L.G. Shagalova. A continuous generalized solution of the Hamilton–Jacobi equation with a three-component Hamiltonian ... P. 257-268

The Cauchy problem for the Hamilton–Jacobi equation of evolution type is studied in the case of one-dimensional state space. The domain in which the equation is considered is divided into three subdomains. In each of these subdomains, the Hamiltonian is continuous, and at their boundaries it suffers a discontinuity in the state variable. The Hamiltonian is convex in the impulse variable, and the dependence on this variable is exponential. We define a continuous generalized solution of the Cauchy problem with a discontinuous Hamiltonian on the basis of the viscous/minimax approach. The proof of the existence of such a generalized solution is constructive. First, a viscosity solution is constructed in the closure of the middle domain. Here, the coercivity of the Hamiltonian with respect to the impulse variable in the middle domain is essential. The solution is then continuously extended to the other two domains. The extensions are constructed by solving variational problems with movable ends based on the method of generalized characteristics. The uniqueness of the generalized solution is proved under the condition that the initial function is globally Lipschitz.

Keywords: Hamilton–Jacobi equation, discontinuous Hamiltonian, generalized solutions, viscosity solutions, method of generalized characteristics

Received September 16, 2021

Revised January 17, 2022

Accepted January 21, 2022

Lyubov Gennad’evna Shagalova, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: shag@imm.uran.ru

REFERENCES

1.   Kruzhkov S.N. Generalized solutions of nonlinear equations of the first order with several variables. I. Mat. Sb. (N.S.), 1966, vol. 70 (112), no. 3, pp. 394–415 (in Russian).

2.   Crandall M.G., Lions P. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 1983, vol. 277, no. 1, pp. 1–42. doi: 10.1090/S0002-9947-1983-0690039-8 

3.   Subbotin A.I. Minimaksnye neravenstva i uravneniya Gamil’tona–Yakobi [Minimax inequalities and Hamilton–Jacobi equations]. Moscow: Nauka Publ., 1991, 216 p. ISBN: 5-02-000139-2 .

4.   Subbotin A.I. Generalized solutions of first-order PDEs. The dynamical optimization perspective. Basel: Birkhäuser, 1995, 312 p. doi: 10.1007/978-1-4612-0847-1 

5.   Capuzzo-Dolcetta I., Lions P. Hamilton–Jacobi equations with state constraints. Trans. Amer. Math. Soc., 1990, vol. 318, no. 2, pp. 643–683. doi: 10.1090/S0002-9947-1990-0951880-0 

6.   Yokoyama E., Giga Y., Rybka P. A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation. Physica D: Nonlinear Phenomena, 2008, vol. 237, no. 22, pp. 2845–2855. doi: 10.1016/j.physd.2008.05.009 .

7.   Saakian D.B., Rozanova O., Akmetzhanov A. Dynamics of the Eigen and the Crow-Kimura models for molecular evolution. Phys. Rev. E, 2008, vol. 78, no. 4, art. no. 041908. doi: 10.1103/PhysRevE.78.041908 

8.   Subbotina N.N., Shagalova L.G. On a solution to the Cauchy problem for the Hamilton–Jacobi equation with state constraints. Trudy Inst. Mat. i Mekh. UrO RAN, 2011, vol. 17, no. 2, pp. 191–208 (in Russian).

9.   Shagalova L.G. Continuous generalized solution of the Hamilton–Jacobi equation with a noncoercive hamiltonian. Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2020, vol. 186, pp. 144–151 (in Russian). doi: 10.36535/0233-6723-2020-186-144-151 .

10.   Clarke F.H. Tonelli’s regurarity theory in the calculus of variations: Recent progress. In: Conti R., De Giorgi E., Giannessi F. (eds), Optimization and Related Fields, 2006, Lecture Notes in Mathematics, vol. 1190, pp. 163–179. doi: 10.1007/BFb0076705 

11.   Courant R., Hilbert D. Methods of mathematical physics. Vol. 2. Partial differential equations. NY: Interscience, 1962, 830 p. ISBN: 9780470179857 . Translated to Russian under the title Uravneniya s chastnymi proizvodnymi. Moscow: Mir Publ., 1964, 832 p.

12.   Subbotina N.N. The method of characteristics for Hamilton–Jacobi equation and its applications in dynamical optimization. J. Math. Sci., 2006, vol. 135, no. 3, pp. 2955–3091. (Modern Math. Appl., vol. 20). doi: 10.1007/s10958-006-0146-2 

13.   Mirică Ş. Generalized solutions by Cauchy’s method of characteristics. Rend. Semin. Mat. Univ. Padova, 1987, vol. 77, pp. 317–350.

14.   Shagalova L. A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum. Cybernetics and Physics. 2021. Vol. 10, no. 4. P. 273–276. doi:10.35470/2226-4116-2021-10-2-273-276 

15.   Bardi M., Evans L.C. On Hopf’s formulas for solutions of Hamilton–Jacobi equations. Nonlinear Analysis: Theory, Methods & Applications, 1984, vol. 8, no. 11, pp. 1373–1381. doi: 10.1016/0362-546X(84)90020-8 

Cite this article as: L.G. Shagalova. A continuous generalized solution of the Hamilton–Jacobi equation with a three-component Hamiltonian, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 1, pp. 257–268.