Open AccessDiscontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity
Author(s) -
Gui–Qiang Chen,
Bo Su
Publication year - 2003
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2003.9.167
Subject(s) - hamilton–jacobi equation , uniqueness , viscosity solution , mathematics , lipschitz continuity , regular polygon , initial value problem , star (game theory) , weak solution , mathematical analysis , pure mathematics , convex function , viscosity , combinatorics , physics , geometry , quantum mechanics
The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with convex Hamiltonians H = H (Du) is established, provided the discontinuous initial value function φ(x) is continuous outside a set Γ of measure zero and satisfies φ(x) ≥ φ**(x) ≡ liminfy→x,y∈ℝd\Γ φ (y). The regularity of discontinuous solutions to Hamilton-Jacobi equations with locally strictly convex Hamiltonians is proved: The discontinuous solutions with almost everywhere continuous initial data satisfying (*) become Lipschitz continuous after finite time. The L1-accessibility of initial data and a comparison principle for discontinuous solutions are shown. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L∞-solutions is also clarified