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June  2019, 8(2): 273-300. doi: 10.3934/eect.2019015

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

Received  October 2018 Revised  January 2019 Published  June 2019 Early access  March 2019

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

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.

Keywords: Convex integration, Monge-Ampère equation, anomalous solutions, Nash-Kuiper algorithm, numerical implementation.
Mathematics Subject Classification: 65M99, 35R25.
Citation: Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015
References:
[1]

V. Borelli, S. Jabrane, F. Lazarus and B. Thibert, Isometric Embeddings of the Square Flat Torus in Ambient Space, Ensaios Matematicos, 2013.  Google Scholar

[2]

V. Borelli, S. Jabrane, F. Lazarus and B. Thibert, Flat tori in three-dimensional space and convex integration, Proc. of the National Acad. of Sciences, 109 (2012), 7218-7223.  doi: 10.1073/pnas.1118478109.  Google Scholar

[3]

T. Buckmaster, C. De Lellis, P. Isett and L. Szekelyhidi Jr., Anomalous dissipation for $ 1/5 $-Hölder Euler flows, Annals of Mathematics, 182 (2015), 127-172.  doi: 10.4007/annals.2015.182.1.3.  Google Scholar

[4]

S. Conti, C. De Lellis and L. Székelyhidi Jr., $ h $-principle and rigidity for $ \mathcal{C}^{1,\alpha} $ isometric embeddings, Nonlinear Partial Differential Equations, 7 (2012), 83-116.  doi: 10.1007/978-3-642-25361-4_5.  Google Scholar

[5]

C. De Lellis and L. Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2), 170 (2009), 1417-1436.  doi: 10.4007/annals.2009.170.1417.  Google Scholar

[6]

C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407.  doi: 10.1007/s00222-012-0429-9.  Google Scholar

[7]

M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-02267-2.  Google Scholar

[8]

P. Isett, A proof of Onsager's conjecture, Ann. of Math. (2), 188 (2018), 871-963.  doi: 10.4007/annals.2018.188.3.4.  Google Scholar

[9]

T. Iwaniec, On the concept of weak Jacobian and Hessian, Report Univ. Jyväskylä, 83 (2001), 181-205.   Google Scholar

[10]

R. L. Jerrard and M. R. Pakzad, Sobolev spaces of isometric immersions of arbitrary dimension and co-dimension, Annali di Matematica Pura ed Applicata, 196 (2017), 687-716.  doi: 10.1007/s10231-016-0591-6.  Google Scholar

[11]

N. H. Kuiper, On $ \mathcal{C}^1 $-isometric imbeddings. Ⅰ, Ⅱ, Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 545–556,683–689.  Google Scholar

[12]

M. Lewicka, L. Mahadevan and M. R. Pakzad, The Monge-Ampère constraint: Matching of isometries, density and regularity and elastic theories of shallow shells, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 34 (2017), 45-67.  doi: 10.1016/j.anihpc.2015.08.005.  Google Scholar

[13]

M. Lewicka and M.R. Pakzad, Convex integration for the Monge-Ampère equation in two dimensions, Analysis and PDE, 10 (2017), 695-727.  doi: 10.2140/apde.2017.10.695.  Google Scholar

[14]

M. Lewicka and M. R. Pakzad, Rigidity and Convexity of the Very Weak Solutions to the Monge-Ampère Equation, in preparation. Google Scholar

[15]

Mpmath is a free (BSD licensed) Python library for real and complex floating-point arithmetic with arbitrary precision., It has been developed by Fredrik Johansson since 2007. http://mpmath.org/ Google Scholar

[16]

J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20-63.  doi: 10.2307/1969989.  Google Scholar

[17]

J. Nash, $ \mathcal{C}^1 $ isometric imbeddings, Ann. Math., 60 (1954), 383-396.  doi: 10.2307/1969840.  Google Scholar

[18]

M. R. Pakzad, On the Sobolev space of isometric immersions, J. Differential Geom., 66 (2004), 47-69.  doi: 10.4310/jdg/1090415029.  Google Scholar

[19]

V. Šverák, On Regularity for the Monge-Ampère Equation without Convexity Assumptions, Heriot-Watt University, 1991. Google Scholar

show all references

References:
[1]

V. Borelli, S. Jabrane, F. Lazarus and B. Thibert, Isometric Embeddings of the Square Flat Torus in Ambient Space, Ensaios Matematicos, 2013.  Google Scholar

[2]

V. Borelli, S. Jabrane, F. Lazarus and B. Thibert, Flat tori in three-dimensional space and convex integration, Proc. of the National Acad. of Sciences, 109 (2012), 7218-7223.  doi: 10.1073/pnas.1118478109.  Google Scholar

[3]

T. Buckmaster, C. De Lellis, P. Isett and L. Szekelyhidi Jr., Anomalous dissipation for $ 1/5 $-Hölder Euler flows, Annals of Mathematics, 182 (2015), 127-172.  doi: 10.4007/annals.2015.182.1.3.  Google Scholar

[4]

S. Conti, C. De Lellis and L. Székelyhidi Jr., $ h $-principle and rigidity for $ \mathcal{C}^{1,\alpha} $ isometric embeddings, Nonlinear Partial Differential Equations, 7 (2012), 83-116.  doi: 10.1007/978-3-642-25361-4_5.  Google Scholar

[5]

C. De Lellis and L. Székelyhidi Jr., The Euler equations as a differential inclusion, Ann. of Math. (2), 170 (2009), 1417-1436.  doi: 10.4007/annals.2009.170.1417.  Google Scholar

[6]

C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407.  doi: 10.1007/s00222-012-0429-9.  Google Scholar

[7]

M. Gromov, Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-02267-2.  Google Scholar

[8]

P. Isett, A proof of Onsager's conjecture, Ann. of Math. (2), 188 (2018), 871-963.  doi: 10.4007/annals.2018.188.3.4.  Google Scholar

[9]

T. Iwaniec, On the concept of weak Jacobian and Hessian, Report Univ. Jyväskylä, 83 (2001), 181-205.   Google Scholar

[10]

R. L. Jerrard and M. R. Pakzad, Sobolev spaces of isometric immersions of arbitrary dimension and co-dimension, Annali di Matematica Pura ed Applicata, 196 (2017), 687-716.  doi: 10.1007/s10231-016-0591-6.  Google Scholar

[11]

N. H. Kuiper, On $ \mathcal{C}^1 $-isometric imbeddings. Ⅰ, Ⅱ, Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 545–556,683–689.  Google Scholar

[12]

M. Lewicka, L. Mahadevan and M. R. Pakzad, The Monge-Ampère constraint: Matching of isometries, density and regularity and elastic theories of shallow shells, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 34 (2017), 45-67.  doi: 10.1016/j.anihpc.2015.08.005.  Google Scholar

[13]

M. Lewicka and M.R. Pakzad, Convex integration for the Monge-Ampère equation in two dimensions, Analysis and PDE, 10 (2017), 695-727.  doi: 10.2140/apde.2017.10.695.  Google Scholar

[14]

M. Lewicka and M. R. Pakzad, Rigidity and Convexity of the Very Weak Solutions to the Monge-Ampère Equation, in preparation. Google Scholar

[15]

Mpmath is a free (BSD licensed) Python library for real and complex floating-point arithmetic with arbitrary precision., It has been developed by Fredrik Johansson since 2007. http://mpmath.org/ Google Scholar

[16]

J. Nash, The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20-63.  doi: 10.2307/1969989.  Google Scholar

[17]

J. Nash, $ \mathcal{C}^1 $ isometric imbeddings, Ann. Math., 60 (1954), 383-396.  doi: 10.2307/1969840.  Google Scholar

[18]

M. R. Pakzad, On the Sobolev space of isometric immersions, J. Differential Geom., 66 (2004), 47-69.  doi: 10.4310/jdg/1090415029.  Google Scholar

[19]

V. Šverák, On Regularity for the Monge-Ampère Equation without Convexity Assumptions, Heriot-Watt University, 1991. Google Scholar

Ω and subsequent one and two corrugations.">Figure 1.  Construction in Example 3.1: the subsolution v0 on Ω and subsequent one and two corrugations.
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Figure 3">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 3.2. The red detail shown in Figure 6">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 7">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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$\sigma$ Example 6.1 Example 6.2 Example 6.3
$10^1$ $0.332 \cdot 10^{-17}$ $0.393 \cdot 10^{-17}$ $0.327 \cdot 10^{-17}$
$10^2$ $0.316 \cdot 10^{-18}$ $0.364 \cdot 10^{-18}$ $0.319 \cdot 10^{-18}$
$10^3$ $0.318 \cdot 10^{-19}$ $0.376 \cdot 10^{-19}$ $0.332 \cdot 10^{-19}$
$10^4$ $0.318 \cdot 10^{-20}$ $0.396 \cdot 10^{-20}$ $0.318 \cdot 10^{-20}$
$10^5$ $0.320 \cdot 10^{-21}$ $0.401 \cdot 10^{-21}$ $0.316 \cdot 10^{-21}$
$10^6$ $0.336 \cdot 10^{-22}$ $0.400 \cdot 10^{-22}$ $0.325 \cdot 10^{-22}$
$10^7$ $0.326 \cdot 10^{-23}$ $0.375 \cdot 10^{-23}$ $0.330 \cdot 10^{-23}$
$10^8$ $0.332 \cdot 10^{-24}$ $0.367 \cdot 10^{-24}$ $0.335 \cdot 10^{-24}$
$10^9$ $0.329 \cdot 10^{-25}$ $0.366 \cdot 10^{-25}$ $0.329 \cdot 10^{-25}$
$10^{10}$ $0.328 \cdot 10^{-26}$ $0.382 \cdot 10^{-26}$ $0.339 \cdot 10^{-26}$
$10^{11}$ $0.338 \cdot 10^{-27}$ $0.399 \cdot 10^{-27}$ $0.326 \cdot 10^{-27}$
$10^{12}$ $0.329 \cdot 10^{-28}$ $0.371 \cdot 10^{-28}$ $0.327 \cdot 10^{-28}$
$10^{13}$ $0.317 \cdot 10^{-29}$ $0.396 \cdot 10^{-29}$ $0.333 \cdot 10^{-29}$
$10^{14}$ $0.320 \cdot 10^{-30}$ $0.388 \cdot 10^{-30}$ $0.325 \cdot 10^{-30}$
$10^{15}$ $0.311 \cdot 10^{-31}$ $0.384 \cdot 10^{-31}$ $0.336 \cdot 10^{-31}$
$10^{16}$ $0.324 \cdot 10^{-32}$ $0.366 \cdot 10^{-32}$ $0.321 \cdot 10^{-32}$
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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$\sigma$ Example 6.1 Example 6.2 Example 6.3
$10^1$ $0.941 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.106 \cdot 10^{-7}$
$10^2$ $0.937 \cdot 10^{-8}$ $0.105 \cdot 10^{-7}$ $0.102 \cdot 10^{-7}$
$10^3$ $0.920 \cdot 10^{-8}$ $0.102 \cdot 10^{-7}$ $0.102 \cdot 10^{-7}$
$10^4$ $0.936 \cdot 10^{-8}$ $0.994 \cdot 10^{-8}$ $0.104 \cdot 10^{-7}$
$10^5$ $0.953 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.104 \cdot 10^{-7}$
$10^6$ $0.935 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.105 \cdot 10^{-7}$
$10^7$ $0.946 \cdot 10^{-8}$ $0.102 \cdot 10^{-7}$ $0.103 \cdot 10^{-7}$
$10^8$ $0.936 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.103 \cdot 10^{-7}$
$10^9$ $0.942 \cdot 10^{-8}$ $0.100 \cdot 10^{-7}$ $0.104 \cdot 10^{-7}$
$10^{10}$ $0.934 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.101 \cdot 10^{-7}$
$10^{11}$ $0.931 \cdot 10^{-8}$ $0.102 \cdot 10^{-7}$ $0.103 \cdot 10^{-7}$
$10^{12}$ $0.926 \cdot 10^{-8}$ $0.101 \cdot 10^{-7}$ $0.104 \cdot 10^{-7}$
$10^{13}$ $0.939 \cdot 10^{-8}$ $0.102 \cdot 10^{-7}$ $0.104 \cdot 10^{-7}$
$10^{14}$ $0.940 \cdot 10^{-8}$ $0.995 \cdot 10^{-8}$ $0.103 \cdot 10^{-7}$
$10^{15}$ $0.945 \cdot 10^{-8}$ $0.986 \cdot 10^{-8}$ $0.102 \cdot 10^{-7}$
$10^{16}$ $0.920 \cdot 10^{-8}$ $0.103 \cdot 10^{-7}$ $0.104 \cdot 10^{-7}$
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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$\sigma$ Example 6.1 Example 6.2 Example 6.3
$10^1$ $0.730 \cdot 10^{-16}$ $0.432 \cdot 10^{-16}$ $0.844 \cdot 10^{-16}$
$10^2$ $0.728 \cdot 10^{-16}$ $0.445 \cdot 10^{-16}$ $0.857 \cdot 10^{-16}$
$10^3$ $0.687 \cdot 10^{-16}$ $0.430 \cdot 10^{-16}$ $0.813 \cdot 10^{-16}$
$10^4$ $0.712 \cdot 10^{-16}$ $0.458 \cdot 10^{-16}$ $0.835 \cdot 10^{-16}$
$10^5$ $0.754 \cdot 10^{-16}$ $0.424 \cdot 10^{-16}$ $0.848 \cdot 10^{-16}$
$10^6$ $0.727 \cdot 10^{-16}$ $0.422 \cdot 10^{-16}$ $0.873 \cdot 10^{-16}$
$10^7$ $0.716 \cdot 10^{-16}$ $0.444 \cdot 10^{-16}$ $0.773 \cdot 10^{-16}$
$10^8$ $0.723 \cdot 10^{-16}$ $0.444 \cdot 10^{-16}$ $0.859 \cdot 10^{-16}$
$10^9$ $0.727 \cdot 10^{-16}$ $0.420 \cdot 10^{-16}$ $0.833 \cdot 10^{-16}$
$10^{10}$ $0.721 \cdot 10^{-16}$ $0.431 \cdot 10^{-16}$ $0.752 \cdot 10^{-16}$
$10^{11}$ $0.698 \cdot 10^{-16}$ $0.478 \cdot 10^{-16}$ $0.808 \cdot 10^{-16}$
$10^{12}$ $0.672 \cdot 10^{-16}$ $0.431 \cdot 10^{-16}$ $0.843 \cdot 10^{-16}$
$10^{13}$ $0.692 \cdot 10^{-16}$ $0.449 \cdot 10^{-16}$ $0.785 \cdot 10^{-16}$
$10^{14}$ $0.733 \cdot 10^{-16}$ $0.414 \cdot 10^{-16}$ $0.789 \cdot 10^{-16}$
$10^{15}$ $0.739 \cdot 10^{-16}$ $0.430 \cdot 10^{-16}$ $0.800 \cdot 10^{-16}$
$10^{16}$ $0.687 \cdot 10^{-16}$ $0.442 \cdot 10^{-16}$ $0.779 \cdot 10^{-16}$
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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$\sigma$ Example 6.1 Example 6.2 Example 6.3
$10^1$ $3.37 \cdot 10^{13}$ $3.05 \cdot 10^{13}$ $3.26 \cdot 10^{13}$
$10^2$ $3.30 \cdot 10^{15}$ $3.00 \cdot 10^{15}$ $3.22 \cdot 10^{15}$
$10^3$ $3.27 \cdot 10^{17}$ $2.98 \cdot 10^{17}$ $3.28 \cdot 10^{17}$
$10^4$ $3.26 \cdot 10^{19}$ $2.98 \cdot 10^{19}$ $3.27 \cdot 10^{19}$
$10^5$ $3.27 \cdot 10^{21}$ $2.99 \cdot 10^{21}$ $3.29 \cdot 10^{21}$
$10^6$ $3.25 \cdot 10^{23}$ $2.99 \cdot 10^{23}$ $3.27 \cdot 10^{23}$
$10^7$ $3.28 \cdot 10^{25}$ $2.99 \cdot 10^{25}$ $3.23 \cdot 10^{25}$
$10^8$ $3.21 \cdot 10^{27}$ $2.99 \cdot 10^{27}$ $3.28 \cdot 10^{27}$
$10^9$ $3.26 \cdot 10^{29}$ $2.98 \cdot 10^{29}$ $3.28 \cdot 10^{29}$
$10^{10}$ $3.30 \cdot 10^{31}$ $2.99 \cdot 10^{31}$ $3.30 \cdot 10^{31}$
$10^{11}$ $3.25 \cdot 10^{33}$ $2.99 \cdot 10^{33}$ $3.23 \cdot 10^{33}$
$10^{12}$ $3.28 \cdot 10^{35}$ $2.99 \cdot 10^{35}$ $3.24 \cdot 10^{35}$
$10^{13}$ $3.23 \cdot 10^{37}$ $2.99 \cdot 10^{37}$ $3.25 \cdot 10^{37}$
$10^{14}$ $3.26 \cdot 10^{39}$ $2.98 \cdot 10^{39}$ $3.25 \cdot 10^{39}$
$10^{15}$ $3.25 \cdot 10^{41}$ $2.99 \cdot 10^{41}$ $3.27 \cdot 10^{41}$
$10^{16}$ $3.29 \cdot 10^{43}$ $2.98 \cdot 10^{43}$ $3.20 \cdot 10^{43}$
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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$\sigma$ 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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