j = 1} X\sp*\sb j X\sb j$ is the subelliptic Laplacian associated to the vector fields $X\sb 1, \dots, X\sb m$.\par Given an open set $U \subset\bbfR\sp n$, and a function $u \in C\sp 1 (U)$, denote by $D\sb{\cal L} u = (X\sb 1u, \dots, X\sb mu)$ the subelliptic gradient of $u$. For $1 < p < \infty$ we consider the functional $$J\sb p(u) = \int\sb U \vert D\sb{\cal L} u \vert\sp p\ dx = \int\sb U \left[ \sum\sp m\sb{j = 1} (X\sb ju)\sp 2 \right]\sp{p/2} dx, $$ and define $S\sp{1,p} (U)$ to be the completion of $C\sp 1\sb 0 (U)$ in the norm generated by $J\sb p$. The Euler equation of $J\sb p$ is $$\sum\sp m\sb{j = 1}X\sp*\sb j \bigl( \vert D\sb{\cal L} u \vert\sp{p-2} X\sb ju \bigr) = 0. \tag 1 $$ We call the operator in (1) the subelliptic $p$-Laplacian. Critical points of $J\sb p$ are (weak) solutions of (1), and vice-versa.\par 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 $S\sp{1,p}$; b) To prove a Harnack type inequality for nonnegative solutions. From b) the Hölder continuity of solutions with respect to the $(X\sb 1, \dots, X\sb m)$-control distance will follow.
[J.Rákosník (Praha)]

MSC 2000:: *65H10 Systems of nonlinear equations (numerical methods)
35B45 A priori estimates
31C45 Nonlinear potential theory, etc.

Keywords: subelliptic $p$-Laplacian; optimal embedding result of Sobolev type; Harnack type inequality

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Zentralblatt MATH Berlin [Germany]