Timeline for A question on the Evans-Krylov theorem and regularity of Monge-Ampere equation
Current License: CC BY-SA 4.0
5 events
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| Mar 3, 2019 at 16:59 | comment | added | Connor Mooney | 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. | |
| Mar 3, 2019 at 11:50 | comment | added | xiaocha123 | 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 | |
| Mar 3, 2019 at 11:39 | comment | added | xiaocha123 | 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. | |
| Mar 3, 2019 at 11:38 | comment | added | xiaocha123 | @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. | |
| Feb 28, 2019 at 15:59 | history | answered | Connor Mooney | CC BY-SA 4.0 |