Search

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x → Ax of the Euclidean case must be replaced by suitable group isomorphisms x → T_A(x), whose differential preserves the first layer of the stratification of Lie(G).

In this article, we describe the pioneering work of Bruno Pini toward the modern Potential Analysis of second order parabolic Partial Differential Equations. We mainly focus on the parabolic Harnack inequality, discovered by Pini in 1954, jointly, and independently, with Jacques Hadamard. Pini made of this inequality one the crucial tools in his construction of a generalized Wiener-type solution to the Dirichlet problem for the Heat equation. To this end, he also used an average operator on the level sets of the Heat kernel, characterizing caloric and sub-caloric functions, in analogy with the classical Gauss–Koebe, Blaschke–Privaloff and Saks Theorems for harmonic and sub-harmonic functions. He also proved a Riesz-type representation Theorem for subcaloric functions. To complete his research design, Pini established the notion of Heat capacity, and proved a Wiener-type criterion for regularity of the boundary points in the generalized Dirichlet problem for the Heat equation.

In this work the authors deal with linear second order partial differential operators of the following type $H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)$ where $X_{1},X_{2},\ldots,X_{q}$ is a system of real Hormander's vector fields in some bounded domain $\Omega\subseteq\mathbb{R}^{n}$, $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that $\lambda^{-1}\vert\xi\vert^{2}\leq\sum_{i,j=1}^{q}a_{ij}(t,x) \xi_{i}\xi_{j}\leq\lambda\vert\xi\vert^{2}\text{}\forall\xi\in\mathbb{R}^{q}, x \in\Omega,t\in(T_{1},T_{2})$ for a suitable constant $\lambda>0$ a for some real numbers $T_{1} < T_{2}$. Table of Contents: Introduction. Part I: Operators with constant coefficients: Overview of Part I; Global extension of Hormander's vector fields and geometric properties of the CC-distance; Global extension of the operator $H_{A}$ and existence of a fundamental solution; Uniform Gevray estimates and upper bounds of fundamental solutions for large $d\left(x,y\right)$; Fractional integrals and uniform $L^{2}$ bounds of fundamental solutions for large $d\left(x,y\right)$; Uniform global upper bounds for fundamental solutions; Uniform lower bounds for fundamental solutions; Uniform upper bounds for the derivatives of the fundamental solutions; Uniform upper bounds on the difference of the fundamental solutions of two operators. Part II: Fundamental solution for operators with Holder continuous coefficients: Assumptions, main results and overview of Part II; Fundamental solution for $H$: the Levi method; The Cauchy problem; Lower bounds for fundamental solutions; Regularity results. Part III: Harnack inequality for operators with Holder continuous coefficients: Overview of Part III; Green function for operators with smooth coefficients on regular domains; Harnack inequality for operators with smooth coefficients; Harnack inequality in the non-smooth case; Epilogue; References. (MEMO/204/961)

We consider a class of degenerate Ornstein–Uhlenbeck operators in $${\mathbb{R}^{N}}$$ , of the kind $$\mathcal{A}\equiv\sum_{i, j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} + \sum_{i, j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}$$ where (a ij ), (b ij ) are constant matrices, (a ij ) is symmetric positive definite on $${\mathbb{R} ^{p_{0}}}$$ (p 0 ≤ N), and (b ij ) is such that $${\mathcal{A}}$$ is hypoelliptic. For this class of operators we prove global L p estimates (1

We establish a covering lemma of Besicovitch type for metric balls in the setting of Holder quasimetric spaces of homogenous type and use it to prove a covering theorem for measurable sets. For families of measurable functions, we introduce the notions of power decay, critical density and double ball property and with the aid of the covering theorem we show how these notions are related. Next we present an axiomatic procedure to establish Harnack inequality that permits to handle both divergence and non divergence linear equations.

We study a notion of link of Lie groups suggested by the structure of the partial differential operators of Kolmogorov type. As an application of our link procedure we construct explicit examples of stratified Lie groups, with dimension and step arbitrarily large. We also give a set of examples of hypoelliptic second-order operators which are left translation invariant and homogeneous of degree two on the previous groups.

We introduce a new class of curvature PDO's describing relevant properties of real hypersurfaces of $C^{n+1}.$ In our setting the pseudoconvexity and the Levi form play the same role as the convexity and the real Hessian matrix play in the real Euclidean one. Our curvature operators are second order fully nonlinear PDO's not elliptic at any point. However, when computed on generalized $s$-pseudoconvex functions, we shall show that their characteristic form is nonnegative definite with kernel of dimension one. Moreover, we shall show that the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurfaces. These properties allow us to prove a strong comparison principle, leading to symmetry theorems for domains with constant curvatures and to identification results for domains with comparable curvatures.

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related “stationary” operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.

In this paper we establish a uniform Harnack inequality for a class of degenerate equations whose prototype is Kolmogorov’s equations in R 3 : D yy u − y D z u − D t u = 0 {{\mathbf {R}}^3}:{D_{{\text {yy}}}}u - {\text {y}}{D_z}u - {D_t}u = 0 . Our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution.

In the first part of this paper we study some differentiabilityperties properties (In Sobolev space setting)of functions belonging to thespace generated by the vector fields in aJ J Dx.connected open subset of Rn ; here the λjs are continuous (but in generalnot smooth) and nonnegative. In particular, under suitable geometricalhypotheses involving the integral curves of we get thenfollowing embedding estimate: for all test functions u and for suitable positive constants e and CP depending only on the "order of degeneration" of the vector fields Such a problem arises naturally in study of the pointewise estimates for the weak solutions of some degenerate elliptic equations with measurable coefficients.

We consider Dirichlet problem for a class of parabolic operators with discontinuous coefficients having the form $$Lu = \sum\limits_{i = 1}^n {\frac{\partial }{{\partial x_i }}\left[ {\sum\limits_{j = 1}^n {a_{ij} (z)\frac{{\partial u}}{{\partial x_j }}} + a_i (z)u} \right]} + \sum\limits_{i = 1}^n {b_i (z)} \frac{{\partial u}}{{\partial x_i }} + c(z)u - \frac{{\partial u}}{{\partial y}}$$ . We prove some necessary and sufficient conditions for a boundary point to be regular for every operator of this kind.

In a bounded set in R n + 1 {{\mathbf {R}}^{n + 1}} we study the problem of the regularity of boundary points for the Dirichlet problem for a parabolic operator with smooth coefficients. We give a geometric characterization, modelled on Wiener’s criterion for Laplace’s equation, of those boundary points that are regular. We also present some important consequences. Here is the main one: a point is regular for a variable coefficient operator if and only if it is regular for the constant coefficient operator obtained by freezing the coefficients at that point.

Let L 1 and L 2 be linear, second order, parabolic operators. We prove some necessary and some sufficient conditions for an L 1 -regular boundary point to be L 2 -regular. In particular we prove the following results: every L 1 -regular boundary points is L 2 -regular, and viceversa, (if and) only if L 1 =L 2 .

Dans R n+1 on considere l'equation parabolique d'ordre 2 Lu=div(Acx, t) ⊇ x u)−D t u=0 ou A(x, t)=(a ij (x, t)) est une fonction matricielle symetrique reelle sur R n+1 a entrees C ∞

Nella presente nota dimostriamo alcuni criteri di regolarita dei punti di frontiera per il primo problema di valori al contorno relativo all'equazione del calore.

Sia P(s, σ)=s2n+σ2n+s2mσ2m con m ed n numeri naturali tali che m

We report on some Liouville-type theorems for a class of linear second-order partial differential equation with nonnegative characteristic form. The theorems we show improve our previous results.

In this note we study the existence and the uniqueness of radially symmetric ground states for the quasilinear elliptic equation $$div\left( {A\left( {|{D_u}|} \right){D_u}} \right) + f(u) = 0$$ (1.1) . Here, by a ground state we mean a nonnegative classical solution of (1.1) on ℝ n , vanishing at infinity but not identically zero. Our results improve those stated in [4]. Complete proofs will appear elsewhere [5].

Let us consider the semilinear equation in ℝ n $$\Delta u + f(u) = 0$$ (1) . In [3], Gidas, Ni and Nirenberg proved that if f(u) = -u + g(u) near u = 0, with, roughly speaking, g(u) = O(u 1+α ) for given α > 0, then the ground states of (1.a) are necessarily spherically symmetric about some point. Throughout this note, a ground state will be a positive, twice differentiate solution of the given equation in ℝ n vanishing at the infinity.

In (1), (2), ~=(x , y, t) denotes the point of R 3, ut is the first derivative of u with respect to t, and analogous notations are used for the other firstand second-order derivatives of u. The notion of Levi curvature for a real manifold was introduced by E.E. Levi in 1909 in order to characterize the holomorphy domains of C 2. Since then, it has played a crucial role in the geometric theory of several complex variables. In looking for the polynomial hull of a graph, Slodkowski and Tomassini implicitly introduced in 1991 the following definition of Levi curvature for Lipschitz-continuous graphs [16].

Papers from the meeting held in Cortona, June 12--17, 2006; We consider homogenization problems with holes for strongly local Dirichlet forms in the cases of Dirichlet and Neumann homogeneous conditions on the boundaries of holes. The main difficulties arise from the absence of a group structure on the underlying space, then from the non-periodic distribution of the holes and from the absence of classical extension results in the holes. The complete proofs of the results will appear in a forthcoming paper

We are concerned with some notions of curvatures associated with pseudoconvexity and the Levi form as the classical Gauss and Mean curvatures are related to the convexity and to the Hessian matrix. In particular, given a prescribed non negative function K, the Levi Monge Ampère equation for the graph of a function is where is the Levi form of the graph u and D u is the Euclidean gradient of u. More generally, we shall consider elementary symmetric functions of the eigenvalues of the Levi form and we shall first show that these curvature equations contain information about the geometric feature of a closed hypersurface. Then, we shall show that the curvature operators lead to a new class of second order fully nonlinear equations whose characteristic form, when computed on generalized pseudoconvex functions, is nonnegative definite with kernel of dimension one. Thus, the equations are not elliptic at any point. However, they have the following redeeming feature: the missing ellipticity direction can be recovered by suitable commutation relations. We shall use this property to study existence, uniqueness and regularity of viscosity solutions of the Dirichlet problem for graphs with prescribed Levi curvature.

The aim of this work is to extend a result by Suzuki and Watson concerning an inverse property for caloric functions. Our result applies, in particular, to the heat operator on stratified Lie groups and to Kolmogorov-Fokker-Planck-type operators. We show that the open sets characterizing the solutions to the involved equations, in terms of suitable average operators, have to be the level sets of the fundamental solutions of the relevant operators. The technique adopted exploits the structure of the propagation sets, i.e., the sets where the solutions to the involved equations attain their maximum.