Regularity of Hamilton–Jacobi Equations when Forward Is Backward
E. N. Barron, P. Cannarsa, R. Jensen
&
C. Sinestrari
Abstract
. We introduce a general principle determining incertain cases the regularity of the viscosity solutions of HamiltonJacobi equations. This principle says that if one can solve theequation forward in time from some initial data and then backward in time resulting in the same initial data, then the solutionmust be
C
1
. Some cases are given when this holds as well as anexample when it does not. Convexity of either the hamiltonianor the initial data plays a crucial role throughout.1.
Introduction
When one tries to solve a ﬁrst order partial diﬀerential equation by themethod of characteristics one sees that if the characteristics do not cross thenthe solution is generally smooth if the initial value is smooth. The characteristicspropagate the initial data throughout the region as time progresses. This leadsone to suspect that if the information of the initial data is not lost in forwardtime, then the characteristics do not cross and hence the solution should besmooth. The problem is how to make sense out of the phrase that the initialinformation is not lost. This paper looks at one approach to this. In our sense,information is not lost if solving the equation forward in time up to time, say
t
=
T
, and then solving the equation backward in time starting with terminaldata at
T
, results in the recovery of the initial data at time
t
= 0.Let us brieﬂy explain what we mean by “forward” and “backward” solutionof a Hamilton–Jacobi equation of the form
u
t
+
H
(
t,x,Du
) = 0. Suppose that
u
(
t,x
) is a function deﬁned in [0
,T
]
×
R
n
, and let us denote by
v
the functionobtained from
u
by “reversing time”, i.e.
v
(
t,x
) =
u
(
T
−
t,x
). Clearly, if
u
is of class
C
1
,
u
is a solution of
u
t
+
H
(
t,x,Du
) = 0 if and only if
v
solves
v
t
−
H
(
T
−
t,x,Dv
) = 0. It is well known, however, that the two properties are nolonger equivalent when dealing with viscosity solutions of the equation. We say
385
386
E. N. Barron, P. Cannarsa, R. Jensen
&
C. Sinestrari
therefore that
u
is a
forward solution
of the equation if it is a viscosity solution inthe usual sense, while we say that
u
is a
backward solution
if
v
(
t,x
) =
u
(
T
−
t,x
)is a viscosity solution of
v
t
−
H
(
T
−
t,x,Dv
) = 0.In this paper we consider the problem
u
t
+
H
(
t,x,Du
) = 0 with some giveninitial data
u
(0
,x
) =
g
(
x
), and we solve it forward in time up to
T
. Then welook at the problem
w
t
+
H
(
t,x,Dw
) = 0, solved backward in time with terminaldata
w
(
T,x
) =
u
(
T,x
). We then address the following two questions.
(i)
Assuming that
w
≡
u
in [0
,T
]
×
R
n
, can we deduce that
u,w
are of class
C
1
?
(ii)
Assuming only that
w
(0
,x
)
≡
u
(0
,x
), can we deduce that
w
≡
u
in[0
,T
]
×
R
n
and that
u,w
are of class
C
1
?In the statements above the regularity of
u
and
w
is meant in (0
,T
)
×
R
n
,since a singularity may be present at the initial or ﬁnal time. Intuitively it seemsthat the answer, at least to question (i), should be aﬃrmative in general. Indeed,we prove that it is aﬃrmative in certain cases, but we also show by examplesthat it is not always aﬃrmative. In Examples 2.6 and 6.2 we provide counterexamples to property (i). The problem in the former of these examples is that theconvexity of the hamiltonian changes as time progresses. At the time when thehamiltonian changes convexity, the characteristics of our solution coalesce andthen fan out instantaneously. This is enough for a corner to develop but notenough to preclude solving forward and backward in time. In the latter examplethe hamiltonian is convex but not strictly convex. In this case, characteristics donot necessarily cross in the presence of a singularity, and so we are able to ﬁnda function which has a corner at all times and still is a forward and backwardsolution. Both types of behavior are excluded when the hamiltonian is strictlyconvex, as we will see in Theorem 2.5. However, even in this case, the answer toquestion (ii) can be negative, as will be shown in Example 6.3. So it seems that,for a general regularity result, a stronger condition is needed which guaranteesthat the characteristics never touch. Indeed the touching of the characteristicsmeans that the characteristic system of ode’s loses uniqueness and that is whatleads to singularities. The problem is to determine a veriﬁable condition on thesolution which is equivalent to no contact of characteristics. But determiningsuch a condition is a diﬃcult problem.In Section 2 we give a precise deﬁnition of a forward and backward solutionand consider question (i). An immediate consequence of assuming that we have afunction which is both a forward and a backward solution of
u
t
+
F
(
t,x,Du
) = 0,with
F
(
t,x,p
) strictly convex and superlinear in
p
, is that this forces
u
to be
C
1
(see Theorem 2.5). Essentially, the strict convexity and the forward–backwardproperty allow us to establish uniform bounds on weak second derivatives andthat is why we have regularity here. Next we consider the case
u
t
+
H
(
t,x,Du
) =0 where the hamiltonian
H
comes from a Mayer problem of optimal control, andso is convex in
Du
, but not strictly convex. Again we assume that we have in
Regularity of Hamilton–Jacobi Equations when Forward is Backward
387hand a solution which is both forward and backward. It is proved in Theorem2.8 that, if this solution is
C
1
both at the initial and ﬁnal time, then it is
C
1
everywhere. The regularity assumption on the initial and ﬁnal data is essentialas shown by the above mentioned Example 6.2.We then turn to study some cases where the stronger property (ii) can beproved. In the third section we consider
u
t
+
H
(
Du
) = 0, with
u
(0
,x
) =
g
(
x
)and
g
convex and smooth, but no convexity assumptions on
H
. Now we havethe Hopf formula [2] (see also [5], [6]) at our disposal and we prove that twosolutions satisfying our backward and forward condition must be smooth andcoincide. A necessary and suﬃcient condition for having
u
(0
,x
) =
w
(0
,x
) underthe assumptions of this section is that
p
→
g
∗
(
p
) +
TH
(
p
) is convex, where
g
∗
is the Legendre–Fenchel conjugate of
g
. Using this property we can prove (seeTheorem 3.1) that the answer to (ii) is aﬃrmative. Furthermore, we see thatsingularities cannot develop until the ﬁrst time
T
when
g
∗
+
TH
(
p
) becomesnonconvex.In the fourth section we show in Theorem 4.1 that if we consider
g
smoothand
H
=
H
(
p
) convex, again two solutions satisfying the backward and forward condition of (ii) must coincide and be smooth. The proof is based on theLax formula. This result will be extended in the next section (Theorem 5.5)to hamiltonians depending also on (
t,x
), but with more restrictive regularityassumptions.In the ﬁfth section we also address a third question, namely:
(iii)
Is it true in general (i.e. without assuming
w
(0
,x
)
≡
u
(0
,x
)) thatthe backward solution
w
of
w
t
+
H
(
t,x,Dw
) = 0 with terminal data
w
(
T,x
) =
u
(
T,x
) is of class
C
1
?This conjecture is motivated by the consideration that in the special case where
H
(
p
) =

p

2
the solution
w
is, at any ﬁxed time
t
∈
(0
,T
), an inf–sup convolutionof the initial value
g
of the forward problem, and therefore is smooth, see Lasryand Lions [13]. We also show that, in the case of a convex hamiltonian, thisconjecture is closely related to the validity of (ii). We prove that the answer to(iii) is aﬃrmative when
H
is strictly convex, independent of
x
, and the spacedimension is one, see Theorem 5.6. In the cases of just convexity of
H
in the gradient, spatial dependence of
H
, or nonconvexity of
H
using the Hopf hypotheses,the property does not hold, as shown by counterexamples 6.1, 6.2 and 6.3. Weleave as an open problem whether strict convexity of
H
in higher dimensions issuﬃcient for (iii) to hold, with
H
=
H
(
t,p
).2.
The case with
u
t
+
H
(
t,x,D
x
u
) = 0In this paper we consider HamiltonJacobi equations of the form
u
t
+
F
(
t,x,Du
) = 0 (
t,x
)
∈
(0
,T
)
×
R
n
.
(2.2.1)
388
E. N. Barron, P. Cannarsa, R. Jensen
&
C. Sinestrari
The assumptions on the function
F
: [0
,T
]
×
R
n
×
R
n
→
R
will vary duringthe paper, but we will always suppose
F
at least continuous. Solutions to thisequation are meant in the viscosity sense (see e.g. [3], [4], [10]).Let us now give the deﬁnition of backward solution of our equation.
Deﬁnition
2.1
.
A function
u
∈
C
([0
,T
]
×
R
n
)
is called a
backward (viscosity) solution
of equation
(2.2.1)
if
v
(
t,x
) :=
u
(
T
−
t,x
)
is a viscosity solution of
v
t
−
F
(
T
−
t,x,Dv
) = 0 (
t,x
)
∈
[0
,T
]
×
R
n
.
By a
forward
solution of the equation we mean a viscosity solution in the usual sense.
The following characterization follows immediately from the deﬁnition.
Proposition
2.2
.
The following properties are equivalent.
(i)
u
is a backward solution of equation
(2.2.1)
.
(ii)
u
is a
(
forward
)
solution of
−
u
t
−
F
(
t,x,Du
) = 0
.
(iii)
If
w
(
t,x
) =
−
u
(
T
−
t, x
)
and
G
(
t,x,p
) =
F
(
T
−
t, x,
−
p
)
, then
w
is a solution of
w
t
+
G
(
t,x,Dw
) = 0
.
(iv)
For any
(
t,x
)
∈
(0
,T
)
×
R
n
we have
p
t
+
F
(
t,x,p
x
)
≥
0
if
(
p
t
,p
x
)
∈
D
+
u
(
t,x
)
p
t
+
F
(
t,x,p
x
)
≤
0
if
(
p
t
,p
x
)
∈
D
−
u
(
t,x
)Of course in (iv) of the proposition, if the inequalities are reversed we getthe deﬁnition of a forward viscosity solution. From the above properties it isclear that the Cauchy problem for equation (2.2.1) with ﬁnal data is well posedin the class of backward solutions.There are several ways to deﬁne the sub and superdiﬀerentials in the statement (iv) but here is the most useful:
D
+
u
(
t,x
) =
{
p
= (
p
t
,p
x
) :
p
=
Dϕ
(
t,x
)
,
∃
ϕ
∈
C
1
, u
−
ϕ
≤
0
,
(
u
−
ϕ
)(
t,x
) = 0
}
,D
−
u
(
t,x
) =
{
p
= (
p
t
,p
x
) :
p
=
Dϕ
(
t,x
)
,
∃
ϕ
∈
C
1
, u
−
ϕ
≥
0
,
(
u
−
ϕ
)(
t,x
) = 0
}
.
Our ﬁrst aim is to prove the regularity of forward and backward solutionswhen
F
is strictly convex with respect to the third argument. Before doingthis, we need to recall the deﬁnition and some basic properties of semiconcavefunctions.
Deﬁnition
2.3
.
A function
u
:
A
→
R
, with
A
⊂
R
n
open, is called
semiconcave
if, for any convex compact set
K
⊂
A
, there exists a nondecreasing function
ω
K
:
R
+
→
R
+
such that
lim
r
→
0
ω
K
(
r
) = 0
and
λu
(
x
) + (1
−
λ
)
u
(
y
)
−
u
(
λx
+ (1
−
λ
)
y
)
≤
λ
(1
−
λ
)

x
−
y

ω
K
(

x
−
y

)(2.2.2)
Regularity of Hamilton–Jacobi Equations when Forward is Backward
389
for any
x,y
∈
K
and
λ
∈
[0
,
1]
.
The function
ω
K
is called a
modulus of semiconcavity
. A function
u
iscalled
semiconvex
if
−
u
is semiconcave. We remark that the previous deﬁnitionis more general than the one often used in the literature, where it is requiredthat
ω
K
(
r
) =
c
K
r
for some
c
K
>
0. We recall the following properties ([3], [7],[1]).
Proposition
2.4
.
(i)
If
u
∈
C
1
(
A
)
, then
u
is semiconcave in
A
, with
ω
K
equal to the modulus of continuity of
Du
in
K
. Conversely, if
u
:
A
→
R
is both semiconcave and semiconvex, then
u
∈
C
1
(
A
)
. If
u
is both semiconcave and semiconvex with
ω
K
linear, then
u
∈
C
1
,
1
loc
(
A
)
.
(ii)
If
u
is semiconcave then
u
is locally Lipschitz continuous and its superdif ferential
D
+
u
is nonempty everywhere.
(iii)
If
u
is semiconcave in
A
and
D
+
u
(
x
)
is a singleton for all
x
∈
A
, then
u
∈
C
1
(
A
)
.
We can now give our ﬁrst regularity result.
Theorem
2.5
.
Let
F
(
t,x,
·
)
be strictly convex and superlinear. Assume in addition that, for any
R >
0
there exists
C
R
>
0
such that

F
(
t,x,p
)
−
F
(
s,y,p
)
≤
C
R
(

t
−
s

+

x
−
y

)
, s,t
∈
[0
,T
]
, x,y
∈
R
n
,

p

< R.
If
u
is a forward and backward locally Lipschitz solution of
u
t
+
F
(
t,x,Du
) = 0
,
(
t,x
)
∈
(0
,T
)
×
R
n
,
then
u
∈
C
1
((0
,T
)
×
R
n
)
.Proof.
Since
u
is a forward solution of the equation,
u
is semiconcave on(0
,T
)
×
R
n
, by Theorem 3.2 in [8] and Proposition 2.5 in [16]. Since
u
is also abackward solution, it satisﬁes by Proposition 2.2(iv)
p
t
+
F
(
t,x,p
x
) = 0
,
∀
(
t,x
)
∈
(0
,T
)
×
R
n
,
∀
(
p
t
,p
x
)
∈
D
+
u
(
t,x
)
.
By the strict convexity of
F
, we deduce that
D
+
u
(
t,x
) is a singleton for any(
t,x
)
∈
(0
,T
)
×
R
n
. Thus, by Proposition 2.4(iii),
u
∈
C
1
((0
,T
)
×
R
n
).
Remark.
The local Lipschitz continuity of the solution
u
in the previous theoremcan be obtained under general conditions on the initial data. It suﬃces forinstance to prescribe an initial value which is lower semicontinuous and boundedfrom below.Of course, the same result holds if the hamiltonian is strictly concave everywhere. On the other hand, the following example shows that, if
H
changes