arXiv subjects

Isidro H. Munive

Publications and source records attributed to Isidro H. Munive.

9 recordsLinked to original sources

arXiv2024

Higher order Boundary Schauder Estimates in Carnot Groups

In his seminal 1981 study D. Jerison showed the remarkable negative phenomenon that there exist, in general, no Schauder estimates near the characteristic boundary in the Heisenberg group $\mathbb H^n$. On the positive side, by adapting tools from Fourier and microlocal analysis, he developed a Schauder theory at a non-characteristic portion of the boundary, based on the non-isotropic Folland-Stein Hölder classes. On the other hand, the 1976 celebrated work of Rothschild and Stein on their lifting theorem established the central position of stratified nilpotent Lie groups (nowadays known as Carnot groups) in the analysis of Hörmander operators but, to present date, there exists no known counterpart of Jerison's results in these sub-Riemannian ambients. In this paper we fill this gap. We prove optimal $Γ^{k,α}$ ($k\geq 2$) Schauder estimates near a $C^{k,α}$ non-characteristic portion of the boundary for $Γ^{k-2, α}$ perturbations of horizontal Laplacians in Carnot groups.

math.AP↗

arXiv2023

Multiplicity of 2-nodal solutions the Yamabe equation

Given any closed Riemannian manifold $(M, g)$, we use the gradient flow method and Sign-Changing Critical Point Theory to prove multiplicity results for 2-nodal solutions of a subcritical Yamabe type equation on $(M, g)$. If $(N, h)$ is a closed Riemannian manifold of constant positive scalar curvature our result gives multiplicity results for the type Yamabe equation on the Riemannian product $(M x N, g + εh)$, for $ε> 0$ small.

math.AP↗

arXiv2022

Borderline gradient continuity for the normalized $p$-parabolic operator

In this paper, we prove gradient continuity estimates for viscosity solutions to $Δ_{p}^N u- u_t= f$ in terms of the scaling critical $L(n+2,1 )$ norm of $f$, where $Δ_{p}^N$ is the game theoretic normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.5 constitutes borderline gradient continuity estimate for $u$ in terms of the modified parabolic Riesz potential $\mathbf{P}^{f}_{n+1}$ as defined in (2.8) below. Moreover, for $f \in L^{m}$ with $m>n+2$, we also obtain Hölder continuity of the spatial gradient of the solution $u$, see Theorem 2.6 below. This improves the gradient Hölder continuity result in [3] which considers bounded $f$. Our main results Theorem 2.5 and Theorem 2.6 are parabolic analogues of those in [9]. Moreover differently from that in [3], our approach is independent of the Ishii-Lions method which is crucially used in [3] to obtain Lipschitz estimates for homogeneous perturbed equations as an intermediate step.

math.AP↗

arXiv2020

Solutions of the Yamabe Equation By Lyapunov-Schmidt Reduction

Given any closed Riemannian manifold $(M,g)$ we use the Lyapunov-Schmidt finite-dimensional reduction method and the classical Morse and Lusternick-Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on $(M,g)$. If $(N,h)$ is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product $(M\times N , g + \ve^2 h )$, for $\ve >0$ small. For example, if $M$ is a closed Riemann surface of genus ${\bf g}$ and $(N,h) = (S^2 , g_0)$ is the round 2-sphere, we prove that for $\ve >0$ small enough and a generic metric $g$ on $M$, the Yamabe equation on $(M\times S^2 , g + \ve^2 g_0 )$ has at least $2 + 2 {\bf g}$ solutions.

math.AP↗

arXiv2019

The Harnack inequality for a class of nonlocal parabolic equations

In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.

math.AP↗

arXiv2019

Gradient continuity estimates for the normalized $p$-Poisson equation

In this paper, we obtain gradient continuity estimates for viscosity solutions of $Δ_{p}^N u= f$ in terms of the scaling critical $L(n,1 )$ norm of $f$, where $Δ_{p}^N$ is the normalized $p-$Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potential $\tilde I^{f}_{q}$. Moreover, for $f \in L^{m}$ with $m>n$, we also obtain $C^{1,α}$ estimates, see Theorem 2.3 below. This improves one of the regularity results in [3], where a $C^{1,α}$ estimate was established depending on the $L^{m}$ norm of $f$ under the additional restriction that $p>2$ and $m > \text{max} (2,n, \frac{p}{2}) $ (see Theorem 1.2 in [3]). We also mention that differently from the approach in [3], which uses methods from divergence form theory and nonlinear potential theory in the proof of Theorem 1.2, our method is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [36]. Moreover, for $f$ continuous, our approach also gives a somewhat different proof of the $C^{1, α}$ regularity result, Theorem 1.1, in [3].

math.AP↗

arXiv2017

Sub-Riemannian curvature of Carnot groups with rank-two distributions

The notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in arXiv:1306.5318, and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler, and sub-Finsler structures. In this work we study the generalized sectional curvature of Carnot groups with rank-two distributions. In particular, we consider the Cartan group and Carnot groups with horizontal distribution of Goursat-type. In these Carnot groups we characterize ample and equiregular geodesics. For Carnot groups with horizontal Goursat distribution we show that their generalized sectional curvatures depend only on the Engel part of the distribution. This family of Carnot groups contains naturally the three-dimensional Heisenberg group, as well as the Engel group. Moreover, we also show that in the Engel and Cartan groups there exist initial covectors for which there is an infinite discrete set of times at which the corresponding ample geodesics are not equiregular.

math.DG↗

arXiv2016

Estimates of the Green function and the initial-Dirichlet problem for the heat equation in sub-Riemannian spaces

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 $p>1$.

math.AP↗

arXiv2014

Volume and distance comparison theorems for sub-Riemannian manifolds

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}.

math.DG↗