The main purpose of this work is to obtain a Harnack inequality and estimates for the Green function for the general class of degenerate elliptic operators described below.

We prove Harnack's inequality for non-negative solutions of some degenerate elliptic operators in divergence form with the lower order term coefficients satisfying a Kato type contition.

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ in the complex tangent space. Their real and imaginary parts span a $2n$-dimensional subspace of the real tangent space, which has dimension $2n+1.$ If the Levi matrix of $M$ is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...

We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation ${\sum }_{i=1}^m\frac{\partial }{\partial x_i}{a}_i(x,u,\nabla u)-c_0{\left|u\right|}^{p-2}u=f(x,u,\nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda \left(\right|u\left|\right){\sum }_{i=1}^m{a}_i(x,u,\eta ){\eta }_i\ge \nu \left(x\right){\left|\eta \right|}^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.

In this paper, we are concerned with the following problem: given a set of smooth vector fields $X_1,\mathrm{\dots },X_m$ on ${\mathbb{R}}^N$, we ask whether there exists a homogeneous Carnot group $\mathbb{G}=(\mathbb{R}{}^N,\circ ,\delta {}_{\lambda })$ such that ${\sum }_i{X}_i^2$ is a sub-Laplacian on $\mathbb{G}$. We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we exhibit several...

First we recall a Faber-Krahn type inequality and an estimate for ${\lambda }_p\left(\Omega \right)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue ${\lambda }_p\left(\Omega \right)$ converges to the Cheeger constant $h\left(\Omega \right)$ as $p\to 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega$ which minimizes the ratio $\left|\partial D\right|/\left|D\right|$ among all simply connected $D\subset \subset \Omega$. As a byproduct we prove that for convex $\Omega$ the Cheeger set $\omega$ is also convex.

Let Ω be an open subset of ${ℝ}^d$ with 0 ∈ Ω. Furthermore, let $H_{\Omega }=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ be a second-order partial differential operator with domain $C_c^{\infty }\left(\Omega \right)$ where the coefficients $c_{ij}\in W_{loc}^{1,\infty }\left(\Omega ̅\right)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=\left(c_{ij}\right)$ satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ∈ Ω. If ${\int }_0^{\infty }ds{s}^{d/2}{e}^{-\lambda \mu \left(s\right)²}<\infty$ for some λ > 0 where $\mu \left(s\right)={\int }_0^{s}dtc{\left(t\right)}^{-1/2}$ then we establish that $H_{\Omega }$ is L₁-unique, i.e. it has a unique L₁-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L₂-extension which generates a submarkovian semigroup. Moreover...