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On a Mathematical model with non-compact boundary conditions describing bacterial population (Ⅱ)
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 [
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. |






Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
Example 6.1 | Example 6.2 | Example 6.3 | |
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