Quasilinear elliptic Hamilton–Jacobi equations on complete manifolds

Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron

doi:10.1016/j.crma.2013.06.007

Partial Differential Equations/Differential Geometry

Quasilinear elliptic Hamilton–Jacobi equations on complete manifolds

Presented by: Haim Brezis

Marie-Françoise Bidaut-Véron ¹; Marta Garcia-Huidobro ²; Laurent Véron ¹

¹ Laboratoire de mathématiques et physique théorique, CNRS UMR 7350, faculté des sciences, 37200 Tours, France
² Departamento de Matematicas, Pontifica Universidad Catolica de Chile, Casilla 307, Correo 2, Santiago de Chile, Chile

Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 445-449.

Abstract
Résumé

Let $(M^{n}, g)$ be an n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor ${Ricc}_{g}$ and sectional curvature ${Sec}_{g}$ . Assume ${Ricc}_{g} ⩾ (1 - n) B^{2}$ , and either $p > 2$ and ${Sec}_{g} (x) = o ({dist}^{2} (x, a))$ when ${dist}^{2} (x, a) \to \infty$ for $a \in M$ , or $1 < p < 2$ and ${Sec}_{g} (x) ⩽ 0$ . If $q > p - 1 > 0$ , any $C^{1}$ solution of (E) $- Δ_{p} u + {| \nabla u |}^{q} = 0$ on M satisfies $| \nabla u (x) | ⩽ c_{n, p, q} B^{\frac{1}{q + 1 - p}}$ for some constant $c_{n, p, q} > 0$ . As a consequence, there exists $c_{n, p} > 0$ such that any positive p-harmonic function v on M satisfies $v (a) e^{- c_{n, p} B dist (x, a)} ⩽ v (x) ⩽ v (a) e^{c_{n, p} B dist (x, a)}$ for any $(a, x) \in M \times M$ .

Soit $(M^{n}, g)$ une variété riemannienne n-dimensionnelle complète, non compacte et connexe de courbures de Ricci ${Ricc}_{g}$ et sectionnelle ${Sec}_{g}$ . On suppose ${Ricc}_{g} ⩾ (1 - n) B^{2}$ et ${Sec}_{g} (x) = o ({dist}^{2} (x, a))$ si ${dist}^{2} (x, a) \to \infty$ pour $a \in M$ si $p > 2$ , ou ${Sec}_{g} (x) ⩽ 0$ si $1 < p < 2$ . Si $q > p - 1 > 0$ , toute solution de classe $C^{1}$ de (E) $- Δ_{p} u + {| \nabla u |}^{q} = 0$ sur M satisfait à $| \nabla u (x) | ⩽ c_{n, p, q} B^{\frac{1}{q + 1 - p}}$ , où $c_{n, p, q} > 0$ est une constante. On en déduit quʼil existe $c_{n, p} > 0$ tel que toute fonction p-harmonique positive v sur M satisfait à lʼencadrement suivant : $v (a) e^{- c_{n, p} B dist (x, a)} ⩽ v (x) ⩽ v (a) e^{c_{n, p} B dist (x, a)}$ pour tout $(a, x) \in M \times M$ .

Received: 2013-06-04
Accepted: 2013-06-24
Published online: 2013-06-01

DOI: 10.1016/j.crma.2013.06.007

Author's affiliations:

Marie-Françoise Bidaut-Véron ¹; Marta Garcia-Huidobro ²; Laurent Véron ¹

¹ Laboratoire de mathématiques et physique théorique, CNRS UMR 7350, faculté des sciences, 37200 Tours, France
² Departamento de Matematicas, Pontifica Universidad Catolica de Chile, Casilla 307, Correo 2, Santiago de Chile, Chile

@article{CRMATH_2013__351_11-12_445_0,
     author = {Marie-Fran\c{c}oise Bidaut-V\'eron and Marta Garcia-Huidobro and Laurent V\'eron},
     title = {Quasilinear elliptic {Hamilton{\textendash}Jacobi} equations on complete manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {445--449},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     year = {2013},
     doi = {10.1016/j.crma.2013.06.007},
     language = {en},
}

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Marie-Françoise Bidaut-Véron; Marta Garcia-Huidobro; Laurent Véron. Quasilinear elliptic Hamilton–Jacobi equations on complete manifolds. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 445-449. doi : 10.1016/j.crma.2013.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.06.007/

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