Figure 1.
Construction in Example 3.1: the subsolution v0 on Ω and subsequent one and two corrugations.
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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
University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA |
In this article, we implement the algorithm based on the convex integration result proved in [
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 and 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
T. Iwaniec,
On the concept of weak Jacobian and Hessian, Report Univ. Jyväskylä, 83 (2001), 181-205.
|
| [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. |
| [11] |
N. H. Kuiper, On $ \mathcal{C}^1 $-isometric imbeddings. Ⅰ, Ⅱ, Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 545–556,683–689. |
| [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. |
| [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. |
| [14] |
M. Lewicka and M. R. Pakzad, Rigidity and Convexity of the Very Weak Solutions to the Monge-Ampère Equation, in preparation. |
| [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/ |
| [16] |
J. Nash,
The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20-63.
doi: 10.2307/1969989. |
| [17] |
J. Nash,
$ \mathcal{C}^1 $ isometric imbeddings, Ann. Math., 60 (1954), 383-396.
doi: 10.2307/1969840. |
| [18] |
M. R. Pakzad,
On the Sobolev space of isometric immersions, J. Differential Geom., 66 (2004), 47-69.
doi: 10.4310/jdg/1090415029. |
| [19] |
V. Šverák, On Regularity for the Monge-Ampère Equation without Convexity Assumptions, Heriot-Watt University, 1991. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
T. Iwaniec,
On the concept of weak Jacobian and Hessian, Report Univ. Jyväskylä, 83 (2001), 181-205.
|
| [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. |
| [11] |
N. H. Kuiper, On $ \mathcal{C}^1 $-isometric imbeddings. Ⅰ, Ⅱ, Nederl. Akad. Wetensch. Proc. Ser. A., 58 (1955), 545–556,683–689. |
| [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. |
| [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. |
| [14] |
M. Lewicka and M. R. Pakzad, Rigidity and Convexity of the Very Weak Solutions to the Monge-Ampère Equation, in preparation. |
| [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/ |
| [16] |
J. Nash,
The imbedding problem for Riemannian manifolds, Ann. Math., 63 (1956), 20-63.
doi: 10.2307/1969989. |
| [17] |
J. Nash,
$ \mathcal{C}^1 $ isometric imbeddings, Ann. Math., 60 (1954), 383-396.
doi: 10.2307/1969840. |
| [18] |
M. R. Pakzad,
On the Sobolev space of isometric immersions, J. Differential Geom., 66 (2004), 47-69.
doi: 10.4310/jdg/1090415029. |
| [19] |
V. Šverák, On Regularity for the Monge-Ampère Equation without Convexity Assumptions, Heriot-Watt University, 1991. |
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 6.
The detail of the three corrugations in Example 3.2
Figure 7.
The three corrugations in Example 6.2
Figure 9.
The corrugations in Example 6.3
Table 1.
Values of the defect $ \|D_3\|_0 $
| Example 6.1 | Example 6.2 | Example 6.3 | |
| Example 6.1 | Example 6.2 | Example 6.3 | |
Table 2.
Values of $ \|\nabla v_3\|_0 $
| Example 6.1 | Example 6.2 | Example 6.3 | |
| Example 6.1 | Example 6.2 | Example 6.3 | |
Table 3.
Values of $ \|\nabla w_3\|_0 $
| Example 6.1 | Example 6.2 | Example 6.3 | |
| Example 6.1 | Example 6.2 | Example 6.3 | |
Table 4.
Values of $ \|\nabla^2 v_3\|_0 $
| Example 6.1 | Example 6.2 | Example 6.3 | |
| Example 6.1 | Example 6.2 | Example 6.3 | |
Table 5.
Values of $ \|\nabla^2 w_3\|_0 $
| Example 6.1 | Example 6.2 | Example 6.3 | |
| Example 6.1 | Example 6.2 | Example 6.3 | |
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