Annales Academię Scientiarum Fennicę
Mathematica
Volumen 37, 2012, 357-373

A GEOMETRIC INEQUALITY FOR STABLE SOLUTIONS OF SEMILINEAR ELLIPTIC PROBLEMS IN THE ENGEL GROUP

Andrea Pinamonti and Enrico Valdinoci

Universitą di Trento, Dipartimento di Matematica
Via Sommarive 14, I-38123 Povo (TN), Italy; pinamonti 'at' science.unitn.it

Universitą di Milano, Dipartimento di Matematica
Via Festa del Perdono 7, I-20122 Milano, Italy; enricovaldinoci 'at' gmail.com

Abstract. We prove that, if E is the Engel group and u is a stable solution of \Delta_Eu = f(u), then

$$\int_{\{\nabla_{\mathbf{E}} u\ne0\}} \left[ |\nabla_{\mathbf{E}} u|^2\Bigg\{\Bigg( p+\frac{\left\langle (Hu)^T\nu, v\right\rangle}{|\nabla_{\mathbf{E}} u|}\Bigg)^2 +h^2\Bigg\}-{\mathcal{J}} \right]\eta^2\leq \int_{\E}|\nabla_{\mathbf{E}}\eta|^2|\nabla_{\mathbf{E}} u|^2$$

for any test function eta \in C₀^\infty(E). Here above, h is the horizontal mean curvature, p is the imaginary curvature and

J := 2(X₃X₂uX₁u - X₃X₁uX₂u) + (X₄u)(X₁u - X₂u)

This can be interpreted as a geometric Poincaré inequality, extending the work of [21, 22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.

2010 Mathematics Subject Classification: Primary 35H20, 35B06, 20F45.

Key words: Rigidity property, symmetry, non-existence results.

Reference to this article: A. Pinamonti and E. Valdinoci: A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group. Ann. Acad. Sci. Fenn. Math. 37 (2012), 357-373.

Full document as PDF file

doi:10.5186/aasfm.2012.3733

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