A question on the Evans-Krylov theorem and regularity of Monge-Ampere equation

Question

In http://ams.rice.edu/leavingmsn?url=https://doi.org/10.1524/anly.1996.16.1.101 Prof. Xu-Jia Wang established the boundary estimates for second derivatives of the solution to classical Dirichlet problem for Monge-Ampere equations, in which the author only assumes $\partial \Omega$ and the boundary data $\varphi$ are both $C^3$. However, the author also gave counterexamples to show that if $\partial\Omega$ or boundary data $\varphi$ is only $C^{2,1}$ smooth, the solution may fail to be $C^2$ smooth near the boundary.

From Wang's counterexamples mentioned above and the approximation lead me to asking a question:

What is the optimal regularity assumptions on the boundary and boundary data such that Evans-Krylov theorem works under these assumptions.

I don't know whether the examples means that the optimal regularity assumption on $\varphi$ and $\partial\Omega$ for deriving $C^{2,\alpha}$ estimate up to boundary via Evans-Krylov theorem are both $C^3$?.

Thanks very much for your attention!

Answer 1

To obtain solutions that are classical up to the boundary, the optimal regularity assumptions are $\varphi,\,\partial \Omega \in C^3$ (and in addition $\Omega$ uniformly convex).

The point is that these hypotheses guarantee boundary $C^2$ estimates, which give global $C^2$ estimates by the concavity of the operator, so the equation becomes uniformly elliptic. One can then rely on the general theory of concave, uniformly elliptic PDE. In Wang's counterexamples, the second derivatives in fact blow up near a boundary point where one of the above conditions is not satisfied. (If they didn't, then the general theory would say the solution is $C^{2,\,\alpha}$ up to the boundary; see the remarks below).

For concave, uniformly elliptic equations of the form $F(D^2u) = f(x)$ (with $f \in C^{\alpha}$) the Evans-Krylov theorem gives ${\it interior} \,\,C^{2,\,\alpha}$ estimates. An interesting observation is that even if $F$ is not concave, if $\varphi,\,\partial \Omega \in C^{2,\,\alpha}$ then the solution is $C^{2,\,\alpha'}$ in a neighborhood of the boundary (see the result of Silvestre and Sirakov here: https://arxiv.org/pdf/1306.6672.pdf ) so we don't need the Evans-Krylov theory near the boundary. Thus, the general theory says that any solution to the Monge-Ampere equation with $C^{2,\,\alpha}$ data is $C^{2,\,\alpha'}$ up to the boundary provided its second derivatives are globally bounded. However, we need stronger hypotheses on the data ($C^3$) to guarantee the global $C^2$ estimate by Wang's examples.

Answer 2

I just want to add and mention the wonderful results by O.Savin in paper "Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampère equation.J. Amer. Math. Soc. 26 (2013)", which improved the earlier result in this direction.

He proved that if $f\in C^{\alpha}$, $\partial\Omega,\varphi\in C^{2,\alpha}$ and $\varphi$ separates quadratically on $\partial\Omega$ from the tangent plane of $u$, then one has $u\in C^{2,\alpha}(\overline{\Omega})$.

In particular, if $\varphi\equiv 0$, then $f\in C^{\alpha}$, $\partial\Omega\in C^{2,\alpha} \Longrightarrow u\in C^{2,\alpha}(\overline{\Omega})$.