A symmetry result for cooperative elliptic systems with singularities

We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the solution may diverge. The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system

Il problema di Brezis-Nirenberg per operatori misti di tipo locale-nonlocale

In this note we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. We develop an existence theory, both in the case of linear and superlinear perturbations; moreover, in the particular case of linear perturbations we also investigate the mixed Sobolev inequality associated with this problem, detecting the optimal constant, which we show that is never achieved.In questa nota presentiamo alcuni risultati di esistenza, nello spirito del noto lavoro di Brezis e Nirenberg (CPAM, 1983), per un problema critico perturbato associato ad un operatore misto di tipo locale-nonlocale. I risultati presentati riguardano sia il caso di perturbazioni lineari, sia il caso di perturbazioni non lineari; nel caso particolare di perturbazioni lineari studiamo anche la disuguaglianza di tipo Sobolev associata al problema, individuandone la costante ottimale e mostrando che essa non è mai assunta

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In the present notes, based on a joint work with prof. E. Lanconelli, we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian(proved by Deny, Hayman and Kennedy) and to the sub-Laplacians on homogeneous Carnot groups (proved by Bonfiglioli and Lanconelli)

Non-divergence operators structured on homogeneous H\"{o}rmander vector fields: heat kernels and global Gaussian bounds

Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying H\"{o}rmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator $$\mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t}$$ where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded H\"{o}lder continuous functions on $\mathbb{R}^{1+n}$, with respect to the "parabolic" distance induced by the vector fields. We prove the existence of a global heat kernel $\Gamma(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,\alpha}(\mathbb{R}^{1+n}\setminus\{(s,y)\})$ for $\mathcal{H}$, such that $\Gamma$ satisfies two-sided Gaussian bounds and $\partial_{t}\Gamma, X_{i}\Gamma,X_{i}X_{j}\Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^n$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H}$, and a standard Harnack inequality for the corresponding stationary operator $$\mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j}.$$ with H\"{o}lder continuos coefficients