Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations
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Introduction
t
Let us consider a Bolza problem of the Calculus of Variations (1.1) min
0
L(y (s), y ′ (s))ds + ϕ(y (t)) : y ∈ W 1,1 (0, t; I Rn ), y (0) = x ,
⋆ ×I Rn → I R+ be a locally Theorem 1.1 Assume that L is continuous. Let W : I R+ Lipschitz viscosity solution of (1.3) which satisfies the initial condition
(1.7)
∀x∈I Rn ,
h→0+,y →x
lim inf W (h, y ) = ϕ(x) .
⋆ ×I Then W = V on I R+ Rn .
To describe the new uniqueness results, we introduce the notion of contingent directional derivatives of a function W : I Rn → I R ∪ {+∞}. These are defined, for n every x ∈ dom(W ) and for every u ∈ I R , by (1.8) D↑ W (x)(u) := lim inf
where the final cost ϕ: I Rn → I R+ ∪ {+∞} is lower semicontinuous and the Lan n grangian L: I R ×I R → I R+ is locally bounded and lower semicontinuous . We assume also that L(x, ·) is convex for every x ∈ I Rn and that the following Tonelli
t
type coercivity assumption is satisfied: there exists a function Θ: I Rn → I R+ such that Θ(u) lim (1.2) = +∞ , ∀ (x, u) ∈ I Rn × I Rn , L(x, u) ≥ Θ(u) . |u|→∞ |u|
y →x
W (y ) − W (x) − p, y − x ≥0 . |y − x| 2
while the superdifferential ∂+ W (x) is defined by ∂+ W (x) := −∂− (−W )(x). ⋆ ×I A continuous function W : I R+ Rn → I R is said to be a viscosity solution to the Hamilton-Jacobi equation (1.3) if the following conditions are satisfied: (1.5) (1.6)
arXiv:math/0006015v1 [math.AP] 2 Jun 2000
Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations
Gianni Dal Maso∗ and H´ el` ene Frankowska†
u∈I Rn
(1.4)
H (x, p) := sup ( p, u − L(x, u)) ,
where ·, · denotes the scalar product in I Rn . In other words, H (x, ·) is the LegendreFenchel conjugate of L(x, ·). It is well known that (1.3) may not have smooth solutions, even if H and ϕ are smooth. To overcome this difficulty, different notions of generalized solutions have been proposed. The notion of viscosity solution can be introduced by means of subdifferentials and superdifferentials. We recall that the subdifferential ∂− W (x) of a function W : I Rn → I R ∪ {+∞} at a point x ∈ dom(W ) is defined by ∂− W (x) := p ∈ I Rn : lim inf
⋆ ×I Theorem 1.2 Assume that L is continuous. Let W : I R+ Rn → I R+ be a continuous viscosity solution of (1.3) which satisfies (1.9), (1.10), and the initial condition ⋆ ×I (1.7). Then W = V on I R+ Rn .
Abstract. We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians. Key words. Discontinuous Lagrangians, Hamilton-Jacobi equations, viability theory, viscosity solutions. AMS-MOS subject classification: 49L20 (primary), 49L25 (secondary).
h→0+ v →u
W (x + hv ) − W (x) . h
The main result of this paper is the following theorem, which shows that the value function V is the unique viscosity solution in the larger class of continuous functions whose contingent derivatives satisfy the following very weak assumptions: (1.9) (1.10) ∀ (t, x) ∈ dom(W ), t > 0, D↑ W (t, x)(0, 0) = 0 , ∀ (t, x) ∈ dom(W ), t > 0, ∃ u ∈ I Rn , D↑ W (t, x)(−1, u) < +∞ .
V (t, x) := min
0
L(y (s), y ′ (s))ds + ϕ(y (t)) : y ∈ W 1,1 (0, t; I Rn ), y (0) = x .
Under our assumptions on L and ϕ the value function is lower semicontinuous on ⋆ ×I ⋆ := I I R+ × I Rn and locally Lipschitz on I R+ Rn , where I R+ R+ \ {0}. Moreover it satisfies the initial condition ∀x∈I Rn ,
SISSA, via Beirut 2-4, 34014 Trieste, Italy CNRS, ERS2064, Centre de Recherche Viabilit´ e, Jeux, Contrˆ ole, Universit´ e de Paris-Dauphine, 75775 Paris Cx 16, France
In [8] and [9] the uniqueness of a bounded uniformly continuous viscosity solution of (1.3) is proved under some assumptions on H , which imply in particular that H is continuous and H (·, p) is uniformly continuous for every p ∈ I Rn . In [10, Theorem 4.5] we proved the following result, that can be applied also to unbounded solutions.
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