An embedding theorem and the harnack inequality for nonlinear subelliptic equations

Capogna, Luca; Danielli, Donatella; Garofalo, Nicola

doi:10.1080/03605309308820992

Cited by 187 publications

(189 citation statements)

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“…If M is a smooth manifold, dµ a smooth volume form, and X is a system of smooth vector fields satisfying Hörmander's finite rank condition rank(Lie{X})(x) = n at every point x ∈ M (see [29]), then the Poincaré inequality is due to Jerison [30] and the doubling condition was established by Nagel, Stein and Wainger in [41]. The PDE (1.4) is sub-elliptic and our results provide a (degenerate) parabolic analogue of the Harnack inequality established by Danielli, Garofalo and one of us in [7]. Theorem 1.1 also covers the case in which dµ can be expressed in local coordinates through a multiple of a smooth volume form times a Muckenhoupt A p weight with respect to the Carnot-Carathéodory metric generated by X.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 62%

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Avelin

Capogna

Citti

et al. 2014

Advances in Mathematics

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Abstract. We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototypewhere p ≥ 2, X = (X 1 , . . . , Xm) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure µ, and X * i denotes the adjoint of X i with respect to µ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M; (ii) a doubling condition for the µ-measure of d−metric balls and (iii) the validity of a Poincaré inequality involving X and µ. Our results extend the recent work in [15], [35], to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.2000 Mathematics Subject Classification.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 62%

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Avelin

Capogna

Citti

et al. 2014

Advances in Mathematics

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“…The imbedding for compactly supported functions is easier and has been obtained in Rotschild and Stein [62], Capogna, Danielli, and Garofalo [5], [6], Danielli [16], and Franchi, Gallot, and Wheeden [24]. Note that Sobolev inequality (11) implies the Poincaré inequality…”

Section: Lemma 24mentioning

confidence: 99%

“…We prove in Lemma 3.2 that such a duality inequality holds true also for general Hörmander vector fields (when there is no theory of Hardy spaces). The second goal is to contribute to the theory of nonlinear subelliptic equationsduring last decade, an area of intensive research; see, e.g., Buckley, Koskela and Lu [4], Capogna, Danielli and Garofalo [5], [7], Citti [11] [52], [53], [54], Vodop'yanov, [74], Vodop'yanov and Chernikov [75], Vodop'yanov and Markina [76], Xu [77], [78], and their references. (We did not mention here any papers concerned with the linear theory of subelliptic equations.…”

Section: Introductionmentioning

confidence: 99%

Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

Hajłasz¹,

Strzelecki²

1998

Mathematische Annalen

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“…The local Hölder regularity of weak solutions to (2) has been established independently in [X1] and [CDG1]. The boundary regularity was studied in [D].…”

Section: Consider the Quasilinear Elliptic Equationmentioning

confidence: 99%

Untitled

Capogna

1996

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We announce the optimal C 1+α regularity of the gradient of weak solutions to a class of quasilinear degenerate elliptic equations in nilpotent stratified Lie groups of step two. As a consequence we also prove a Liouville type theorem for 1-quasiconformal mappings between domains of the Heisenberg group H n . Statement of the resultsConsider the quasilinear elliptic equationwhere, for every η ∈ R n , and almost every x, ξ ∈ R n . The sharp C 1+α regularity of weak solutions to (1) is one of the pillars on which the modern theory of quasilinear partial differential equations rests. The ideas on which its proof is based form a recurring theme in nonlinear analysis: first use difference quotients to prove that the weak solutions admit second (weak) derivatives, then differentiate the equations and observe that the derivatives of the solution are themselves solutions to some linear partial differential equations, whose coefficients are not very regular. At this point the regularity theory for linear equations with nonsmooth coefficients provides the final step in the proof of the Hölder continuity of the gradient of weak solutions to (1). The observation that one could reduce the study of quasilinear equations to studying linear equations with "bad" coefficients goes back to the pioneering work of Morrey, and has been developed by many mathematicians in the last thirty years. Although (1) is a relatively simple elliptic equation, the regularity theorem has far-reaching applications in calculus of variations (see, for instance, [Gi]) and in the theory of quasiconformal mappings in space [G1].Two natural and inter-related questions arise: Can the ellipticity hypothesis in (1) be somewhat weakened, and still expect regularity of the gradient of the solutions? Is this "new" problem of some geometric relevance? In this announcement we provide positive answers to both questions.

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An embedding theorem and the harnack inequality for nonlinear subelliptic equations

Cited by 187 publications

References 4 publications

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Subelliptic p -harmonic maps into spheres and the ghost of Hardy spaces

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