A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group. (English) Zbl 1381.35202
Summary: We prove that, if \(\mathbf E\) is the Engel group and \(u\) is a stable solution of \(\Delta_{\mathbf E}u = f(u)\), then
\[ \int_{\{\nabla_{\mathbf E}u\neq 0\}} \left[| \nabla_{\mathbf E}u|^2\left\{\left(p+\frac{\langle (Hu)^T\nu, v\rangle}{| \nabla_{\mathbf E}u|}\right)^2+h^2\right\}^2-\mathcal J\right] \,\eta^2\leq \int_{\mathbf E} | \nabla_{\mathbf E}\eta|^2 | \nabla_{\mathbf E}u|^2 \]
for any test function \(\eta\in C_0^\infty(\mathbf E)\). Here above, \(h\) is the horizontal mean curvature, \(p\) is the imaginary curvature and
\[ \mathcal J:= 2(X_3X_2uX_1u - X_3X_1uX_2u) + (X_4u)(X_1u - X_2u). \]
This can be interpreted as a geometric Poincaré inequality, extending the work of P. Sternberg and K. Zumbrun [J. Reine Angew. Math. 503, 63–85 (1998; Zbl 0967.53006), Arch. Ration. Mech. Anal. 141, 375–400 (1998; Zbl 0911.49025)] and F. Ferrari and the second author [Math. Ann. 343, No. 2, 351–370 (2009; Zbl 1173.35057)] to stratified groups of step 3. As an application, we provide a non-existence result.
\[ \int_{\{\nabla_{\mathbf E}u\neq 0\}} \left[| \nabla_{\mathbf E}u|^2\left\{\left(p+\frac{\langle (Hu)^T\nu, v\rangle}{| \nabla_{\mathbf E}u|}\right)^2+h^2\right\}^2-\mathcal J\right] \,\eta^2\leq \int_{\mathbf E} | \nabla_{\mathbf E}\eta|^2 | \nabla_{\mathbf E}u|^2 \]
for any test function \(\eta\in C_0^\infty(\mathbf E)\). Here above, \(h\) is the horizontal mean curvature, \(p\) is the imaginary curvature and
\[ \mathcal J:= 2(X_3X_2uX_1u - X_3X_1uX_2u) + (X_4u)(X_1u - X_2u). \]
This can be interpreted as a geometric Poincaré inequality, extending the work of P. Sternberg and K. Zumbrun [J. Reine Angew. Math. 503, 63–85 (1998; Zbl 0967.53006), Arch. Ration. Mech. Anal. 141, 375–400 (1998; Zbl 0911.49025)] and F. Ferrari and the second author [Math. Ann. 343, No. 2, 351–370 (2009; Zbl 1173.35057)] to stratified groups of step 3. As an application, we provide a non-existence result.
MSC:
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35H20 | Subelliptic equations |
| 35J61 | Semilinear elliptic equations |
