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We prove a strong comparison principle for the solution of the Levi equation $L\left(u\right)=\sum {}_{i=1}{}^n\left(\right(1+u{}_t{}^2\left)\right(u{}_{x_i{x}_i}+u{}_{y_i{y}_i})+(u{}_{x_i}{}^2+u{}_{y_i}{}^2)u{}_{tt}+2(u{}_{y_i}-u{}_{x_i}u{}_t)u{}_{x_{i}t}-2(u{}_{x_i}+u{}_{y_i}u{}_t)u{}_{y_{i}t}+k(x,y,t\left)\right(1+\left|Du\right|{}^2){}^{3/2}=0$, applying Bony Propagation Principle.

In this paper we study heat kernels associated with a Carnot group G, endowed with a family of collapsing left-invariant Riemannian metrics σε which converge in the Gromov- Hausdorff sense to a sub-Riemannian structure on G as ε→ 0. The main new contribution are Gaussian-type bounds on the heat kernel for the σε metrics which are stable as ε→0 and extend the previous time-independent estimates in [16]. As an application we study well posedness of the total variation flow of graph surfaces over a...

In this paper a $C^{\mathrm{\infty }}$-regularity result for the strong viscosity solutions to the prescribed Levi-curvature equation is announced. As an application, starting from a result by Z. Slodkowski and G. Tomassini, the $C^{\mathrm{\infty }}$-solvability of the Dirichlet problem related to the same equation is showed.