Corrugated versus smooth uniqueness and stability of negatively curved isometric immersions
Author:
Cleopatra Christoforou
Journal:
Quart. Appl. Math. 81 (2023), 533-551
MSC (2020):
Primary 53C42, 35L65, 35A02, 58K25, 57R42, 53C21, 35B35, 53C45
DOI:
https://doi.org/10.1090/qam/1663
Published electronically:
March 30, 2023
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Additional Information
Abstract: We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into $\mathbb {R}^3$. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a $C^{1,1}$ isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in $L^2$.
References
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- Cleopatra Christoforou, BV weak solutions to Gauss-Codazzi system for isometric immersions, J. Differential Equations 252 (2012), no. 3, 2845–2863. MR 2860643, DOI 10.1016/j.jde.2011.08.046
- Cleopatra Christoforou, The relative entropy method for inhomogeneous systems of balance laws, Quart. Appl. Math. 79 (2021), no. 2, 201–227. MR 4246491, DOI 10.1090/qam/1577
- Cleopatra Christoforou and Marshall Slemrod, Isometric immersions via compensated compactness for slowly decaying negative Gauss curvature and rough data, Z. Angew. Math. Phys. 66 (2015), no. 6, 3109–3122. MR 3428456, DOI 10.1007/s00033-015-0591-1
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- Sophia Demoulini, David M. A. Stuart, and Athanasios E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 927–961. MR 2960036, DOI 10.1007/s00205-012-0523-6
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- K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–1688. MR 285799, DOI 10.1073/pnas.68.8.1686
- Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006. MR 2261749, DOI 10.1090/surv/130
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- Siran Li and Marshall Slemrod, From the Nash-Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: analogues, examples, and extensions, J. Math. Phys. 64 (2023), no. 1, Paper No. 011511, 29. MR 4538702, DOI 10.1063/5.0100212
- Sorin Mardare, The fundamental theorem of surface theory for surfaces with little regularity, J. Elasticity 73 (2003), no. 1-3, 251–290 (2004). MR 2057747, DOI 10.1023/B:ELAS.0000029986.60986.8c
- Sorin Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry, J. Math. Pures Appl. (9) 84 (2005), no. 12, 1659–1692 (English, with English and French summaries). MR 2180386, DOI 10.1016/j.matpur.2005.08.002
- Alexey Miroshnikov and Konstantina Trivisa, Relative entropy in hyperbolic relaxation for balance laws, Commun. Math. Sci. 12 (2014), no. 6, 1017–1043. MR 3194369, DOI 10.4310/CMS.2014.v12.n6.a2
- Denis Serre and Alexis F. Vasseur, $L^2$-type contraction for systems of conservation laws, J. Éc. polytech. Math. 1 (2014), 1–28 (English, with English and French summaries). MR 3322780, DOI 10.5802/jep.1
- Athanasios E. Tzavaras, Relative entropy in hyperbolic relaxation, Commun. Math. Sci. 3 (2005), no. 2, 119–132. MR 2164193, DOI 10.4310/CMS.2005.v3.n2.a2
References
- Wentao Cao, Feimin Huang, and Dehua Wang, Isometric immersions of surfaces with two classes of metrics and negative Gauss curvature, Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1431–1457. MR 3401012, DOI 10.1007/s00205-015-0885-7
- Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang, Isometric immersions and compensated compactness, Comm. Math. Phys. 294 (2010), no. 2, 411–437. MR 2579461, DOI 10.1007/s00220-009-0955-5
- Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang, Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1843–1852. MR 2587469, DOI 10.1090/S0002-9939-09-10187-9
- Cleopatra Christoforou, BV weak solutions to Gauss-Codazzi system for isometric immersions, J. Differential Equations 252 (2012), no. 3, 2845–2863. MR 2860643, DOI 10.1016/j.jde.2011.08.046
- Cleopatra Christoforou, The relative entropy method for inhomogeneous systems of balance laws, Quart. Appl. Math. 79 (2021), no. 2, 201–227. MR 4246491, DOI 10.1090/qam/1577
- Cleopatra Christoforou and Marshall Slemrod, Isometric immersions via compensated compactness for slowly decaying negative Gauss curvature and rough data, Z. Angew. Math. Phys. 66 (2015), no. 6, 3109–3122. MR 3428456, DOI 10.1007/s00033-015-0591-1
- Cleopatra Christoforou and Marshall Slemrod, On the decay rate of the Gauss curvature for isometric immersions, Bull. Braz. Math. Soc. (N.S.) 47 (2016), no. 1, 255–265. MR 3475693, DOI 10.1007/s00574-016-0136-z
- Yann Brenier, Camillo De Lellis, and László Székelyhidi Jr., Weak-strong uniqueness for measure-valued solutions, Comm. Math. Phys. 305 (2011), no. 2, 351–361. MR 2805464, DOI 10.1007/s00220-011-1267-0
- Cleopatra Christoforou and Athanasios E. Tzavaras, Relative entropy for hyperbolic-parabolic systems and application to the constitutive theory of thermoviscoelasticity, Arch. Ration. Mech. Anal. 229 (2018), no. 1, 1–52. MR 3799089, DOI 10.1007/s00205-017-1212-2
- Bernard D. Coleman and Walter Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rational Mech. Anal. 13 (1963), 167–178. MR 153153, DOI 10.1007/BF01262690
- C. M. Dafermos, The second law of thermodynamics and stability, Arch. Rational Mech. Anal. 70 (1979), no. 2, 167–179. MR 546634, DOI 10.1007/BF00250353
- C. M. Dafermos, Stability of motions of thermoelastic fluids, J. Thermal Stresses 2 (1979), 127–134.
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010. MR 2574377, DOI 10.1007/978-3-642-04048-1
- Sophia Demoulini, David M. A. Stuart, and Athanasios E. Tzavaras, Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics, Arch. Ration. Mech. Anal. 205 (2012), no. 3, 927–961. MR 2960036, DOI 10.1007/s00205-012-0523-6
- Ronald J. DiPerna, Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), no. 1, 137–188. MR 523630, DOI 10.1512/iumj.1979.28.28011
- Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty. MR 1138207, DOI 10.1007/978-1-4757-2201-7
- K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 1686–1688. MR 285799, DOI 10.1073/pnas.68.8.1686
- Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006. MR 2261749, DOI 10.1090/surv/130
- Jia Xing Hong, Realization in ${\mathbf {R}}^3$ of complete Riemannian manifolds with negative curvature, Comm. Anal. Geom. 1 (1993), no. 3-4, 487–514. MR 1266477, DOI 10.4310/CAG.1993.v1.n4.a1
- Siran Li and Marshall Slemrod, From the Nash–Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions: Analogues, examples, and extensions, J. Math. Phys. 64 (2023), no. 1, Paper No. 011511, 29. MR 4538702, DOI 10.1063/5.0100212
- Sorin Mardare, The fundamental theorem of surface theory for surfaces with little regularity, J. Elasticity 73 (2003), no. 1-3, 251–290 (2004). MR 2057747, DOI 10.1023/B:ELAS.0000029986.60986.8c
- Sorin Mardare, On Pfaff systems with $L^p$ coefficients and their applications in differential geometry, J. Math. Pures Appl. (9) 84 (2005), no. 12, 1659–1692 (English, with English and French summaries). MR 2180386, DOI 10.1016/j.matpur.2005.08.002
- Alexey Miroshnikov and Konstantina Trivisa, Relative entropy in hyperbolic relaxation for balance laws, Commun. Math. Sci. 12 (2014), no. 6, 1017–1043. MR 3194369, DOI 10.4310/CMS.2014.v12.n6.a2
- Denis Serre and Alexis F. Vasseur, $L^2$-type contraction for systems of conservation laws, J. Éc. polytech. Math. 1 (2014), 1–28 (English, with English and French summaries). MR 3322780, DOI 10.5802/jep.1
- Athanasios E. Tzavaras, Relative entropy in hyperbolic relaxation, Commun. Math. Sci. 3 (2005), no. 2, 119–132. MR 2164193
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Additional Information
Cleopatra Christoforou
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus
MR Author ID:
776610
ORCID:
0000-0003-4467-3322
Email:
christoforou.cleopatra@ucy.ac.cy
Keywords:
Isometric immersions,
metric,
stability,
uniqueness,
relative entropy,
corrugated,
curvature
Received by editor(s):
January 24, 2023
Received by editor(s) in revised form:
February 4, 2023
Published electronically:
March 30, 2023
Dedicated:
Dedicated to my advisor Constantine Dafermos on the occasion of his 80th birthday
Article copyright:
© Copyright 2023
Brown University
