no. 7
The Inhomogeneous Dirichlet problem for natural operators on manifolds
Harvey, F. Reese1; Lawson, H. Blaine Jr2
1 Department of Mathematics RICE University Houston, TX 77005-1982 (USA)
2 Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651 (USA)
Annales de l'Institut Fourier, Riemannian Geometry. Past, Present and Future an homage to Marcel Berger December 6–9, 2017, IHES, Bures-sur-Yvette, Volume 69 (2019) no. 7, pp. 3017-3064.

We discuss the inhomogeneous Dirichlet problem written locally as:

f(x,u,Du,D2u)=ψ(x)

where f is a “natural” differential operator on a manifold X, with a restricted domain F in the space of 2-jets. “Naturality” refers to operators that arise intrinsically from a given geometry on X. Importantly, the equation need not be convex and can be highly degenerate. Furthermore, ψ can take the values of f on ∂F.

A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption.

The main theorem covers many geometric equations, for example: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex and symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established.

There are also results where ψ is a delta function.

Il s’agit du problème de Dirichlet inhomogène  :

f(x,u,Du,D2u)=ψ(x)

sur une variété X où f est un opérateur différentiel « naturel » sur un domaine F dans l’espace de 2-jets. Des opérateurs naturels viennent intrinsèquement d’une géometrie donnée sur X. Un point important est que l’équation n’est pas nécessairement convexe et pourrait être très dégénérée. De plus, les valeurs de ψ peuvent toucher f(∂F).

Le nouvel outil principal est l’idée de jet-équivalence locale qui donne une comparaison faible locale, puis une comparaison sous conditions nécessaires globales.

Le théorème principal s’applique à plusieurs équations géometriques, par exemple  : des opérateurs invariants orthogonalement sur une variété riemannienne, des opérateurs G-invariants sur une G-variété, des opérateurs sur une variété quasi-complexe ou symplectique. Il s’applique aussi à toutes les branches de ces équations. Des résultats d’existence et d’unicité sont établis.

Il y a aussi des résultats lorsque ψ est une fonction delta.

Published online:
DOI: 10.5802/aif.3344
Classification: 35A99, 53C15, 53C38
Keywords: Inhomogenous Dirichlet Problem, Geometric Operators on Manifolds
Mots-clés : Problème de Dirichlet inhomogène, Opérateurs géometriques sur les variétés

Harvey, F. Reese 1; Lawson, H. Blaine Jr 2

1 Department of Mathematics RICE University Houston, TX 77005-1982 (USA)
2 Department of Mathematics Stony Brook University Stony Brook, NY 11794-3651 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Harvey, F. Reese; Lawson, H. Blaine Jr. The Inhomogeneous Dirichlet problem for natural operators on manifolds. Annales de l'Institut Fourier, Riemannian Geometry. Past, Present and Future an homage to Marcel Berger December 6–9, 2017, IHES, Bures-sur-Yvette, Volume 69 (2019) no. 7, pp. 3017-3064. doi : 10.5802/aif.3344. https://aif.centre-mersenne.org/articles/10.5802/aif.3344/

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