The representation formula for solutions of some class Hamilton–Jacobi equations

We consider the Cauchy problem for Hamilton–Jacobi equation of the form ut +H(ux) = 0, (1) u(0, x) = φ(x) (2) in domain S = {(t, x): t > 0, x ∈ R} with the lower semicontinious (lsc) initial function φ. For Hamilton H is convex with respect to ux, A. Douglis, S.N. Kruzkov first defined the notion of the generalized (semiconcave) solution of (1), (2). Definition 1. The Lipschits continuous function u(t, x) in ST is called the generalized (semiconcave) solution of (1), (2) if u(t, x) solves (1) a.e. on ST , satisfies (2), and for ∀l ∈ R, ∃Cδ ≻ 0, that the inequality u(t, x+ l)− 2u(t, x) + u(t, x− l) 6 Cδ|l| , (3) holds, when (t, x) ∈ S T = {(t, x): 0 < δ 6 t 6 T, x ∈ R }. E. Hopf gave [2] the representation formula for the semiconcave (1), (2) solutions. Theorem 1. Suppose H(p) is convex and satisfies the coercivity condition


Introduction
We consider the Cauchy problem for Hamilton-Jacobi equation of the form in domain S = {(t, x): t > 0, x ∈ R n } with the lower semicontinious (lsc) initial function ϕ.
If is H(p) strictly convex, S.N. Kruzkov proved [3], that formula (5) gives the semiconcave solution, when ϕ(x) is bounded and lsc on R n . The solution in this case satisfies initial condition in the sense The function is called the fundamental solution of (1).

The calculation of fundamental solutions
In order to define a function Φ(q) we need to solve the equation with respect y. In general we can not do it. It can be done when hamiltonian has the form where a i , p ∈ R n , b i ∈ R. We define the fundamental solution and prove the representation formula (5) for solutions of Notice, that the coercivity condition (4) for the hamiltonian (8) is not satisfied. Let x ∈ R. For the linear equation where a i = const, the Legrendre tranform of H(p) = a i p + b i is and the fundamental solution The solution can be represented by formula This solution does not satisfied semiconcave property (3), when ϕ(x) = |x|. Thus we need to consider the other class of generalized solutions of (1), (2), which has been defined in [1].

Definition 2.
A lsc function u on S with values in R ∪ {+∞} is a lsc solution of (1), (2), if for all (p t , p x ) ∈ D − u(t, x) (superdifferential), when u(t, x) < +∞, and We use the theorem which was proved in this paper. Let H be finite, continuous and convex. Then u defined by formula (5) is the unique lsc solution of (1), (2), that is bounded from below by a function of linear growth.
For the hamiltonians (8), suppose a i+1 ≻ a i , the Legrendre transform is Then the function is convex, satisfies a.e. (9) in {(t, x): x ∈ [ξ +a 1 t, ξ +a m t]}and the initial condition (6), thus, from the last theorem we have, that it is the unique fundamental solution of (9).

Example 1. Suppose we have the Cauchy problem
and the viscosity solution can be represented by formula It is clear, that if we construct the Legrendre transform of hamiltonian (8), then we easy define the fundamental solution. Next we explain, how we can define the Legrendre transform, when x ∈ R n , n > 1.