Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

Capogna, Luca; Citti, Giovanna

doi:10.1007/s13373-015-0076-8

Cited by 19 publications

(22 citation statements)

References 65 publications

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“…We recall the results of Capogna and Han [14] for uniformly subelliptic operators, of Bramanti and Brandolini [5] for heat-type operators, and the results of Lunardi [30] and Gutiérrez and Lanconelli [24], which apply to a large class of squares of vector fields plus a drift term. Schauder estimates uniform in ǫ have been proved by the authors in [10] in the setting of Carnot Groups and by two of us in [7] in the setting of general Hörmander type vector fields.…”

Section: Stability Of Schauder Estimatesmentioning

confidence: 96%

“…We prove that weak solutions of such PDE satisfy a Harnack inequality and consequently obtain C 1,α interior estimates for the original solution u ǫ , which are uniform in ǫ > 0. At this point one rewrites the PDE in (1.6) in non-divergence form and invokes the stable Schauder estimates for subriemannian equations (see [10], [7]) to prove local higher regularity and long time existence. …”

Section: 2mentioning

confidence: 99%

“…In general we have sup ǫ>0 d ǫ (x, y) = d 0 (x, y) and it is well known that (G, d ǫ ) converges in the Gromov-Hausdorff sense as ǫ → 0 to the sub-Riemannian space (G, d 0 ). (See for instance [7,25] and references therein).…”

Section: Structure Stability In the Riemannian Limitmentioning

confidence: 99%

“…Stability of the homogenous structure as ǫ → 0. If we denote by dx the Haar measure in G, and by |Ω| the corresponding measure of a subset Ω ⊂ G, then Rea and two of the authors have proved in [7,11] that Proposition 2.1. There exist constants C, R > 0 independent of ǫ such that for every x ∈ G and R > r > 0,…”

Section: Structure Stability In the Riemannian Limitmentioning

confidence: 99%

“…Following [28,Chapter 12 where g = C 1 ( 2C 0 λw k + 1), for the constants λ, C 0 , C 1 from Lemma 4.3. In view of the weak Harnack inequality [7,Proposition 7.6], one has that for some constant C 4 > 0 independent of ǫ and for Q − ǫ (r) = {(x, t) ∈ Q| d ǫ (x, x 0 ) < r and t 0 − 3r 2 < t < t 0 < 2t 2 }, Applying Schauder estimates (see [10] for the Carnot groups setting and [7] for the general Lie group case). we immediately deduce the proof of Theorem 1.3.…”

Section: 2mentioning

confidence: 99%

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Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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Abstract. We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ǫ collapsing to a subRiemannian metric σ 0 as ǫ → 0. We establish C k,α estimates for this flow, that are uniform as ǫ → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in [10] to the total variation flow. IntroductionThe mean curvature flow is the motion of a surface where each points is moving in the direction of the normal with speed equal to the mean curvature In the case where the evolution of graphs S t = {(x, u(x, t))} ⊂ R n × R is considered, then, provided enough regularity is assumed, the function u satisfies the equationGiven appropriate boundary/initial conditions, global in time solutions asymptotically converge to minimal graphs.In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The so-called sub-Riemannian Hörmander type setting [22], [40]. In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ 0 ) that arises as limit of collapsing left-invariant tame Riemannian structures (G, σ ǫ ).Our approach to the existence of global (in time) smooth solutions is based on a Riemannian approximation scheme. We study graph solutions of the mean curvature flow in the Riemannian spaces (G, σ ǫ ) where G is a group and σ ǫ is a family of Riemannian metrics that 'collapse' as ǫ → 0 to a sub-Riemannian metric σ 0 in G.

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Section: Stability Of Schauder Estimatesmentioning

confidence: 96%

Section: 2mentioning

confidence: 99%

Section: Structure Stability In the Riemannian Limitmentioning

confidence: 99%

Section: Structure Stability In the Riemannian Limitmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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Left-invariant geometries on SU(2) are uniformly doubling

Eldredge¹,

Gordina²,

Saloff‐Coste³

2018

Geom. Funct. Anal.

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A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on SU(2). We show that left-invariant geometries on SU(2) are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.1991 Mathematics Subject Classification. Primary 53C21; Secondary 35K08, 53C17, 58J35, 58J60, 22C05, 22E30.

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Schauder estimates at the boundary for sub-laplacians in Carnot groups

2019

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In this paper we prove Schauder estimates at the boundary for sub-Laplacian type operators in Carnot groups. While internal Schauder estimates have been deeply studied, up to now subriemannian estimates at the boundary are known only in the Heisenberg groups. The proof of these estimates in the Heisenberg setting, due to Jerison ([32]), is based on the Fourier transform technique and can not be repeated in general Lie groups. After the result of Jerison no new contribution to the boundary problem has been provided. In this paper we introduce a new approach, which allows to build a Poisson kernel starting from the fundamental solution, from which we deduce the Schauder estimates at non characteristic boundary points.1991 Mathematics Subject Classification. 35R03, 35B65, 35J25.

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Regularity for subelliptic PDE through uniform estimates in multi-scale geometries

Cited by 19 publications

References 65 publications

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Left-invariant geometries on SU(2) are uniformly doubling

Schauder estimates at the boundary for sub-laplacians in Carnot groups

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