Timeline for Degenerate Monge-Ampere equation on a bounded domain with $C^{2,1}$ boundary
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 9, 2018 at 16:13 | comment | added | user126920 | Let me make clear that I haven't read the paper very carefully. There could be a much better explanation for why you don't need to worry about the $C^3$ norm of the boundary. | |
| Oct 9, 2018 at 16:11 | comment | added | user126920 | This sort of approximation is commonly left implicit, which one could argue is a bit sketchy, but on the other hand, it's tedious to write down and no new ideas are involved. Having said that, I agree the requirement that each $\Omega_k$ be convex makes things somewhat tricky. | |
| Oct 9, 2018 at 12:55 | comment | added | xiaocha123 | However, in my opinion, the author did not mention definitely what are the smooth exhausting domains used by himself (maybe I made the mistake here). Hence, in my opinion, it seems that the approximating process in the paper is different from that is presented in Gilbarg-Trudinger's book (see Chapter 8, probably, Theroem 8.34). Precisely, the approximating Dirichlet problems (55) in the paper are still defined on $\Omega$, rather than on certain smooth and strictly convex exhausting domains of $\Omega$ itself. That is the part which confuses me. Thanks! | |
| Oct 9, 2018 at 12:45 | comment | added | xiaocha123 | But there is still a problem confusing me. Firstly, I shall present my own understanding: the author verified that the approximating functions $f_{\epsilon,\rho}$ satisfy Condition (C) near the boundary $\partial \Omega$ such that one can obtain the approximating Dirichlet problems (55) by letting $\rho \rightarrow 0^+$, that is, $$ \det (D^2 u_\epsilon)=f_\epsilon \mbox{ in } \Omega, \mbox{ } u_\epsilon =0 \mbox{ on } \partial\Omega.$$ | |
| Oct 9, 2018 at 12:07 | comment | added | xiaocha123 | @Stanley Snelson Thanks very much for your helpful comment! I believe that the approximation procedure you informed me is a good approach. | |
| Oct 8, 2018 at 15:20 | comment | added | user126920 | The actual steps of the approximation procedure would be very similar to the one for linear elliptic equations, which you can find in Chapter 6 of Gilbarg-Trudinger. (I don't have the book in front of me, so I can't give a more precise reference.) | |
| Oct 8, 2018 at 15:07 | comment | added | user126920 | The key point is that the estimates don't depend quantitatively on the $C^3$ regularity of the boundary, so you can approximate $\Omega$ with smoother $\Omega_k$ whose boundaries converge in the $C^{2,1}$ norm, and take the limit. You can probably do both approximations at once (letting $\varepsilon = 1/k$) or to keep it simpler, you could do the entire proof as written for $C^3$ domains, get a priori estimates, and finally do the domain approximation. | |
| Oct 8, 2018 at 14:15 | history | edited | xiaocha123 | CC BY-SA 4.0 |
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| Oct 8, 2018 at 14:07 | history | edited | xiaocha123 | CC BY-SA 4.0 |
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| Oct 8, 2018 at 13:59 | history | edited | xiaocha123 | CC BY-SA 4.0 |
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| Oct 8, 2018 at 13:55 | review | First posts | |||
| Oct 8, 2018 at 14:12 | |||||
| Oct 8, 2018 at 13:54 | history | asked | xiaocha123 | CC BY-SA 4.0 |