Problems involving Hamilton-Jacobi equations arise in many contexts. We will investigate the equation $u_t+H(Du{x,t})$=0 in R^n×(0,∞). First, although classical analysis of the associated problem is limited to local considerations owing to the crossing of characteristics, we derive the Hopf-Lax formula by the calculus of variations and define a weak solution of the Hamilton-Jacobi equation on $R^n×[0, ∞)$ and then we will show the uniqueness of the weak solution under proper assumptions of the Hamiltonian H and the initial data. Next we will investigate the viscosity solution of the Hamiltonian equation. M. G. Crandall and P. L. Lions introduced the notion of viscosity solution of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differential anywhere. The value of this concept is established by the fact that very general existence, uniqueness results hold for viscosity solutions of many problems arising in fields of application. Here we look more closely at the Hopf-Lax formula and two kinds of solutions of the Hamiltonian equation.
해밀터니안과 초기 데이터의 적절한 가정아래 홉-랙스 공식에 의해 정의된 함수는 전체 위치 공간과 시간 영역에서 해밀턴-자코비 방정식의 유일한 약화된 해(weak solution)가 된다. 또한 이 함수는 점성 해(viscosity solution)를 정의하는 중요한 두 가지 성질들을 전체 위치 공간과 시간 영역에서 만족하고 유한 시간 영역에서는 유일한 점성 해(viscosity solution)가 된다.
