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!


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 $f\in C^{\alpha}$, $\partial\Omega\in C^{2,\alpha} \Longrightarrow u\in C^{2,\alpha}(\overline{\Omega})$.

  • $\begingroup$ Yes, thank you for adding this. $\endgroup$ – Connor Mooney Feb 28 '19 at 16:32
  • $\begingroup$ Thank you very much $\endgroup$ – xiaocha123 Mar 2 '19 at 7:50

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.

  • $\begingroup$ @John Wung Thanks for your helful and detailed answer. I still have a question on the related topic. As we know many authors studied the solvability of Monge-Ampere equation and general fully nonlinear equations. For instance, in the Theorem 1.2 of `On the Dirichlet problem for Hessian equations' projecteuclid.org/euclid.acta/1485890887 Prof. Turdinger proved the existence of $C^{3,\alpha}$ solutions for Dirichlet problem of equations on $\Omega\subset\mathbb{R}^n$, provided that $\partial\Omega$ and boundary data are both $C^{3,1}$, and other assumptions hold. $\endgroup$ – xiaocha123 Mar 3 '19 at 11:38
  • $\begingroup$ The boundedness of second estimates in the paper depends on the fourth order derivatives of $\partial\Omega$ and boundary data. As you pointed out, in Wang's examples, the second derivatives in fact blow up near a boundary point where one of the $C^3$ conditions is not satisfied (the 2-d Monge-Ampere equation in Wang's example is hence not uniformly elliptic). That is why the solution does not lie in $C^2$ near the boundary. $\endgroup$ – xiaocha123 Mar 3 '19 at 11:39
  • $\begingroup$ Comparing these work, I do not understand what is the difference between the two cases: the $C^{2,1}$ or $C^{3,1}$ regularity assumption. Namely, if the second derivatives blow up near the boundary point at which one of the $C^{4}$ conditions is not satisfied, then what happen and how does one get the $C^{3,\alpha}$ solutions? This puzzles me now. I appreciate the further information and explanation. Best regards $\endgroup$ – xiaocha123 Mar 3 '19 at 11:50
  • $\begingroup$ With $C^{3,\,1}$ data there are boundary $C^2$ estimates. The hypotheses on the data needed for boundary $C^2$ estimates were later relaxed to $C^3$ by Wang (the estimates depend on the modulus of continuity of the third derivatives of the data, boundedness of fourth derivatives is not needed), and more generally to any condition that implies quadratic separation of the boundary data from the tangent planes (in particular $C^3$ data, but not $C^{2,\,1}$) by Savin. $\endgroup$ – Connor Mooney Mar 3 '19 at 16:59

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