# laplace equation on manifolds with boundary

in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f$ if and only if $\int f dvol = 0$. The solution is unique up to a constant. If $f \in C^{r + \alpha}$ ($r \geq 0$ a integer or $r=+\infty$, $0 < \alpha < 1$), then $\varphi \in C^{r+2+\alpha}$. in this theorem the manifolds do not have boundary.

My question: are there similar results with riemannian manifolds with boundary ?

william

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To be more specific, you mean Aubin's book Some Nonlinear Problems in Riemannian Geometry –  Ben McKay Mar 4 '12 at 11:25
no I mean: Non linear Analysis on manifolds. Monge-Ampere equations. –  william Mar 4 '12 at 11:54

Similar results hold for manifolds with boundary, but you need to include boundary conditions. The most common boundary conditions in the case of Laplacian are the Dirichlet and the Neumann conditions.

The only tricky part in the case with boundary is regularity along the boundary. (In the regularirty results you need to include assumptions on the regularity of the boundary data. Two good references are

Gilbarg & Trudinger: Elliptic partial differential equations of second order

or

C. Morrey: Multiple integrals in the calculus of variations.

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