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

  • * Corresponding author: Marta Lewicka

    * Corresponding author: Marta Lewicka
The authors have been partially supported by the NSF awards DMS-1406730 and DMS-1613153.
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  • 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.

    Mathematics Subject Classification: 65M99, 35R25.

    Citation:

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

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

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

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

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

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

    Figure 7.  The three corrugations in Example 6.2

    Figure 8.  The detail of Figure 7

    Figure 9.  The corrugations in Example 6.3

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

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

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

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

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

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

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

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

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

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