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(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 Xf 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.

Classification:  35A99,  53C15,  53C38
Keywords: Inhomogenous Dirichlet Problem, Geometric Operators on Manifolds
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     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}},
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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, to appear, 48 p.

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