On the Dirichlet Problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion

We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan

Lp{L^p}-Liouville Theorems for Invariant Partial Differential Operators in Rn{\mathbb{R}^n}

We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are given

Weighted Lp{L^p}-Liouville Theorems for Hypoelliptic Partial Differential Operators on Lie Groups

We prove weighted $L^p$-Liouville theorems for a class of second order hypoelliptic partial differential operators $\mathcal{L}$ on Lie groups $\mathbb{G}$ whose underlying manifold is $n$-dimensional space. We show that a natural weight is the right-invariant measure $\check{H}$ of $\mathbb{G}$. We also prove Liouville-type theorems for $C^2$ subsolutions in $L^p(\mathbb{G},\check{H})$. We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator $\mathcal{L}-\partial_t$

Lp - Liouville theorems for invariant evolution equations

Some Liouville-type theorems in Lebesgue spaces for several classes of evolution equations are presented. The involved operators are left invariant with respect to Lie group composition laws. Results for both solutions and sub-solutions are given

On the Dirichlet problem in cylindrical domains for evolution Ole\v{\i}nik--Radkevi\v{c} PDE's: a Tikhonov-type theorem

We consider the linear second order PDO's $$\mathscr{L} = \mathscr{L}_0 - \partial_t : = \sum_{i,j =1}^N \partial_{x_i}(a_{i,j} \partial_{x_j} ) - \sum_{j=i}^N b_j \partial_{x_j} - \partial _t,$$and assume that $\mathscr{L}_0$ has nonnegative characteristic form and satisfies the Ole\v{\i}nik--Radkevi\v{c} rank hypoellipticity condition. These hypotheses allow the construction of Perron-Wiener solutions of the Dirichlet problems for $\mathscr{L}$ and $\mathscr{L}_0$ on bounded open subsets of $\mathbb R^{N+1}$ and of $\mathbb R^{N}$, respectively. Our main result is the following Tikhonov-type theorem: Let $\mathcal{O}:= \Omega \times ]0, T[$ be a bounded cylindrical domain of $\mathbb R^{N+1}$, $\Omega \subset \mathbb R^{N},$ $x_0 \in \partial \Omega$ and $0 < t_0 < T.$ Then $z_0 = (x_0, t_0) \in \partial \mathcal{O}$ is $\mathscr{L}$-regular for $\mathcal{O}$ if and only if $x_0$ is $\mathscr{L}_0$-regular for $\Omega$. As an application, we derive a boundary regularity criterion for degenerate Ornstein--Uhlenbeck operators

On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$\partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \qquad (x,t) \in \mathbb{R}^N \times\, ]- \infty ,T[,$$ proved by a functional analytic approach based on Choquet theory. As a consequence, we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.Comment: The results of the present version recover most of the ones in the previous version, but, on top of it, this new version contains some further new and interesting result

Una base di insiemi risolutivi per l'equazione del calore: una costruzione elementare

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.Con un semplice espediente preso dalla teoria dei polinomi calorici, dimostriamo l'esistenza di una base della topologia euclidea i cui elementi sono insiemi risolutivi per l'equazione del calore. Questo risultato può essere utilizzato per costruire, con un approccio elementare, la soluzione di Perron del problema di Dirichlet calorico su arbitrari insiemi aperti limitati dello spazio-tempo euclideo

Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators

We consider non-negative solutions (Formula presented.) of second order hypoelliptic equations(Formula presented.) where \u3a9 is a bounded open subset of (Formula presented.) and x denotes the point of \u3a9. For any fixed x0 08 \u3a9, we prove a Harnack inequality of this type(Formula presented.) where K is any compact subset of the interior of the (Formula presented.)-propagation set ofx0 and the constant CK does not depend on u

Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type Theorem

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form

On the Perron solution of the caloric Dirichlet problem: an elementary approach

By an easy trick taken from caloric polynomial theory, we construct a family ℬ of almost regular domains for the caloric Dirichlet problem. ℬ is a basis of the Euclidean topology. This allows to build, with a basically elementary procedure, the Perron solution to the caloric Dirichlet problem on every bounded domain