ANNALES DE L'INSTITUT FOURIER

The Inhomogeneous Dirichlet problem for natural operators on manifolds
Annales de l'Institut Fourier to appear, , 48 p.

We discuss the inhomogeneous Dirichlet problem written locally as:

$f\left(x,u,Du,{D}^{2}u\right)=\psi \left(x\right)$

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, $\psi$ can take the values of $f$ on $\partial F$.

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 $\psi$ is a delta function.

Il s’agit du problème de Dirichlet inhomogène  :

$f\left(x,u,Du,{D}^{2}u\right)=\psi \left(x\right)$

sur une variété $X$$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 $\psi$ peuvent toucher $f\left(\partial F\right)$.

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 $\psi$ est une fonction delta.

Classification:  35A99,  53C15,  53C38
Keywords: Inhomogenous Dirichlet Problem, Geometric Operators on Manifolds
@unpublished{AIF_0__0_0_A23_0,
author = {Harvey, F. Reese and Lawson, H. Blaine},
title = {The Inhomogeneous Dirichlet problem for natural operators on manifolds},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Harvey, F. Reese; Lawson, H. Blaine Jr. The Inhomogeneous Dirichlet problem for natural operators on manifolds. Annales de l'Institut Fourier, to appear, 48 p.

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