aa r X i v : . [ m a t h . A P ] D ec HOPF-LAX FORMULA AND GENERALIZED CHARACTERISTICS
NGUYEN HOANG
Abstract.
We study some differential properties of viscosity solution for Hamilton -Jacobi equations defined by Hopf-Lax formula u ( t, x ) = min y ∈ R n (cid:8) σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)(cid:9) . A generalized form of characteristics of the Cauchy problem is taken into account thecontext. Then we examine the strip of differentiability of the viscosity solution givenby the function u ( t, x ) . Introduction
The global studies of Hamilton-Jacobi equations have spent more than a half cen-tury with several notions of solutions including classical and generalized solutions.Among explicit representations of the solutions, two formulas Hopf-Lax and Hopf playimportant roles. Many qualitative as well as quantitative properties of solutions arestudied via these formulas. The first formula was established in 1957 by Lax [13]for 1-dimensional case and then in 1965, Hopf [11] proved both formulas for multidi-mensional case. Until now, they turned to be classical. Nevertheless, recently severalpapers pursuit to exploit the mentioned formulas and get new results, e.g., [1, 2],...Consider the Cauchy problem for Hamilton-Jacobi equations of the form(1.1) ∂u∂t + H ( t, x, D x u ) = 0 , ( t, x ) ∈ Ω = (0 , T ) × R n , (1.2) u (0 , x ) = σ ( x ) , x ∈ R n . If the Hamiltonian is independent of ( t, x ) i.e., H ( t, x, p ) = H ( p ) and H ( p ) is convexand superlinear, σ is Lipschitz on R n , then the Hopf-Lax formula for solution of theproblem (1.1)-(1.2) is defined by(1.3) u ( t, x ) = min y ∈ R n n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o , here H ∗ denotes the Fenchel conjugate of the convex function H. This formula gives the unique viscosity solution to the above Cauchy problem [4, 9,7, 17]. If H = H ( t, x, p ) is a convex function with respect to the last variable, then therepresentation of viscosity solution of the problem (1.1)-(1.2) (which can be consideredas a general form of Hopf-Lax formula) is established and studied thoroughly; see [9, 7]and references therein. Especially, in the monograph [7], the analysis of regularity Key words and phrases.
Hamilton-Jacobi equation, Hopf-Lax formula, generalized characteristics,viscosity solution, semiconvexity, semiconcavity.This research is partially supported by the NAFOSTED, Vietnam, under grant properties of the viscosity solution is studied comprehensively where method of calculusof variation is used.This paper is devoted to investigate some differential properties of formula (1.3) fora class of convex Hamiltonians H = H ( p ) . We define and then use the “generalizedcharacteristics” for nonsmooth initial data to connect Hopf-Lax formula, the way issomewhat similar to the case of classical solution of the problem since the solution u ( t, x ) may be of class C whereas the initial may not be. Although several differentialresults are known in the general cases for H = H ( t, x, p ) under strong assumptions,e.g., H ( t, x, p ) is of class C and the Hessian D H is definite positive; see [7], it is notconvenient to apply them to the case H = H ( p ) under minimum hypotheses. Therefore,in this paper we attempt to exploit tools and technique of Calculus and PDEs to dealwith the specific case and not to infer any result from Calculus of Variation. Severalproperties here can not directly deduced by the general ones.The structure of the paper is as follows. In section 2, we first collect some factsrelated to semidifferentials of a function and viscosity solution of the Cauchy problemfor Hamilton-Jacobi equations defined by Hopf-Lax formula for convex/concave Hamil-tonians. We show that the Hopf-Lax formula inherits the semiconvexity of initial datain short time under suitable assumption on H. In section 3, we introduce a kind of gen-eralized characteristics of the Cauchy problem using the semidifferentials of the intialdata. Then we present some differential properties of Hopf-Lax formula related to thegeneralized characteristics.In section 4, we study the regularity of viscosity solution u ( t, x ) defined by Hopf-Lax formula and verify some sufficient conditions such that the function u ( t, x ) isdifferentiable on the regions of the form (0 , t ∗ ) × R n . Finally, we rededuce a result onthe backward and forward solution based on Hopf-Lax formula, presented in [6] as anapplication of this section.We use the following notations. Let T be a positive number, Ω = (0 , T ) × R n ; | . | and h ., . i be the Euclidean norm and the scalar product in R n , respectively. For a function u ( t, x ) defined on Ω , we denote D x u = ( u x , . . . , u x n ) and Du = ( u t , D x u ) . The Hessianof a function h ( x ) on R n will be denoted by D h ( x ) . Preliminaries and Hopf-Lax formulas
Assumptions.
Consider the following problem (
H, σ )(2.1) ∂u∂t + H ( D x u ) = 0 , ( t, x ) ∈ Ω = (0 , T ) × R n , (2.2) u (0 , x ) = σ ( x ) , x ∈ R n . We suppose that the following conditions hold for H ( p ) and σ ( x ).(H0) H ( p ) is strictly convex on R n and superlinear, i.e., lim k p k→∞ H ( p ) k p k = + ∞ . (H1) σ ( x ) is continuous on R n and for every ( t , x ) ∈ [0 , T ) × R n there exist positiveconstants r, N such that (2.3) min k y k≤ N n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o < σ ( z ) + tH ∗ (cid:0) x − zt (cid:1) as k z k > N, | t − t | + k x − x k < r, t = 0 , where H ∗ is the Fenchel conjugate functionof H, i.e., H ∗ ( z ) = max y ∈ R n {h z, y i − H ( y ) } , ∀ z ∈ R n . Denote ζ ( t, x, y ) = σ ( y ) + tH ∗ (cid:0) x − yt (cid:1) , ( t, x, y ) ∈ (0 , T ) × R n × R n , and(2.4) ℓ ( t, x ) = { y ∈ R n | ζ ( t, x, y ) = min k y k≤ N ζ ( t, x, y ) = min R n ζ ( t, x, y ) } , then ℓ ( t, x ) is a compact set in R n by the hypothesis (H1) and the lower semi-continuityof the function ζ ( t, x, . ) . Remark .
1. The assumption (H1) can be considered as a compatible condition forthe Hamiltonian H ( p ) and initial data σ ( x ) .
2. Assume (H0) holds and σ ( x ) is Lipschitz continuous on R n . Then condition (H1)is automatically satisfied. In the sequel, dependent on the context, some time we useassumption (H0), (H1) or (H0) and σ ( x ) is Lipschitz continuous. Definition 2.2.
Assume (H0), (H1). Then the function u ( t, x ) defined on Ω by theformula(2.5) u ( t, x ) = min y ∈ R n n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o is called Hopf-Lax formula of the problem (2.1)-(2.2). Remark . Analogously, we can establish the Hopf-Lax formula for concave Hamil-tonian as follows. Let K : R n → R be a concave function. Assume that the convexfunction H ( p ) = − K ( p ) and − σ ( x ) satisfy the condition (H0), (H1). Then the Hopf-Lax formula for the Cauchy problem ( K, σ ) can be represented as(2.6) u ( t, x ) = max y ∈ R n n σ ( y ) − t ( − K ) ∗ (cid:0) y − xt (cid:1)o . Sub- and superdifferential of a function.
Let O be and open set of R m andlet v : O → R be a function and y ∈ O . We denote by D + v ( y ) (resp. D − v ( y )) the setof all p ∈ R m defined by: D + v ( y ) = { p ∈ R m | lim sup h → v ( y + h ) − v ( y ) − h p , h i| h | ≤ } (resp. D − v ( y ) = { p ∈ R m | lim inf h → v ( y + h ) − v ( y ) − h p , h i| h | ≥ } ) . NGUYEN HOANG
We call D + v ( y ) (resp. D − v ( y )) superdifferential (resp. subdifferential) of the func-tion v ( y ) at y ∈ O . In general, D + v ( y ) or D − v ( y ) is called the semidifferential of v at y ∈ O . Note that, if v is a convex function, then the subdifferential D − v ( x ) coincides withsubgradient ∂v ( x ) of the function v. The following proposition presents some elementary properties of semidifferentialsof functions that are necessary in the sequel.
Proposition 2.4.
Let u, v be functions defined on an open set
O ⊂ R m . (i) Suppose that the function v attains a minimum (resp. maximum) at y ∈ O . Then ∈ D − v ( y ) (resp. ∈ D + v ( y )) . (ii) If the semidifferentials of u and v at y are not empty, then D ± u ( y ) + D ± v ( y ) ⊂ D ± ( u + v )( y ) . (iii) Suppose that, at a point y ∈ O , one of two functions u, v is differentiable andthe semidifferential of the other is not empty. Then we have D ± u ( y ) + D ± v ( y ) = D ± ( u + v )( y ) . Proof. (i) Suppose that v attains a minimum at y . Then for any small enough h, wehave v ( y + h ) ≥ v ( y ) . Thenlim inf h → v ( y + h ) − v ( y ) | h | ≥ . Thus, 0 ∈ D − v ( y ) . (ii) We check for the case of subdifferential since the other is the same. Take p ∈ D − u ( y ) , q ∈ D − v ( y ) , then(2.7) lim inf h → ( u + v )( y + h ) − ( u + v )( y ) − h p + q, h i| h | ≥ lim inf h → ( u )( y + h ) − ( u )( y ) − h p, h i| h | + lim inf h → ( v )( y + h ) − ( v )( y ) − h q, h i| h | ≥ . Thus, D − u ( y ) + D − v ( y ) ⊂ D − ( u + v )( y ) . (iii) Now assume that v is differentiable at y ∈ O and D − u ( y ) = ∅ , then D − ( u + v )( y ) = ∅ . Take s ∈ D − ( u + v )( y ) and set p = s − q where q = Dv ( y ) , we have u ( y + h ) − u ( y ) − h p, h i| h | = ( u + v )( y + h ) − ( u + v )( y ) − h p + q, h i| h |− v ( y + h ) − v ( y ) − h q, h i| h | . Taking lim inf of both sides as h → , we see that p ∈ D − u ( y ) . Thus s = p + q, and we get that D − ( u + v )( y ) = D − u ( y ) + { Dv ( y ) } . (cid:3) Definition 2.5.
Let O be a convex subset of R m and let v : O → R be a continuousfunction. (a) The function v is called semiconcave with linear modulus if there is a constant C > λv ( y ) + (1 − λ ) v ( y ) − v ( λy + (1 − λ ) y ) ≤ λ (1 − λ ) C | y − y | for any y , y in O and for any λ ∈ [0 , . Then the number C is called a semiconcavityconstant of v. The function v is semiconvex if and only if − v is semiconcave.(b) The function v is called uniformly convex with constant Λ > v ( y ) − Λ | y | , y ∈O is a convex function.con Remark .
1. The theory of semiconcave functions has been fully studied since thelast decades of previous century. The reader is referred to the monograph [7] for acomprehensive development of the topic.2. The notion of semiconcavity (resp. uniform convexity) is a special case of the no-tion σ -smoothness (resp. ρ -convexity) of a function, see [3]. The following propositionis extracted from Prop. 2.6 of the just cited article. Proposition 2.7.
Let v : R m → R be a convex function. Moreover,(i) Suppose that v is uniformly convex with a constant C > . Then the Fenchelconjugate function v ∗ is a semiconcave function with a semiconcavity constant C > . (ii) Suppose that v is a semiconcave function with a semiconcavity constant C ∗ > . Then v ∗ is a uniformly convex function with a constant C ∗ . Definition 2.8.
A continuous functions u : [0 , T ] × R n → R is called a viscosity solution of the Cauchy problem (1.1)-(1.2) on Ω = (0 , T ) × R n , provided that the following hold:(i) lim ( t,x ) → (0 ,x ) u ( t, x ) = σ ( x ) for all x ∈ R n ;(ii) For each ( t , x ) ∈ Ω and ( p, q ) ∈ D + u ( t , x ) , then(2.8) p + H ( t , x , q ) ≤ , and for each ( t , x ) ∈ Ω and ( p, q ) ∈ D − u ( t , x ) , then(2.9) p + H ( t , x , q ) ≥ . If the continuous function u satisfies (i) and (ii)-(2.8) (resp. (ii)-(2.9)), then it iscalled a viscosity subsolution (resp. supersolution) of the problem (1.1)-(1.2). It isnoted that, there are several propositions which are equivalent to this definition, e.g.,the notion of C -test function is used instead of semidifferentials, see [8].We collect here some important properties of Hopf-Lax formula, given by the follow-ing theorem. The proof is as the one of cited papers with minor adjustment. Theorem 2.9. (see [9, 7, 14] ). Assume (H0), (H1). Then(a) The function u ( t, x ) defined by Hopf-Lax formula u ( t, x ) = min y ∈ R n n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o is a viscosity solution of the problem (2.1)-(2.2). Moreover, it is locally Lipschitzcontinuous on Ω . NGUYEN HOANG (b) Suppose that σ ( x ) is semiconcave on R n or H is uniformly convex. Then for all t ∈ (0 , T ] , the solution u ( t, x ) is semiconcave in x on R n . (c) The solution u ( t, x ) is continuously differentiable in some open V ⊂ Ω if andonly if ℓ ( t, x ) is a singleton for all ( t, x ) ∈ V. Remark . From (c), we note that, if u ( t, x ) is differentiable at ( t , x ) then ℓ ( t , x ) = { p } is a singleton; see [14]. On the other hand, the sufficient condition in b) will beimproved in the theorem below. Theorem 2.11.
Assume (H0), (H1). Let ( t , x ) ∈ Ω such that ℓ ( t , x ) is a singleton.Then Hopf-Lax formula u ( t, x ) defined by (2.5) is differentiable at ( t , x ) .Proof. Suppose that ℓ ( t , x ) = { y } . We denote p t = H ∗ ( x − y t ) − t h x − y , H ∗ z ( x − y t ) i ; p = H ∗ z ( x − y t )and let α = lim inf ( h,k ) → (0 , u ( t + h, x + k ) − u ( t , x ) − p t h − h p, k i p h + | k | , for ( h, k ) ∈ R × R n . Then there exists a sequence ( h m , k m ) → (0 ,
0) such that lim m →∞ Φ m = α, whereΦ m = u ( t + h m , x + k m ) − u ( t , x ) − p t h m − h p, k m i p h m + | k m | . For each m ∈ N , we take y m ∈ ℓ ( t + h m , x + k m ) then(2.10) Φ m ≥ ϕ ( t + h m , x + k m , y m ) − ϕ ( t , x , y m ) − p t h m − h p, k m i p h m + | k m | ≥ ( t + h m ) H ∗ ( x + k m − y m t + h m ) − t H ∗ ( x − y m t ) − p t h m − h p, k m i p h m + | k m | , Applying the mean value theorem for the functions ( t, x ) tH ∗ ( x − y m t ) on the linesegment [( t , x − y m ) , ( t + h m , x + k m − y m )] we can see that the numerator of righthand side of (2.10) is equal to the following expression:Ψ m = h m (cid:16) H ∗ ( x + k ∗ m − y m t + h ∗ m ) − t + h ∗ m h H ∗ z ( x + k ∗ m − y m t + h ∗ m ) , x + k ∗ m − y m i − p t (cid:17) + k m (cid:16) H ∗ z ( x + k ∗ m − y m t + h ∗ m ) − p (cid:17) , where | h ∗ m | ≤ | h m | , | k ∗ m | ≤ | k m | . Taking into account the assumption (H1), it is easy to see that, for ( h m , k m ) smallenough, the sequence ( y m ) m is bounded, then we can choose a subsequence also denotedby ( y m ) m such that y m → y ∗ as m → ∞ . Since the set-valued mapping ( t, x ) ℓ ( t, x )is upper semicontinuous [18], then y ∗ ∈ ℓ ( t , x ) , that is y = y ∗ . Now, letting m → ∞ we have α = lim m →∞ Φ m ≥ lim m →∞ Ψ m p h m + | k m | = 0 . On the other hand, let β = lim sup ( h,k ) → (0 , u ( t + h, x + k ) − u ( t , x ) − p t h − h p, k i p h + | k | . We have, for y ∈ ℓ ( t , x ) , then u ( t , x ) = σ ( y ) + t H ∗ ( x − y t ) , and u ( t + h, x + k ) ≤ σ ( y ) + ( t + h ) H ∗ ( x + k − y t + h ) . Thus, applying the mean value theorem as above, we have u ( t + h, x + k ) − u ( t , x ) ≤ ( t + h ) H ∗ ( x + k − y t + h ) − t H ∗ ( x − y t ) ≤ h (cid:16) H ∗ z ( x + k ∗ − y t + h ∗ ) − t + h h x + k ∗ − y , H ∗ z ( x + k ∗ − y t + h ∗ ) − p t (cid:17) + k (cid:16) H ∗ z ( x + k ∗ − y t + h ∗ ) − p (cid:17) = Ψ( h, k ) , where | h ∗ | ≤ | h | , | k ∗ | ≤ | k | . Therefore β ≤ lim sup ( h,k ) → (0 , Ψ( h, k ) p h + | k | = 0 . Thus, lim ( h,k ) → (0 , u ( t + h, x + k ) − u ( t , x ) − p t h − h p, k i p h + | k | = 0 . The theorem is then proved. (cid:3)
Remark . We suppose that for fixed t ∈ (0 , T ] , the function u ( t, · ) is differentiableat x ∈ R n . Then ℓ ( t , x ) is a singleton, see [18]. By above theorem, we deduce that u ( t, x ) is also differentiable at ( t , x ) as a function of two variables. Theorem 2.13.
Assume (H0), (H1). In addition, let H = H ( p ) be a semiconcavefunction with the semiconcavity constant θ − > and let σ be a semiconvex functionwith constant B > . Then there exists t ∗ ∈ (0 , T ) such that for all t ∈ (0 , t ∗ ) , thefunction v ( x ) = u ( t , x ) is semiconvex, where u ( t, x ) is the Hopf-Lax formula definedby (2.5).Proof. By assumption and Prop. 2.7, we first note that the Legendre conjugate function H ∗ is a uniformly convex function with constant θ > . Therefore the function φ ( p ) = H ∗ ( p ) − θ | p | is convex and then, for all a, b ∈ R n we have(2.11) H ∗ ( a ) + H ∗ ( b ) − H ∗ ( a + b ≥ θ | a | + | b | − | a + b | ) = θ | a − b | . Next, we follow the argument in the proof of Theorem 3.5.3 (iv) [7] with an appropriateadjustment. Take γ > θγ > B and then choose t ∗ ∈ (0 , T ) such that NGUYEN HOANG < γt ∗ ≤ . Let t ∈ (0 , t ∗ ) , x , x ∈ R n , pick out y ∈ ℓ ( t , x ) , y ∈ ℓ ( t , x ); usingthe inequality (2.11) we have u ( t , x ) + u ( t , x ) − u ( t , x + x ≥ σ ( y ) + t H ∗ ( x − y t ) + σ ( y )+ t H ∗ ( x − y t ) − σ ( y + y − t H ∗ ( x − y + x − y t ) ≥ σ ( y )+ σ ( y ) − σ ( y + y t (cid:0) H ∗ ( x − y t ) + H ∗ ( x − y t ) − H ∗ ( x − y + x − y t ) (cid:1) ≥ − B | y − y | + θ t | ( x − x ) − ( y − y ) | ≥ − B | y − y | + 1 γt Λ4 (cid:0) | x − x | + | y − y | − h x − x , y − y i (cid:1) . Using the obvious inequality 2 h a, b i ≤ | a | ǫ + ǫ | b | for ǫ > , we see thatΛ4 (cid:0) | x − x | + | y − y | − h x − x , y − y i (cid:1) ≥ Λ4 (cid:0) | x − x | + | y − y | − ΛΛ − B | x − x | − Λ − B Λ | y − y | (cid:1) = − Λ B Λ − B (cid:12)(cid:12) x − x (cid:12)(cid:12) + B (cid:12)(cid:12) y − y (cid:12)(cid:12) . Therefore, u ( t , x ) + u ( t , x ) − u ( t , x + x ≥ − Λ Bγt (Λ − B ) (cid:12)(cid:12) x − x (cid:12)(cid:12) + ( Bγt − B ) | y − y | ≥ − Λ Bγt (Λ − B ) (cid:12)(cid:12) x − x (cid:12)(cid:12) . Thus, the function v ( x ) = u ( t , x ) is a semiconvex function. (cid:3) Generalized characteristics
In this section we focus on the study of the differentiability of function u ( t, x ) givenby Hopf-Lax formula on the characteristics. To this aim, let us recall the Cauchymethod of characteristics for problem (2.1)-(2.2).We first suppose that H ( p ) and σ ( x ) are of class C . The characteristic differential equations of problem (2.1)-(2.2) is as follows(3.1) ˙ x = H p ; ˙ v = h H p , p i − H ; ˙ p = 0with initial conditions(3.2) x (0 , y ) = y ; v (0 , y ) = σ ( y ) ; p (0 , y ) = σ y ( y ) , y ∈ R n . Then a characteristic strip of the problem (2.1)-(2.2) (i.e., a solution of the systemof differential equations (3.1) - (3.2)) is defined by(3.3) x = x ( t, y ) = y + tH p ( σ y ( y ) ,v = v ( t, y ) = σ ( y ) + t (cid:0) h H p ( σ y ( y )) , σ y ( y ) i − H ( σ y ( y )) (cid:1) ,p = p ( t, y ) = σ y ( y ) . The first component of solutions (3.3) is called the characteristic curve (briefly, char-acteristics) emanating from (0 , y ) , y ∈ R n , i.e., the curve defined by(3.4) C : x = x ( t, y ) = y + tH p ( σ y ( y )) , t ∈ [0 , T ] . Let t ∈ (0 , T ] . If for any t ∈ (0 , t ) such that x ( t, · ) : R n → R n is a diffeomorphism,then u ( t, x ) = v ( t, x − ( t, x )) is a C solution of the problem on the region (0 , t ) × R n . Now we assume that σ is merely a continuous function on R n . We use followingnotation D σ ( y ) = ( D + σ ( y ) ∪ D − σ ( y ) , if D + σ ( y ) ∪ D − σ ( y ) = ∅ , { } , otherwise.It is known that, the subset of R n such that D + σ ( y ) = ∅ (resp. D − σ ( y ) = ∅ ) isdense in R n . Now, the initial condition p (0 , y ) = σ y ( y ) can be replaced as follows p (0 , y ) = q, for some q ∈ D σ ( y ) , then a generalized characteristics emanating from (0 , y ) is defined by the curve C : x = x ( t, y ) = y + tH p ( q ) , q ∈ D σ ( y ) . For each ( t, x ) ∈ (0 , T ] × R n , we denote by ℓ ∗ ( t, x ) the set of y ∈ R n such that thereis a generalized characteristic curve starting from (0 , y ) , y ∈ R n goes through the point( t, x ) . In other words, y ∈ ℓ ∗ ( t, x ) if and only if x = y + tH p ( q ) , for some q ∈ D σ ( y ) . Using the initial condition q ∈ D σ ( y ) for p (0 , y ), we see that in general, there is abundle of generalized characteristics that emanates at an initial point (0 , y ) , y ∈ R n . Nevertheless, in the following, we can single out “right” characteristics going througha point ( t , x ) , t > σ is of class C . The following theorem is known under assumptions that
H, σ are of class C , e.g.,see [7, 14]. It remains true for the case of generalized characteristic curves and theproof is similar to the earlier one. For the reader’s convenience, we write down theargument. Theorem 3.1.
Let H ∈ C ( R n ) and σ ∈ C ( R n ) satisfying (H0), (H1) and let ( t, x ) ∈ (0 , T ] × R n . Then ℓ ( t, x ) ⊂ ℓ ∗ ( t, x ) and the solution u ( t, x ) of the problem (2.1)- (2.2)given by Hopf-Lax formula (2.5) can be defined as follows u ( t, x ) = min y ∈ R n n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o = min y ∈ ℓ ∗ ( t,x ) n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o . Proof.
Let ζ ( t, x, y ) = σ ( y ) + tH ∗ (cid:0) x − yt (cid:1) , ( t, x ) ∈ Ω , y ∈ R n . Then u ( t, x ) = min y ∈ R n ζ ( t, x, y ) . Since H is a strictly convex and superlinear function, then H ∗ is continuously differ-entiable on R n . Therefore the minimum of ζ ( t, x, . ) is attained at some y ∈ R n whichis a stationary point of ζ ( t, x, . ) . Then by Propositions 2.4 we get0 ∈ D − y ζ ( t, x, y ) = D − y σ ( y ) − H ∗ z (cid:0) x − yt (cid:1) . Thus, H ∗ z (cid:0) x − yt (cid:1) = q ∈ D − σ ( y ) . From this equality and by a differential property of theFenchel conjugate functions, we have H p ( q ) = x − yt . Consequently, x = y + tH p ( q ) , where q ∈ D − σ ( y ) , thus y ∈ ℓ ∗ ( t, x ). Since then, wehave min y ∈ R n n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o = min y ∈ ℓ ∗ ( t,x ) n σ ( y ) + tH ∗ (cid:0) x − yt (cid:1)o and ℓ ( t, x ) ⊂ ℓ ∗ ( t, x ) . (cid:3) Corollary 3.2.
Let ( t , x ) ∈ (0 , T ] × R n . Then for each y ∈ ℓ ( t , x ) , there is a uniquecharacteristic curve of the form C : x = x ( t, y ) = y + tH p ( q ) starting at (0 , y ) and goesthrough the point ( t , x ) , where q = H ∗ z (cid:0) x − yt (cid:1) ∈ D − σ ( y ) . Proof.
Since ∅ 6 = ℓ ( t , x ) ⊂ ℓ ∗ ( t , x ) , we take y ∈ ℓ ( t , x ) , then H ∗ z (cid:0) x − yt (cid:1) = q ∈ D − σ ( y ) . Let C : x = x ( t, y ) = y + tH p ( q ) be the corresponding generalized character-istic curve defined by y and q, we rewrite C as x = x ( t, y ) = y + t ( x − yt ) . Thus it isobvious that C goes through the point ( t , x ) and it is unique since C is the straightline joining two points (0 , y ) and ( t , x ) . (cid:3) Now, let C : x = x ( t, y ) = y + tH p ( q ) , q ∈ D σ ( y ) be a generalized characteristiccurve starting from y ∈ R n . Suppose that C goes through a point ( t , x ) , t > x = y + t H p ( q ) . Thus, C can be rewritten as the form x = x ( t ) = x + ( t − t ) H p ( q )where q ∈ D σ ( y ) and y verifies the equality y = x − t H p ( q ) , i.e., y ∈ ℓ ∗ ( t , x ) . FromCorollary 3.2, we get the following definition.
Definition 3.3.
We say that the characteristic curve C : x = x ( t, y ) = y + tH p ( q ) , q ∈ D σ ( y ) is of the type (I) at point ( t , x ) ∈ (0 , T ] × R n , if y ∈ ℓ ( t , x ) . If y ∈ ℓ ∗ ( t , x ) \ ℓ ( t , x ) then C is said to be the type (II) at ( t , x ) . To analyze properties of type of characteristic curves at a point ( t , x ) ∈ (0 , T ] × R n we need the following lemmas. Lemma 3.4. [9]
For each x ∈ R n and ≤ s < t ≤ T, the Hopf-Lax formula (2.5) canbe represented as u ( t, x ) = min y ∈ R n n u ( s, y ) + ( t − s ) H ∗ (cid:16) x − yt − s (cid:17)o . A useful property of strictly convex function can be deduced from the following.
Lemma 3.5.
Let v be a convex function and D = dom v ⊂ R n . Suppose that thereexist p, p ∈ D, p = p and y ∈ ∂v ( p ) such that h y, p − p i = v ( p ) − v ( p ) . Then for all z in the straight line segment [ p, p ] we have v ( z ) = h y, z i − h y, p i + v ( p ) . Moreover, y ∈ ∂v ( z ) for all z ∈ [ p, p ] . Proof.
See [15]. (cid:3)
Now we investigate properties of characteristic curves of type (I), (II) at ( t , x ) givenby the following theorems. Theorem 3.6.
Assume (H0), (H1). Let ( t , x ) ∈ (0 , T ] × R n , q = H ∗ z ( x − y t ) ∈ D − σ ( y ) where y ∈ ℓ ( t , x ) and let C : x = x ( t ) = x + ( t − t ) H p ( q ) , be a generalized characteristic curve of type (I) at ( t , x ) . Then y ∈ ℓ ( t, x ) for all ( t, x ) ∈ C , < t ≤ t . Moreover, ℓ ( t, x ) = { y } for all ( t, x ) ∈ C , ≤ t < t . Proof.
Let ( t , x ) ∈ C , < t ≤ t . By Hopf-Lax formula, one has u ( t , x ) = σ ( y ) + t H ∗ ( x − y t ) , y ∈ ℓ ( t , x ) . By Lemma 3.4, u ( t , x ) = min z ∈ R n { u ( t , z ) + ( t − t ) H ∗ ( x − zt − t ) } thus, u ( t , x ) ≤ u ( t , x ) + ( t − t ) H ∗ ( x − x t − t ) . Therefore, σ ( y ) + t H ∗ ( x − y t ) ≤ u ( t , x ) + ( t − t ) H ∗ ( x − x t − t ) . Since the points (0 , y ) , ( t , x ) , ( t , x ) belong to C , we have x − x t − t = x − y t = x − y t = H p ( q ) , then σ ( y ) + t H ∗ ( x − y t ) ≤ u ( t , x ) . Thus, y ∈ ℓ ( t , x ) . Next, we prove that ℓ ( t, x ) = { y } for all ( t, x ) ∈ C , < t < t . To this end, take anarbitrary y ∈ R n , y = y and denote by η ( t, y ) = ζ ( t, x, y ) − ζ ( t, x, y ) , ( t, x ) ∈ C , t ∈ (0 , t ] , where ζ ( t, x, y ) = σ ( y ) − tH ∗ ( x − yt ) . Then(3.5) η ( t, y ) = σ ( y ) − σ ( y ) + t (cid:0) H ∗ ( x − yt ) − H ∗ ( x − y t ) (cid:1) . for t ∈ (0 , t ] , x = x ( t ) . We shall prove that η ( t, y ) > t ∈ (0 , t ) . It is obviously that, η ( t , y ) ≥ . Note that x = x ( t ) = x + ( t − t ) H p ( q ) , thus y = x (0) = x − t H p ( q ) . Then from (3.5) with a simple calculation, we obtain η ′ t ( t, y ) = H ∗ ( x − yt ) − H ∗ ( x − y t ) + 1 t D y − y , H ∗ z ( x − yt ) E . Since H ∗ is a strictly convex function and y = y , we use a monotone property ofsubdifferential of convex function and Lemma 3.5 to get H ∗ ( x − yt ) − H ∗ ( x − y t ) < t D y − y, H ∗ z ( x − yt ) E . Thus, η ′ t ( t, y ) < , ∀ t ∈ (0 , t ) and the function η ( t, y ) is strictly decreasing on theinterval [0 , t ] . Therefore, η ( t, y ) > η ( t , y ) ≥ , ∀ t ∈ [0 , t ) . This means that, if y = y then y / ∈ ℓ ( t , x ) . The theorem now is proved. (cid:3)
Example.
Consider the following Cauchy problems ( ∂u∂t + | u x | = 0 , ( t, x ) ∈ Ω = (0 , T ) × R ,u (0 , x ) = σ ( x ) , x ∈ R . (i) Let σ ( x ) = −| x | . Then Hopf formula gives viscosity u ( t, x ) = min y ∈ R {−| y | + 12 t ( x − y ) } = −| x | − t . For ( t , x ) = (1 , , we see that ℓ (1 ,
0) = { , − } and D − σ (0) = ∅ , D + σ (0) = [ − , . The characteristic curves C of the problem have the form x = x ( t, y ) = y + tq, q ∈ D σ ( y ) . Suppose that C goes through (1 , , then 0 = y + q. If C starts from (0 , q = 0 ∈ D σ (0) , otherwise, where y = 0 , then q = ± , thus y = ∓ . Thus,the characteristic curves going through (1 ,
0) consist of two of type (I): x = (1 , t ) =1 − t, x = x ( − , t ) = − t and one of type (II) is x = x (0 , t ) = 0 . Note that a bundleof characteristic curves emanating from (0 ,
0) is x = x ( t,
0) = αt, where α ∈ [ − , . (ii) Let σ ( x ) = −| x | . Then Hopf formula gives a viscosity u ( t, x ) = x − , x > t − x − t , x < t x t , | x | ≤ t, and u ( t, x ) is of class C (Ω) . For any ( t , x ) ∈ Ω there is a unique characteristic curvegoing through this point, since u ( t, x ) is differentiable at there.In this case, a bundle of characteristic curves emanating from (0 ,
0) is also x = x ( t,
0) = αt, where α ∈ [ − , . All characteristic curves in the bundle are of type (II),except x = x ( t,
0) = 0 is of type (I). By above examples, we see that, if the characteristic curve C is of type (II) at ( t , x )then it may be of type (II) at any point ( t, x ) ∈ C , ≤ t ≤ t . Nevertheless, if thegiven data are of class C ( R n ) , we have the following: Theorem 3.7.
Assume (H0), (H1). In addition, suppose that
H, σ are of class C . Take ( t , x ) ∈ Ω and let C : x = x ( t ) = x + tH p ( σ y ( y )) be a characteristic curve oftype (II) at ( t , x ) . Then there exists θ ∈ (0 , t ) such that C is of type (I) at ( θ, x ( θ )) and C is of type (II) for all point ( t, x ) ∈ C , t ∈ ( θ, t ] . Proof.
The proof of this theorem is similar to the one of Thm. 2.11 in [15]. (cid:3)
For a locally Lipschitz function, the usage of notions of sub- and superdifferential aswell as reachable gradients to study its differentiability is effective. We use Theorems3.6, 2.11 to establish a relationship between ℓ ( t , x ) and the set of reachable gradients. Definition 3.8.
Let v = v ( t, x ) : Ω → R and let ( t , x ) ∈ Ω . We define the set D ∗ v ( t , x ) of reachable gradients of v ( t, x ) at ( t , x ) as follows: R n +1 ∋ ( p, q ) ∈ D ∗ v ( t , x ) if and only if there exists a sequence ( t k , x k ) k ⊂ Ω \{ ( t , x ) } such that v ( t, x ) is differentiable at ( t k , x k ) and,( t k , x k ) → ( t , x ) , ( v t ( t k , x k ) , v x ( t k , x k )) → ( p, q ) as k → ∞ . If v ( t, x ) is a locally Lipschitz function, then D ∗ v ( t, x ) = ∅ and it is a compact set,see [7], p.54.Now let u ( t, x ) be the Hopf-Lax formula defined by (2.5) and let ( t , x ) ∈ Ω . Wedenote by H ( t , x ) = { ( − H ( q ) , q ) } where q = H ∗ z ( x − y t ) ∈ D − σ ( y ) , and y ∈ ℓ ( t , x ) . Then a relationship between D ∗ u ( t , x ) and the set ℓ ( t , x ) is given by the followingtheorem. Theorem 3.9.
Assume (H0) and (H1). Let u ( t, x ) be the Hopf-Lax formula for problem(2.1)-(2.2). Then for all ( t , x ) ∈ Ω , we have D ∗ u ( t , x ) = H ( t , x ) . Proof.
Let ( p , q ) be an element of H ( t , x ) , then p = − H ( q ) where q = H ∗ z ( x − y t ) ∈ D − σ ( y ) for some y ∈ ℓ ( t , x ) . Let C be the characteristic curve of type (I) at( t , x ) starting form y . By Theorem (3.6), the solution u ( t, x ) is differentiable atall points ( t, x ) ∈ C , t ∈ [0 , t ) . Put t k = t − /k, then C ∋ ( t k , x k ) → ( t , x )and ( u t ( t k , x k ) , u x ( t k , x k )) = ( − H ( q ) , q ) → ( − H ( q ) , q ) ∈ D ∗ u ( t , x ) as k → ∞ . Therefore, H ( t , x ) ⊂ D ∗ u ( t , x ) . On the other hand, let ( p, q ) ∈ D ∗ u ( t , x ) and ( t k , x k ) k ⊂ Ω \ { ( t , x ) } such that u ( t, x ) is differentiable at ( t k , x k ) and ( t k , x k ) → ( t , x ) , ( u t ( t k , x k ) , u x ( t k , x k )) → ( p, q ) as k → ∞ . For each k ∈ N , we take y k ∈ ℓ ( t k , x k ) such that ( u t ( t k , x k ) , u x ( t k , x k )) =( − H ( q k ) , q k ) where q k = H ∗ z ( x k − y k t k ) ∈ D − σ ( y k ) . Since ( y k ) k is bounded, we can assumethat y k converges to y . Since multivalued function ℓ ( t, x ) is u.s.c, then letting k → ∞ , we see that y ∈ ℓ ( t , x ) and p = lim k →∞ − H ( q k ) = − H ( q ) . Thus ( p, q ) ∈ H ( t , x ) . The theorem is then proved. (cid:3)
Remark . From Theorems 2.11 and 3.9, we see that if u ( t, x ) is differentiable at( t, x ) then Du ( t, x ) = ( u t ( t, x ) , D x u ( t, x )) = ( H ∗ ( x − y t ) − t h x − y , H ∗ z ( x − y t ) i , H ∗ z ( x − y t ) , for the unique y ∈ ℓ ( t, x ) . If u ( t, x ) is not differentiable at ( t, x ) , then D + u ( t, x ) = co H ( t, x ) and D − u ( t, x ) = ∅ . Therefore it may be helpful to use this result to study some properties of derivativesof the solution of the problem (2.1)-(2.2) defined by Hopf-Lax formula.4.
Regularity of Hopf-Lax formula
In this section we will study the strips of the form V = (0 , t ∗ ) × R n ⊂ Ω such that u ( t, x ) is continuously differentiable on them. Theorem 4.1.
Assume (H0), (H1). Let u ( t, x ) be the viscosity solution of problem(2.1)-(2.2) defined by Hopf-type formula (2.5). Suppose that there exists t ∗ ∈ (0 , T ) such that the mapping: y x ( t ∗ , y ) = y + t ∗ H p ( σ y ( y )) is injective. Then u ( t, x ) iscontinuously differentiable on the open strip (0 , t ∗ ) × R n . Proof.
Let ( t , x ) ∈ (0 , t ∗ ) × R n and let C : x = x + ( t − t ) H p ( p )where p ∈ D σ ( y ) , y ∈ ℓ ( t , x ) , be the characteristic curve going through ( t , x )defined as in Corollary 3.2.Let ( t ∗ , x ∗ ) be the intersection point of C and plane ∆ t ∗ : t = t ∗ . By assumption, themapping y x ( t ∗ , y ) is injective and ℓ ( t ∗ , x ∗ ) = ∅ , so there is unique a characteristiccurve passing ( t ∗ , x ∗ ) . This characteristic curve is exactly C . Therefore, we can rewrite C as follows: x = x ∗ + ( t ∗ − t ) H p ( p ∗ )where p ∗ ∈ D − σ ( y ∗ ) is unique defined with respect to y ∗ ∈ ℓ ( t ∗ , x ∗ ) . Since ℓ ∗ ( t ∗ , x ∗ ) is a singleton, so is ℓ ( t ∗ , x ∗ ) . Consequently, C is of type (I) at ( t ∗ , x ∗ )and ℓ ( t, x ) = { y ∗ } for all ( t, x ) ∈ (0 , t ∗ ) × R n , particularly at ( t , x ) and then, y ∗ = y . Applying Theorems 2.11, 2.8 b) we see that u ( t, x ) is of class C in (0 , t ∗ ) × R n . (cid:3) Note that at some point ( t , x ) ∈ Ω where u ( t, x ) is differentiable there may be morethan one characteristic curve goes through, that is ℓ ∗ ( t , x ) may not be a singleton.Next, we have: Theorem 4.2.
Assume (H0), (H1). Moreover, let σ be Lipschitz and of class C ( R n ) . Suppose that ℓ ( t ∗ , x ) is a singleton for every point of the plane ∆ t ∗ = { ( t ∗ , x ) ∈ R n +1 : x ∈ R n } , < t ∗ ≤ T. Then the function u ( t, x ) defined by Hopf-Lax formula (2.5) iscontinuously differentiable on the open strip (0 , t ∗ ) × R n . Proof.
Let ( t , x ) ∈ (0 , t ∗ ) × R n . Since σ ( x ) is Lipschitz on R n then there exists m > | σ y ( y ) | ≤ m for all y ∈ R n . By assumption, for each z ∈ R n , there exists unique p ( z ) ∈ ℓ ( t ∗ , z ) such that C z : x ( t ) = z +( t − t ∗ ) H p ( σ ( p ( z )) is the unique characteristic curve of type (I) going through ( t ∗ , z ) . Since the multivalued function z ℓ ( t ∗ , z ) is u.s.c; see [ ? ] and, by assumption ℓ ( t ∗ , z ) = { p ( z ) } is a singleton for all z ∈ R n , then z p ( z ) is continuous.Consider the following mapping Λ : R n → R n defined byΛ( z ) = x − ( t ∗ − t ) H p ( σ y ( p ( z )) , ∀ z ∈ R n , where p ( z ) ∈ ℓ ( t ∗ , z ) . Then Λ( z ) is also continuous on R n . Since σ y ( p ( z )) is bounded and H p ( p ) is continuous, there exists M > | Λ( z ) − x | ≤ | t ∗ − t || H p ( σ y ( p ( z )) | ≤ M, ∀ z ∈ R n . Therefore Λ is a continuous function from the closed ball B ′ ( x , M ) into itself. ByBrouwer theorem, Λ has a fixed point x ∗ ∈ B ′ ( x , M ) , i.e., Λ( x ∗ ) = x ∗ , hence x = x ∗ + ( t ∗ − t ) H p ( σ y ( p ( x ∗ ))In other words, there exists a characteristic curve C of the type (I) at ( t ∗ , x ∗ )described as in Theorem 3.6 passing ( t , x ). Since ℓ ( t ∗ , x ∗ ) is a singleton, so is ℓ ( t , x ) . Applying Theorem 2.11, we see that u ( t, x ) is continuously differentiable in(0 , t ∗ ) × R n . (cid:3) We note that the hypotheses of above theorems are equivalent to the fact that,there is a unique characteristic curve of type (I) at points ( t ∗ , x ∗ ) , x ∗ ∈ R n goingthrough ( t , x ) . This is also equivalent to the fact that the function ϕ ( x ) = u ( t ∗ , x ) isdifferentiable on R n . In general, at some point ( t , x ) ∈ (0 , t ∗ ) × R n where u ( t, x ) isdifferentiable there may be more than one characteristic curves of type (I) or (II) atpoint ( t ∗ , x ∗ ) as above going through ( t , x ) , that is ℓ ∗ ( t ∗ , x ∗ ) may not be a singleton.Even neither is ℓ ( t ∗ , x ∗ ) . Nevertheless, we have:
Theorem 4.3.
Assume (H0), (H1). Let u ( t, x ) be the viscosity solution of the problem(2.1)-(2.2) defined by Hopf-Lax formula. Suppose that there exists t ∗ ∈ (0 , T ) suchthat all characteristic curves passing ( t ∗ , x ) , x ∈ R n are of type (I). Then u ( t, x ) iscontinuously differentiable on the open strip (0 , t ∗ ) × R n . Proof.
We argue similarly to the proof of Theorem 4.1. Let ( t , x ) ∈ (0 , t ∗ ) × R n andlet C : x = x + ( t − t ) H p ( p )where p ∈ D − σ ( y ) , y ∈ ℓ ( t , x ) be the characteristic curve going through ( t , x )defined as above.Let ( t ∗ , x ∗ ) be the intersection point of C and plane ∆ t ∗ : t = t ∗ . Then we have x ∗ = x + ( t ∗ − t ) H p ( p )Therefore, we can rewrite C as x = x ∗ − ( t ∗ − t ) H p ( p ) + ( t − t ) H p ( p ) = x ∗ + ( t − t ∗ ) H p ( p )to see that it is also a characteristic curve passing( t ∗ , x ∗ ) . By assumption, C is of type(I) at this point, so it is of type (I) for all ( t, x ) ∈ C , ≤ t < t ∗ , by Theorem 3.6. Thus, ℓ ( t , x ) is a singleton. As before, we come to the conclusion of the theorem. (cid:3) The following corollary establishes the existence of a strip of differentiability of vis-cosity solution of the problem (2.1)-(2.2) defined by Hopf-Lax formula.
Corollary 4.4.
Assume (H0), (H1). In addition, suppose that the function H = H ( p ) is both uniformly convex and semiconcave; σ is semiconvex. Then there is the greatestnumber t ∗ ∈ (0 , T ] such that the Hopf-Lax formula is of class C ((0 , t ∗ ) × R n ) . Proof.
Since H is uniformly convex, then for each t ∈ (0 , T ] , the function u ( t, · ) where u ( t, x ) defined by Hopf-Lax formula, is a semiconcave, see [9]. On the other hand,by Theorem 2.7, there exists t ∈ (0 , T ] such that u ( t , · ) is semiconvex. Therefore, u ( t , · ) is differentiable on the plane t = t , and in virtue of Theorem 4.2, u ( t, x ) iscontinuously differentiable on the strip (0 , t ) × R n . Let t ∗ = sup { t ∈ (0 , T ] | u ( t, · ) is differentiable on R n } . Then arguing as above, wesee that u ( t, x ) ∈ C ((0 , t ∗ ) × R n ) . (cid:3) Remark .
1. If H ∈ C ( R n ) and there is α > α I ≤ D H ≤ αI where D H is the Hessian of H and I is the ( n × n )-unit matrix, then H is both uniformlyconvex and semiconcave.2. The above result can be considered as a version of the existence of the classicalsolution of the problem (2.1)-(2.2) under assumption that the data are only of class C . (cf. Corollary 1.5.5, [7].) An Application:
When backward solution is forward solution
It should be noted that, a function u ( t, x ) ∈ C ([0 , T ] × R n ) is a classical solution ofthe equation u t + H ( t, x, D x u ) = 0 if and only if, the function v ( t, x ) = u ( T − t, x ) is aclassical solution of the equation v t − H ( T − t, x, D x v ) = 0 . This statement is not truein general, if u ( t, x ) is merely a viscosity solution of the equation.In [6] the authors introduce the notion “forward” and “backward” viscosity solutionof a Hamilton-Jacobi equation of the form u t + H ( t, x, Du ) = 0 as follows. The function u ( t, x ) ∈ C ([0 , T ] × R n ) is called a forward solution of the equation u t + H ( t, x, D x u ) = 0if it is a viscosity in the usual sense, while u ( t, x ) is said to be a backward solution ofthe equation, if v ( t, x ) = u ( T − t, x ) is a viscosity solution of the equation v t − H ( T − t, x, D x v ) = 0 . For a comprehensive knowledge of this topic, we refer the readers to the paper justmentioned above. Here we would like review the case u t + H ( Du ) = 0 where theviscosity solution is defined by Hopf-Lax formula.Following the presentation in [6], sect. 4, let function w ( t, x ) be a backward viscositysolution of the problem w t + H ( Dw ) = 0 with terminal condition w ( T, x ) = g ( x ) , i.e.,the function v ( t, x ) = w ( T − t, x ) is the viscosity solution of the problem v t − H ( Dv ) = 0with initial condition v (0 , x ) = g ( x ) . We reproduce the following theorem and giveanother proof for continuous differentiability of u ( t, x ) in the statement (ii). Note that,the origin proof is rather long but direct, ours seems to be “shorter” since it is inheritedseveral intermediate results. Theorem 4.6.
Assume H : R n → R is of class C ( R n ) , convex and superlinear, g ∈ C ( R n ) and is Lipschitz. Let w ( t, x ) be the backward viscosity solution of the problem w t + H ( Dw ) = 0; w ( T, x ) = g ( x ) and let u ( t, x ) be the (forward) viscositysolution of the problem u t + H ( Du ) = 0; u (0 , x ) = w (0 , x ) . Then(i) A necessary and sufficient condition for u ( T, x ) = g ( x ) is (4.1) g ( x ) = min z max y n g ( y ) − T H ∗ (cid:16) y − zT (cid:17) + T H ∗ (cid:16) x − zT (cid:17)o . (ii) If (4.1) holds, then u ( t, x ) = w ( t, x ) , ( t, x ) ∈ [0 , T ] × R n and u is continuouslydifferentiable on (0 , T ] × R n . Proof.
The proof of (i) and equality u ( t, x ) = w ( t, x ) is taken from [6]. We only checkthat the function u = u ( t, x ) is continuously differentiable in (0 , T ] × R n . Let u ( t, x ) bea unique viscosity of the problem (2.1)-(2.2), where σ ( x ) = w (0 , x ) defined by Hopf-Laxformula. By assumption, u ( T, x ) = g ( x ) is a differentiable function on R n , we applyTheorem 4.2 to get the desire result. (cid:3) Corollary 4.7.
Let the assumptions of Theorem 4.6 hold and let g ( x ) be a semiconvexfunction. Suppose that H ∈ C ( R n ) and is uniformly convex. If u ( T, x ) = g ( x ) then u ( t, x ) = w ( t, x ) , ( t, x ) ∈ [0 , T ] × R n and u is continuously differentiable on (0 , T ] × R n . Proof.
Following [9], u ( t, x ) is semiconcave in x for fixed t ∈ (0 , T ] . Thus, u ( T, x ) = g ( x )is both semiconvex and semiconcave, then u ( T, x ) is of class C ( R n ) . Applying Theorem4.2, we see that u ( t, x ) ∈ C ((0 , T ]) × R n ) . (cid:3) References [1] Aubin, J.P.,
Lax-Hopf formula and Max-Plus properties of solutions to Hamilton-Jacobi equations,
Nonlinear Differential Equations and Applications NoDEA (2012), 1-25.[2] Avantaggiati, A., and P. Loreti,
Lax type formulas with lower semicontinuous initial data andhypercontractivity results
Nonlinear Differential Equations and Applications NoDEA (2012), 1-27.[3] Az´e, Dominique, and J-P. Penot,
Uniformly convex and uniformly smooth convex functions,
Annales de la facult´e des sciences de Toulouse, Universit´e Paul Sabatier, Vol. 4, No. 4, (1995),pp. 705-730.[4] Bardi M. and L.C. Evans,
On Hopf ’s formulas for solutions of Hamilton-Jacobi equations,
Non-linear. Anal. TMA, 8(1984), No 11, pp. 1373-1381.[5] Bardi M., Capuzzo Dolcetta I., “Optimal control and viscosity solutions of Hamilton-Jacobiequations”, Birkhauser, Boston, 1997.[6] Barron E.N., Cannarsa P., Jensen R. & Sinestrari C.,
Regularity of Hamilton-Jacbi equationswhen forward is backward,
Indiana University Math. Journal, , 385-409, (1999).[7] Cannarsa P. & Sinestrari C., “Semiconcave functions, Hamilton-Jacobi equations and optimalcontrol”, Birkhauser, Boston 2004.[8] Crandall M.G. and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations,
Trans. Amer.Math. Soc. 277 (1983), 1-42.[9] Evans L. C., “Partial Differential Equations”, Graduate Studies in Math., Vol. 19, AMS, Provi-dence, Rhode Island, 1998.[10] Fleming W. H.,
The Cauchy Problem for a nonlinear first order partial differential Equations,
J.Diff. Eqs. (1969), 515-530.[11] Hopf E., Generalized solutions of non-linear equations of first order,
J. Math. Mech. (1965),951-973.[12] Lions, J. P., “Generalized solutions of Hamilton-Jacobi equations.” Research Notes in Math. 69,Pitman, Boston, MA, 1982. [13] Lax P. D., Hyperbolic systems of conservation laws II , Comm. Pure Appl. Math. (1957), 537-566.[14] Nguyen Hoang, Regularity of generalized solutions of Hamilton-Jacobi equations , Nonlinear Anal.59 (2004), 745-757[15] Nguyen Hoang,
Regularity of viscosity solutions defined by Hopf-type formula for Hamilton -Jacobi equations, arXiv preprint, arXiv:1208.3288 (2012).[16] Rockafellar T., “Convex Analysis”, Princeton Univ. Press, 1970.[17] Str¨omberg, T.,
The Hopf-Lax formula gives the unique viscosity solution,
Differential and IntegralEquations Explicit global Lipschitz solutions to first-ordernonlinear partial differential equations.
Viet. J. Math , No. 2 (1999), 93-114.[19] Tran Duc Van, Mikio Tsuji, Nguyen Duy Thai Son, “The characteristic method and its general-izations for first order nonlinear PDEs”, Chapman & Hall/CRC, 2000. Department of Mathematics, College of Education, Hue University, 32 LeLoi, Hue,Viet Nam
E-mail address ::