Browse issues
no. 4
Uniqueness of semi-concave weak Solutions for Hamilton–Jacobi Equations
Victor Issa1
1 École Normale Supérieure de Lyon, Lyon, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 877-900

It is well known that when the nonlinearity is convex, the Hamilton–Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton–Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.

Il est bien connu que, lorsque la non-linéarité est convexe, l’équation d’Hamilton–Jacobi admet une unique solution faible semi-convexe et que cette solution est la solution de viscosité. Dans cet article, motivé par des problèmes liés aux verres de spin, nous montrons que si l’équation d’Hamilton–Jacobi avec non-linéarité strictement convexe et condition initiale régulière admet une solution faible semi-concave, alors celle-ci est la solution de viscosité.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1825
Classification: 35F21, 35D30, 35D40
Keywords: Hamilton–Jacobi equations, viscosity solutions, weak solutions
Mots-clés : équation d’Hamilton–Jacobi, solutions de viscosité, solutions faibles

Victor Issa 1

1 École Normale Supérieure de Lyon, Lyon, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2025_6_34_4_877_0,
     author = {Victor Issa},
     title = {Uniqueness of semi-concave weak {Solutions} for {Hamilton{\textendash}Jacobi} {Equations}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {877--900},
     year = {2025},
     publisher = {Universit\'e de Toulouse, Toulouse},
     volume = {Ser. 6, 34},
     number = {4},
     doi = {10.5802/afst.1825},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1825/}
}
TY  - JOUR
AU  - Victor Issa
TI  - Uniqueness of semi-concave weak Solutions for Hamilton–Jacobi Equations
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2025
SP  - 877
EP  - 900
VL  - 34
IS  - 4
PB  - Université de Toulouse, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1825/
DO  - 10.5802/afst.1825
LA  - en
ID  - AFST_2025_6_34_4_877_0
ER  - 
%0 Journal Article
%A Victor Issa
%T Uniqueness of semi-concave weak Solutions for Hamilton–Jacobi Equations
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2025
%P 877-900
%V 34
%N 4
%I Université de Toulouse, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1825/
%R 10.5802/afst.1825
%G en
%F AFST_2025_6_34_4_877_0
Victor Issa. Uniqueness of semi-concave weak Solutions for Hamilton–Jacobi Equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 34 (2025) no. 4, pp. 877-900. doi: 10.5802/afst.1825

[1] Luigi Ambrosio Transport Equation and Cauchy Problem for BV Vector fields, Invent. Math., Volume 158 (2004) no. 2, pp. 227-260 | DOI | MR | Zbl

[2] Guy Barles An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications, Lecture Notes in Mathematics, 2074, Springer, 2013, pp. 49-109 | DOI | Zbl

[3] Francois Bouchut; Francois James; Simona Mancini Uniqueness and weak stability for multidimensional transport equations with one-sided Lipschitz coefficient, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 4 (2005) no. 1, pp. 1-25 | Zbl | MR

[4] Piermarco Cannarsa; Carlo Sinestrari Semiconcave functions, Hamilton–Jacobi equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser, 2004, xii+304 pages | Zbl | DOI | MR

[5] Hong-Bin Chen; Jean-Christophe Mourrat On the free energy of vector spin glasses with non-convex interactions (2023) | arXiv

[6] Hong-Bin Chen; Jean-Christophe Mourrat; Jiaming Xia Statistical inference of finite-rank tensors, Ann. Henri Lebesgue, Volume 5 (2022), pp. 1161-1189 | DOI | MR | Zbl | Numdam

[7] Lucilla Corrias; Maurizio Falcone; Roberto Natalini Numerical schemes for conservation laws via Hamilton–Jacobi equations, Math. Comput., Volume 64 (1995) no. 210, pp. 555-580 | DOI | MR

[8] Lawrence C. Evans Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, 2010, xxi+749 pages | MR | Zbl

[9] Pierre-Louis Lions Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathematics, 69, Pitman, 1982, 317 pages | MR | Zbl

[10] Pierre-Louis Lions; Benjamin Seeger Transport equations and flows with one-sided Lipschitz velocity fields (2023) | arXiv

[11] Jean-Christophe Mourrat Nonconvex interactions in mean-field spin glasses, Probab. Math. Phys., Volume 2 (2021) no. 2, pp. 61-119 | DOI | Zbl

[12] Jean-Christophe Mourrat The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space, Can. J. Math., Volume 74 (2022) no. 3, pp. 607-629 | DOI | MR | Zbl

[13] Jean-Christophe Mourrat Free energy upper bound for mean-field vector spin glasses, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 59 (2023) no. 3, pp. 1143-1182 | MR | Zbl

[14] Dmitry Panchenko The Sherrington–Kirkpatrick Model, Springer Monographs in Mathematics, Springer, 2013, xii+156 pages | DOI | Zbl

[15] G. Parisi Infinite Number of Order Parameters for Spin-Glasses, Phys. Rev. Lett., Volume 43 (1979) no. 23, pp. 1754-1756 | DOI

[16] Denis Serre Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, 2003, xxii+263 pages | Zbl

Cited by Sources: