7
$\begingroup$

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 ?

hope for answers.

william

Improve this question
$\endgroup$
2
  • $\begingroup$ To be more specific, you mean Aubin's book Some Nonlinear Problems in Riemannian Geometry $\endgroup$
    – Ben McKay
    Mar 4, 2012 at 11:25
  • 1
    $\begingroup$ no I mean: Non linear Analysis on manifolds. Monge-Ampere equations. $\endgroup$
    – william
    Mar 4, 2012 at 11:54

1 Answer 1

Reset to default
8
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.