Visualization of the convex integration solutions to the Monge-Ampère equation

Luca Codenotti; Marta Lewicka

doi:10.3934/eect.2019015

Article Contents

2019, Volume 8, Issue 2: 273-300. Doi: 10.3934/eect.2019015

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Visualization of the convex integration solutions to the Monge-Ampère equation

Luca Codenotti and
Marta Lewicka^,

University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

^* Corresponding author: Marta Lewicka
^* Corresponding author: Marta Lewicka

Received: October 2018

Revised: January 2019

Early access: March 2019

Published: June 2019

The authors have been partially supported by the NSF awards DMS-1406730 and DMS-1613153.

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Abstract

In this article, we implement the algorithm based on the convex integration result proved in [13] and obtain visualizations of the first iterations of the Nash-Kuiper scheme, approximating the anomalous solutions to the Monge-Ampère equation in two dimensions.

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Mathematics Subject Classification: 65M99, 35R25.

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Figure 1. Construction in Example 3.1: the subsolution v₀ on Ω and subsequent one and two corrugations.

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Figure 2. Two corrugations in Example 3.1. The red detail is shown in Figure 3

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Figure 3. The detail of the three corrugations in Example 3.1

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Figure 4. Construction in Example 3.2: the original function $ v_0 $ and subsequent one and two corrugations

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Figure 5. Two corrugations in Figure 3.2. The red detail shown in Figure 6

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Figure 6. The detail of the three corrugations in Example 3.2

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Figure 7. The three corrugations in Example 6.2

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Figure 8. The detail of Figure 7

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Figure 9. The corrugations in Example 6.3

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Table 1. Values of the defect $ \|D_3\|_0 $

$\sigma$	Example 6.1	Example 6.2	Example 6.3
$10^1$	$0.332 \cdot 10^{-17}$	$0.393 \cdot 10^{-17}$	$0.327 \cdot 10^{-17}$
$10^2$	$0.316 \cdot 10^{-18}$	$0.364 \cdot 10^{-18}$	$0.319 \cdot 10^{-18}$
$10^3$	$0.318 \cdot 10^{-19}$	$0.376 \cdot 10^{-19}$	$0.332 \cdot 10^{-19}$
$10^4$	$0.318 \cdot 10^{-20}$	$0.396 \cdot 10^{-20}$	$0.318 \cdot 10^{-20}$
$10^5$	$0.320 \cdot 10^{-21}$	$0.401 \cdot 10^{-21}$	$0.316 \cdot 10^{-21}$
$10^6$	$0.336 \cdot 10^{-22}$	$0.400 \cdot 10^{-22}$	$0.325 \cdot 10^{-22}$
$10^7$	$0.326 \cdot 10^{-23}$	$0.375 \cdot 10^{-23}$	$0.330 \cdot 10^{-23}$
$10^8$	$0.332 \cdot 10^{-24}$	$0.367 \cdot 10^{-24}$	$0.335 \cdot 10^{-24}$
$10^9$	$0.329 \cdot 10^{-25}$	$0.366 \cdot 10^{-25}$	$0.329 \cdot 10^{-25}$
$10^{10}$	$0.328 \cdot 10^{-26}$	$0.382 \cdot 10^{-26}$	$0.339 \cdot 10^{-26}$
$10^{11}$	$0.338 \cdot 10^{-27}$	$0.399 \cdot 10^{-27}$	$0.326 \cdot 10^{-27}$
$10^{12}$	$0.329 \cdot 10^{-28}$	$0.371 \cdot 10^{-28}$	$0.327 \cdot 10^{-28}$
$10^{13}$	$0.317 \cdot 10^{-29}$	$0.396 \cdot 10^{-29}$	$0.333 \cdot 10^{-29}$
$10^{14}$	$0.320 \cdot 10^{-30}$	$0.388 \cdot 10^{-30}$	$0.325 \cdot 10^{-30}$
$10^{15}$	$0.311 \cdot 10^{-31}$	$0.384 \cdot 10^{-31}$	$0.336 \cdot 10^{-31}$
$10^{16}$	$0.324 \cdot 10^{-32}$	$0.366 \cdot 10^{-32}$	$0.321 \cdot 10^{-32}$

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Table 2. Values of $ \|\nabla v_3\|_0 $

$\sigma$	Example 6.1	Example 6.2	Example 6.3
$10^1$	$0.941 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.106 \cdot 10^{-7}$
$10^2$	$0.937 \cdot 10^{-8}$	$0.105 \cdot 10^{-7}$	$0.102 \cdot 10^{-7}$
$10^3$	$0.920 \cdot 10^{-8}$	$0.102 \cdot 10^{-7}$	$0.102 \cdot 10^{-7}$
$10^4$	$0.936 \cdot 10^{-8}$	$0.994 \cdot 10^{-8}$	$0.104 \cdot 10^{-7}$
$10^5$	$0.953 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.104 \cdot 10^{-7}$
$10^6$	$0.935 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.105 \cdot 10^{-7}$
$10^7$	$0.946 \cdot 10^{-8}$	$0.102 \cdot 10^{-7}$	$0.103 \cdot 10^{-7}$
$10^8$	$0.936 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.103 \cdot 10^{-7}$
$10^9$	$0.942 \cdot 10^{-8}$	$0.100 \cdot 10^{-7}$	$0.104 \cdot 10^{-7}$
$10^{10}$	$0.934 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.101 \cdot 10^{-7}$
$10^{11}$	$0.931 \cdot 10^{-8}$	$0.102 \cdot 10^{-7}$	$0.103 \cdot 10^{-7}$
$10^{12}$	$0.926 \cdot 10^{-8}$	$0.101 \cdot 10^{-7}$	$0.104 \cdot 10^{-7}$
$10^{13}$	$0.939 \cdot 10^{-8}$	$0.102 \cdot 10^{-7}$	$0.104 \cdot 10^{-7}$
$10^{14}$	$0.940 \cdot 10^{-8}$	$0.995 \cdot 10^{-8}$	$0.103 \cdot 10^{-7}$
$10^{15}$	$0.945 \cdot 10^{-8}$	$0.986 \cdot 10^{-8}$	$0.102 \cdot 10^{-7}$
$10^{16}$	$0.920 \cdot 10^{-8}$	$0.103 \cdot 10^{-7}$	$0.104 \cdot 10^{-7}$

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Table 3. Values of $ \|\nabla w_3\|_0 $

$\sigma$	Example 6.1	Example 6.2	Example 6.3
$10^1$	$0.730 \cdot 10^{-16}$	$0.432 \cdot 10^{-16}$	$0.844 \cdot 10^{-16}$
$10^2$	$0.728 \cdot 10^{-16}$	$0.445 \cdot 10^{-16}$	$0.857 \cdot 10^{-16}$
$10^3$	$0.687 \cdot 10^{-16}$	$0.430 \cdot 10^{-16}$	$0.813 \cdot 10^{-16}$
$10^4$	$0.712 \cdot 10^{-16}$	$0.458 \cdot 10^{-16}$	$0.835 \cdot 10^{-16}$
$10^5$	$0.754 \cdot 10^{-16}$	$0.424 \cdot 10^{-16}$	$0.848 \cdot 10^{-16}$
$10^6$	$0.727 \cdot 10^{-16}$	$0.422 \cdot 10^{-16}$	$0.873 \cdot 10^{-16}$
$10^7$	$0.716 \cdot 10^{-16}$	$0.444 \cdot 10^{-16}$	$0.773 \cdot 10^{-16}$
$10^8$	$0.723 \cdot 10^{-16}$	$0.444 \cdot 10^{-16}$	$0.859 \cdot 10^{-16}$
$10^9$	$0.727 \cdot 10^{-16}$	$0.420 \cdot 10^{-16}$	$0.833 \cdot 10^{-16}$
$10^{10}$	$0.721 \cdot 10^{-16}$	$0.431 \cdot 10^{-16}$	$0.752 \cdot 10^{-16}$
$10^{11}$	$0.698 \cdot 10^{-16}$	$0.478 \cdot 10^{-16}$	$0.808 \cdot 10^{-16}$
$10^{12}$	$0.672 \cdot 10^{-16}$	$0.431 \cdot 10^{-16}$	$0.843 \cdot 10^{-16}$
$10^{13}$	$0.692 \cdot 10^{-16}$	$0.449 \cdot 10^{-16}$	$0.785 \cdot 10^{-16}$
$10^{14}$	$0.733 \cdot 10^{-16}$	$0.414 \cdot 10^{-16}$	$0.789 \cdot 10^{-16}$
$10^{15}$	$0.739 \cdot 10^{-16}$	$0.430 \cdot 10^{-16}$	$0.800 \cdot 10^{-16}$
$10^{16}$	$0.687 \cdot 10^{-16}$	$0.442 \cdot 10^{-16}$	$0.779 \cdot 10^{-16}$

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Table 4. Values of $ \|\nabla^2 v_3\|_0 $

$\sigma$	Example 6.1	Example 6.2	Example 6.3
$10^1$	$3.37 \cdot 10^{13}$	$3.05 \cdot 10^{13}$	$3.26 \cdot 10^{13}$
$10^2$	$3.30 \cdot 10^{15}$	$3.00 \cdot 10^{15}$	$3.22 \cdot 10^{15}$
$10^3$	$3.27 \cdot 10^{17}$	$2.98 \cdot 10^{17}$	$3.28 \cdot 10^{17}$
$10^4$	$3.26 \cdot 10^{19}$	$2.98 \cdot 10^{19}$	$3.27 \cdot 10^{19}$
$10^5$	$3.27 \cdot 10^{21}$	$2.99 \cdot 10^{21}$	$3.29 \cdot 10^{21}$
$10^6$	$3.25 \cdot 10^{23}$	$2.99 \cdot 10^{23}$	$3.27 \cdot 10^{23}$
$10^7$	$3.28 \cdot 10^{25}$	$2.99 \cdot 10^{25}$	$3.23 \cdot 10^{25}$
$10^8$	$3.21 \cdot 10^{27}$	$2.99 \cdot 10^{27}$	$3.28 \cdot 10^{27}$
$10^9$	$3.26 \cdot 10^{29}$	$2.98 \cdot 10^{29}$	$3.28 \cdot 10^{29}$
$10^{10}$	$3.30 \cdot 10^{31}$	$2.99 \cdot 10^{31}$	$3.30 \cdot 10^{31}$
$10^{11}$	$3.25 \cdot 10^{33}$	$2.99 \cdot 10^{33}$	$3.23 \cdot 10^{33}$
$10^{12}$	$3.28 \cdot 10^{35}$	$2.99 \cdot 10^{35}$	$3.24 \cdot 10^{35}$
$10^{13}$	$3.23 \cdot 10^{37}$	$2.99 \cdot 10^{37}$	$3.25 \cdot 10^{37}$
$10^{14}$	$3.26 \cdot 10^{39}$	$2.98 \cdot 10^{39}$	$3.25 \cdot 10^{39}$
$10^{15}$	$3.25 \cdot 10^{41}$	$2.99 \cdot 10^{41}$	$3.27 \cdot 10^{41}$
$10^{16}$	$3.29 \cdot 10^{43}$	$2.98 \cdot 10^{43}$	$3.20 \cdot 10^{43}$

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Table 5. Values of $ \|\nabla^2 w_3\|_0 $

$\sigma$	Example 6.1	Example 6.2	Example 6.3
$10^1$	$3.11 \cdot 10^{5}$	$2.80 \cdot 10^{5}$	$2.12 \cdot 10^{5}$
$10^2$	$3.12 \cdot 10^{7}$	$2.82 \cdot 10^{7}$	$2.05 \cdot 10^{7}$
$10^3$	$3.20 \cdot 10^{9}$	$2.87 \cdot 10^{9}$	$2.23 \cdot 10^{9}$
$10^4$	$3.17 \cdot 10^{11}$	$2.60 \cdot 10^{11}$	$2.18 \cdot 10^{11}$
$10^5$	$3.12 \cdot 10^{13}$	$2.85 \cdot 10^{13}$	$2.15 \cdot 10^{13}$
$10^6$	$3.18 \cdot 10^{15}$	$2.80 \cdot 10^{15}$	$2.08 \cdot 10^{15}$
$10^7$	$3.12 \cdot 10^{17}$	$2.79 \cdot 10^{17}$	$2.18 \cdot 10^{17}$
$10^8$	$3.17 \cdot 10^{19}$	$2.73 \cdot 10^{19}$	$2.18 \cdot 10^{19}$
$10^9$	$3.27 \cdot 10^{21}$	$2.74 \cdot 10^{21}$	$2.18 \cdot 10^{21}$
$10^{10}$	$3.31 \cdot 10^{23}$	$2.75 \cdot 10^{23}$	$2.10 \cdot 10^{23}$
$10^{11}$	$3.21 \cdot 10^{25}$	$2.65 \cdot 10^{25}$	$2.13 \cdot 10^{25}$
$10^{12}$	$3.25 \cdot 10^{27}$	$2.69 \cdot 10^{27}$	$2.19 \cdot 10^{27}$
$10^{13}$	$3.10 \cdot 10^{29}$	$2.79 \cdot 10^{29}$	$2.08 \cdot 10^{29}$
$10^{14}$	$3.08 \cdot 10^{31}$	$2.71 \cdot 10^{31}$	$2.21 \cdot 10^{31}$
$10^{15}$	$3.29 \cdot 10^{33}$	$2.79 \cdot 10^{33}$	$2.20 \cdot 10^{33}$
$10^{16}$	$3.39 \cdot 10^{35}$	$2.72 \cdot 10^{35}$	$2.22 \cdot 10^{35}$

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