arXiv subjects
Publications and source records attributed to Nicola Garofalo.
The aim of this paper is to prove $Γ^{1,α}$ Schauder estimates near a $C^{1,α}$ non-characteristic portion of the boundary for $Γ^{0, α}$ perturbations of horizontal Laplaceans in Carnot groups. This situation of minimally smooth domains presents itself naturally in the study of subelliptic free boundary problems of obstacle type, see [15].
Two elementary and classical results about the Bessel quotient $y_ν= \frac{I_{ν+1}}{I_ν}$ state that on the half-line $(0,\infty)$ one has for $ν\ge -1/2$: \begin{itemize} \item[(i)] $0 < y_ν< 1$; \item[(ii)] $y_ν$ is strictly increasing. \end{itemize} In this paper we show that (i) and (ii) have some nontrivial and interesting applications to pde's. As a consequence of them, we establish some sharp new results for a class of degenerate partial differential equations of parabolic type in $\Rnp\times (0,\infty)$ which arise in connection with the analysis of the fractional heat operator $(\p_t - Δ)^s$ in $\Rn\times (0,\infty)$, see Theorems 1.2, 1.4, 1.5 and 1.7 below.
In this note I present some properties of sub-Laplaceans associated with a collection of smooth vector fields satisfying Hörmander's finite rank assumption. One notable aspect of the paper is the development of the fractional powers of sub-Laplaceans as Dirichlet-to-Neumann maps of an extension problem inspired to the famous 2007 work of Caffarelli and Silvestre for the standard Laplacean. A key tool is an extension problem for the fractional heat equation for which I compute the relevant Poisson kernel. I then use the latter to: 1) find the Poisson kernel for the time-independent case; and 2) solve the extension problem.
We study the strong unique continuation property backwards in time for the nonlocal equation in $\mathbb{R}^{n} \times \mathbb{R}$ \begin{equation}\label{one} (\partial_t - Δ)^{s} u = V(x,t)u \end{equation} for $s \in (0,1)$. Our main result Theorem 1.2 can be thought of as the nonlocal counterpart of the result obtained by Poon for the case when $s=1$. In order to prove Theorem 1.2 we develop the regularity theory of the extension problem for the equation above. With such theory in hands we establish: i) a basic monotonicity result for an adjusted frequency function which plays a central role in this paper, see Theorem 1.3; ii) an extensive blowup analysis of the so-called Almgren rescalings associated with the extension problem. We feel that our work will also be of interest e.g. in the study of certain basic open questions in free boundary problems, as well as in nonlocal segregation problems.
In this note we present some of the most basic aspects of the fractional Laplacean with a self-contained and purely didactic intent, and with a somewhat different slant from the several excellent existing references. Given the interest that nonlocal operators have generated since the 2007 extension paper of Caffarelli and Silvestre, we feel it is appropriate offering to young researchers a quick additional guide to the subject which, we hope, will nicely complement the existing ones.
We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.
We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc, as well as in a strip.
Based on a variant of the frequency function approach of Almgren, we establish an optimal bound on the vanishing order of solutions to stationary Schrödinger equations associated to a class of subelliptic equations with variable coefficients whose model is the so-called Baouendi-Grushin operator. Such bound provides a quantitative form of strong unique continuation that can be thought of as an analogue of the recent results of Bakri and Zhu for the standard Laplacian.
Based on a variant of the frequency function approach of Almgren, we establish an optimal upper bound on the vanishing order of solutions to variable coefficient Schrödinger equations at a portion of the boundary of a $C^{1,Dini}$ domain. Such bound provides a quantitative form of strong unique continuation at the boundary. It can be thought of as a boundary analogue of an interior result recently obtained by Bakri and Zhu for the standard Laplacian.
In a cylinder $D_T = Ω\times (0,T)$, where $Ω\subset \mathbb{R}^n$, we examine the relation between the $L$-caloric measure, $dω^{(x,t)}$, where $L$ is the heat operator associated with a system of vector fields of Hörmander type, and the measure $dσ_X\times dt$, where $dσ_X$ is the intrinsic $X$-perimeter measure. The latter constitutes the appropriate replacement for the standard surface measure on the boundary and plays a central role in sub-Riemannian geometric measure theory. Under suitable assumptions on the domain $Ω$ we establish the mutual absolute continuity of $dω^{(x,t)}$ and $dσ_X\times dt$. We also derive the solvability of the initial-Dirichlet problem for $L$ with boundary data in appropriate $ L^p$ spaces, for every $p>1$.
We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.
We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ α}$ domain is $H^{k+ α}$ up to the boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+α}$ if the domain is assumed to be space-time $C^{1, α}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DS1], and we closely follow the ideas in that paper. We also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary.
We continue the study of Modica type gradient estimates for non-homogeneous parabolic equations initiated in \cite{BG}. First, we show that for the parabolic minimal surface equation with a semilinear force term if a certain gradient estimate is satisfied at $t=0$, then it holds for all later times $t>0$. We then establish analogous results for reaction-diffusion equations such as \eqref{e0} below in $\Om \times [0, T]$, where $\Om$ is an epigraph such that the mean curvature of $\partial \Om$ is nonnegative. We then turn our attention to settings where such gradient estimates are valid without any a priori information on whether the estimate holds at some earlier time. Quite remarkably (see Theorem \ref{main3}, Theorem \ref{main5} and Theorem \ref{T:ricci}), this is is true for $\Rn \times (-\infty, 0]$ and $\Om \times (-\infty, 0]$, where $\Om$ is an epigraph satisfying the geometric assumption mentioned above, and for $M \times (-\infty,0]$, where $M$ is a connected, compact Riemannian manifold with nonnegative Ricci tensor. As a consequence of the gradient estimate \eqref{mo2}, we establish a rigidity result (see Theorem \ref{main6} below) for solutions to \eqref{e0} which is the analogue of Theorem 5.1 in \cite{CGS}. Finally, motivated by Theorem \ref{main6}, we close the paper by proposing a parabolic version of the famous conjecture of De Giorgi also known as the $\ve$-version of the Bernstein theorem.
In this paper we establish the $C^{1,β}$ regularity of the regular part of the free boundary in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is a continuation of the recent paper [GSVG14], where two of us established the interior optimal regularity of the solution. Two of the central results of the present work are a new monotonicity formula and a new epiperimetric inequality.
In this paper, we study an inhomogeneous variant of the normalized $p$-Laplacian evolution which has been recently treated in \cite{BG1}, \cite{Do}, \cite{MPR} and \cite{Ju}. We show that if the initial datum satisfies the pointwise gradient estimate \eqref{e:main1} a.e., then the unique solution to the Cauchy problem \eqref{main5} satisfies the same gradient estimate a.e. for all later times, see \eqref{e:main} below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation \eqref{main5} was first obtained in \cite{CGS}. Such estimate generalizes one obtained by L. Modica for the Laplacian, and it has connections to a famous conjecture of De Giorgi.
Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that $(M,d)$ be complete. We assume that $\M$ is also equipped with another first-order differential bilinear form $Γ^Z$ and we assume that $Γ$ and $Γ^Z$ satisfy the Hypothesis below. With these forms we introduce in \eqref{cdi} below a generalization of the curvature-dimension inequality from Riemannian geometry, see Definition \ref{D:cdi}. In our main results we prove that, using solely \eqref{cdi}, one can develop a theory which parallels the celebrated works of Yau, and Li-Yau on complete manifolds with Ricci bounded from below. We also obtain an analogue of the Bonnet-Myers theorem. In Section \ref{S:appendix} we construct large classes of sub-Riemannian manifolds with transverse symmetries which satisfy the generalized curvature-dimension inequality \eqref{cdi}. Such classes include all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is bounded from below, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is bounded from below.
In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in \cite{BG1} and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of \cite{BG1}.
Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is non negative.