Partial Differential Equations/Differential Geometry
Quasilinear elliptic Hamilton–Jacobi equations on complete manifolds
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 445-449.

Let (Mn,g) be an n-dimensional complete, non-compact and connected Riemannian manifold, with Ricci tensor Riccg and sectional curvature Secg. Assume Riccg(1n)B2, and either p>2 and Secg(x)=o(dist2(x,a)) when dist2(x,a) for aM, or 1<p<2 and Secg(x)0. If q>p1>0, any C1 solution of (E) Δpu+|u|q=0 on M satisfies |u(x)|cn,p,qB1q+1p for some constant cn,p,q>0. As a consequence, there exists cn,p>0 such that any positive p-harmonic function v on M satisfies v(a)ecn,pBdist(x,a)v(x)v(a)ecn,pBdist(x,a) for any (a,x)M×M.

Soit (Mn,g) une variété riemannienne n-dimensionnelle complète, non compacte et connexe de courbures de Ricci Riccg et sectionnelle Secg. On suppose Riccg(1n)B2 et Secg(x)=o(dist2(x,a)) si dist2(x,a) pour aM si p>2, ou Secg(x)0 si 1<p<2. Si q>p1>0, toute solution de classe C1 de (E) Δpu+|u|q=0 sur M satisfait à |u(x)|cn,p,qB1q+1p, où cn,p,q>0 est une constante. On en déduit quʼil existe cn,p>0 tel que toute fonction p-harmonique positive v sur M satisfait à lʼencadrement suivant : v(a)ecn,pBdist(x,a)v(x)v(a)ecn,pBdist(x,a) pour tout (a,x)M×M.

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Published online:
DOI: 10.1016/j.crma.2013.06.007
Bidaut-Véron, Marie-Françoise 1; Garcia-Huidobro, Marta 2; Véron, Laurent 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
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Bidaut-Véron, Marie-Françoise; Garcia-Huidobro, Marta; Véron, Laurent. 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. http://www.numdam.org/articles/10.1016/j.crma.2013.06.007/

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