2 deleted 2 characters in body

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

1

# 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 ???