Ishii, Hitoshi

EPrint Series of Department of Mathematics, Hokkaido University

Asymptotic solutions of Hamilton-Jacobi equations
with convex Hamiltonians
Hitoshi ISHII∗
Abstract
The following discussion is based on joint work with N. Ichihara, especially on [14].
We are concerned with the Cauchy problem for the Hamilton-Jacobi equationut +H(x,Du) = 0 in Rn × (0,+∞),u( · , 0) = u0 on Rn, (1)
and study the long time behavior of the solution of (1).
We assume throughout the following (A1)–(A5).
(A1) H ∈ C(Rn × Rn),
(A2) inf{H(x, p) | x ∈ B(0, r), |p| ≥ R} −→ +∞ as R→ +∞ for every r > 0,
(A3) H(x, p) is convex with respect to p for every x ∈ Rn,
(A4) for each φ ∈ S−H , there exist C > 0 and ψ ∈ S−H−C such that
lim
|x|→∞
(φ− ψ)(x) =∞,
(A5) u0 ∈ C(Rn).
Here and henceforth, we denote by SH (resp., S−H , S+H) the set of all continuous viscosity
solutions (resp., subsolutions, supersolutions) of H(x, Du(x)) = 0 in Rn. Similarly,
given a domain Ω, we denote by SH(Ω) (resp., S−H(Ω), S+H(Ω)) the set of all continuous
viscosity solutions (resp., subsolutions, supersolutions) of H(x, Du(x)) = 0 in Ω.
Note that hypotheses (A1)–(A5) are not enough to assure the unique solvability of
(1) in the sense of viscosity solution. To start our discussion with this generality, we
∗E-mail: hitoshi.ishii@waseda.jp. Department of Mathematics, Faculty of Education and Inte-
grated Arts and Sciences, Waseda University. Supported in part by Grant-in-Aids for Scientific Re-
search, No. 18204009, JSPS.
2000 MSC : Primary 35B40; Secondary 35F25, 35B37.
Keywords : Hamilton-Jacobi equations, asymptotic solutions, long-time behavior, representation
formula.
1
define the (unique) solution of (1) as follows. For any ψ ∈ C(Rn) and t ≥ 0, we define
the function Ttψ on Rn by
Ttψ(x) = inf
{∫ 0
−t
L(η(s), η˙(s)), ds+ ψ(η(−t)) ∣∣ η ∈ C([−t, 0]; x)}, (2)
and refer the function u(x, t) := Ttu0(x) as the solution of (1). Here L is the Lagrangian
ofH defined by L(x, ξ) = sup{ξ·p−H(x, p)} for (x, ξ) ∈ R2n and C([a, b];x), with a < b,
denotes the space of all absolutely continuous functions (called curves) η : [a, b]→ Rn
(i.e. η ∈ AC([a, b])) such that η(b) = x. Also, C((−∞, a];x) denotes the space of all
functions η ∈ C((−∞, 0]) such that η ∈ C([c, a];x) for all c < a.
We remark here that Ttψ(x) is well-defined with the following interpretation and
Ttψ(x) ∈ [−∞, 0) for all (x, t) ∈ Rn× [0, ∞). To see this, we fix ψ ∈ C(Rn). Note that
L(x, ξ) is lower semicontinuous on R2n, due to the assumption that H ∈ C(R2n), and
hence that the function s 7→ L(η(s), η˙(s)) is Lebesgue measurable on (−∞, 0) for any
η ∈ AC((−∞, 0]). Noting that H(x, 0) = − infξ∈Rn L(x, ξ) for all x ∈ Rn, we observe
that, for each t > 0, r > 0 and η ∈ AC([−t, 0]) satisfying η([−t, 0]) ⊂ B(0, r), we have
L(η(s), η˙(s)) ≥ −maxx∈B(0, r)H(x, 0). Therefore, for η ∈ AC([−t, 0]), it is natural to
set ∫ 0
−t
L(η(s), η˙(s)) ds =∞
if s 7→ L(η(s), η˙(s)) is not integrable on (−t, 0). With this interpretation, Ttψ(x) is
well-defined for all (x, t) ∈ Rn × [0, ∞). Next we see that for all (x, t) ∈ Rn × [0, ∞),
Ttψ(x) ≤
∫ 0
−t
L(x, 0) ds+ ψ(x) = L(x, 0)t+ ψ(x) <∞.
Now, let us consider an example where Ttu0(x) = −∞ for some (x, t). Let n = 1,
H(p) = (1/2)|p|2 and u0(x) = −|x|2. Then the Lagrangian L of H is given by L(ξ) =
(1/2)|ξ|2. Consider the curve η ∈ C((−∞, 0]; 0) given by η(s) = cs, with c > 0, and
observe that for any t > 0,
Ttu0(0) ≤
∫ 0
−t
L(η˙(s)) ds+ u0(η(−t)) = c
2t
2
− (ct)2 = c
2t
2
(1− 2t),
which implies that Ttu0(0) = −∞ if t > 1/2. We recall that if the function u(x, t) :=
Ttu0(x) is continuous on an open set U ⊂ Rn × (0, ∞), then u is a viscosity solution
of ut +H(x,Du) = 0 in U .
Our objective is to investigate the long-time behavior of the solution of (1). More
precisely, we are concerned with the convergence of the form
u(x, t) + at− φ(x) −→ 0 in C(Rn) as t→∞ (3)
2
for some a ∈ R and φ ∈ C(Rn), where C(Rn) is equipped with the topology of locally
uniform convergence. Note that if u satisfies (1) in the viscosity sense, then the function
φ(x)− at, which we call an asymptotic solution of (1), enjoys the following stationary
Hamilton-Jacobi equation in the viscosity sense:
H(x,Dφ) = a in Rn. (4)
Thus a natural question related to (3) is the additive eigenvalue problem for H which
seeks for a pair (a, φ) ∈ R × C(Rn) such that φ is a solution of (4), i.e., φ ∈ SH−a.
The additive eigenvalue problem appears in ergodic control, in which it is called the
ergodic control problem, and in homogenization, in which it is called the cell problem.
A standard approach to (3) is first to solve the additive eigenvalue problem for
H and then to try to prove the convergence (3) for each fixed solution of the additive
eigenvalue problem for H. However, to simplify our presentation, we will deal only with
the latter step in the above approach and investigate if (3) holds for a fixed a ∈ R.
We assume moreover that a = 0. Indeed, convergence (3) with general a is equivalent
to (3) with a = 0 once H and u are replaced by H − a and u(x, t) + at, respectively.
Then, one of fundamental observations is the following.
Proposition 1. Let
u−0 (x) := sup{φ(x) |φ ∈ S−H , φ ≤ u0 in Rn},
u∞(x) := inf{ψ(x) |ψ ∈ SH , ψ ≥ u−0 in Rn}.
Under the additional assumption
(A6) −∞ < u−0 (x) ≤ u∞(x) <∞ for all x ∈ Rn,
we have
lim inf
t→∞
u(x, t) = u∞(x) for all x ∈ Rn,
The condition (A6) is equivalent to saying that
{φ ∈ S−H | φ ≤ u0 in Rn} 6= ∅ and {φ ∈ SH | φ ≥ u−0 in Rn} 6= ∅.
Thus our purpose here is to show the convergence:
u(·, t) −→ u∞ in C(Rn) as t→∞. (5)
Asymptotic problems of this type has been studied intensively in the last decade.
As one of the most typical cases, it was proved that if H satisfies (A1)–(A3) and H(x, p)
is Zn-periodic with respect to x and is strictly convex with respect to p, then for each
Zn-periodic initial function u0 ∈ BUC(Rn) there exists a solution (a, φ) ∈ R × C(Rn)
3
of the additive eigenvalue problem for H such that (3) is valid and the constant a is
determined independently of u0. We refer to the literatures [2, 4, 5, 8, 9, 18, 19, 20]
and references therein for more details. Remark that [2, 19] deal with non-convex
Hamiltonians whereas most of others are concerned only with convex ones.
It has also been of interest in recent years on the long-time behavior of viscosity
solutions to (1) that are not necessarily spatially periodic. As far as non-periodic
solutions are concerned, the above (A1)-(A6) are insufficient to obtain the convergence
(3) even if we admit strict convexity for H in any sense (see [3, 12]). The papers
[1, 11, 12, 15] deal with some situations in which the solution of (1) has indeed the
desired convergence of the form (3) for a suitable (a, φ).
Motivated by these earlier results, given a point z ∈ Rn, we will introduce three
general criteria (C1), (C2) and (C3), each of which, together mostly with (A1)–(A6)
above, guarantees the pointwise convergence of the solution u of (1):
u(z, t) −→ u∞(z) as t −→∞. (6)
We then apply these criteria to obtain general convergence results of the form (5) and
apply them to several examples. Our results cover most of existing results, and, on the
other hand, involve a few observations which seem to be new.
One of our new observations is concerned with strict convexity for H. As pointed
out in several literatures (see e.g. [2]), it is necessary in some situations to require a
sort of strict convexity for H, as a genuine nonlinearity for H, so that the solution of
(1) converges to an asymptotic solution as t → ∞. We use condition (A7)+ or (A7)−
which guarantees, respectively, strict convexity of H(x, p) in p at the zero level-set of H
“upward” or “downward” (see below). We point out here that some of our convergence
results involving (A7)− are not covered by the previous results. Another important
feature here is in the observations of a “switch-back” motion of nearly optimal curves
for the variational formula (2) for Ttu0 in some situations. In one-dimensional case we
have already encountered such a switch-back motion (see [13]), and here we investigate
with greater generality the cases where such switch-back motions appear.
We use the dynamical approach in our investigations here which is based on tools
from weak KAM theory (see e.g. [8, 10, 6, 7]) such as extremal curves, Aubry sets and
representations formulas for solutions. We state our general criteria for the convergence
to asymptotic solutions in terms of extremal curves, which seems inevitable to attain
further generality.
Now, we introduce our criteria for the pointwise convergence. We begin with exis-
tence of extremal curves.
Theorem 2. Let φ ∈ SH and z ∈ Rn. Then there exists a curve γ ∈ C((−∞, 0]; z)
4
such that for all t > 0, ∫ 0
−t
L(γ(s), γ˙(s)) ds = φ(z)− φ(γ(−t)). (7)
A curve γ ∈ C((−∞, 0]; z) is said to be extremal for φ at z if it satisfies (7) for all
t > 0. We denote by Ez(φ) the set of all extremal curves for φ at z. Also, we use the
notation: E(φ) := ⋃x∈Rn Ex(φ) and E := ⋃φ∈SH E(φ).
We fix any γ ∈ Ez and introduce the first criterion
(C1) lim
t→∞
(u0 − u∞)(γ(−t)) = 0.
Note that u−0 ≤ u0 in Rn by definition and that
lim inf
t→∞
(u∞ − u−0 )(γ(−t)) = 0.
Hence condition (C1) is equivalent to the condition
lim
t→∞
(u0 − u−0 )(γ(−t)) = 0. (8)
Theorem 3. Under condition (C1), the convergence (6) holds.
Next, we introduce our second criterion.
(C2) For each ε > 0 there exists a τ > 0 such that for any t > 0 and for some
η ∈ AC([−t, 0]),
η(−t) = η(0) = γ(−τ) and
∫ 0
−t
L(η(s), η˙(s)) ds < ε.
Theorem 4. Under condition (C2), the convergence (6) holds.
Our third criterion is the following.
(C3) For any ε > 0, there exists a τ > 0 and for each t ≥ τ , a σ(t) ∈ [0, τ ] such that
u∞(γ(−t)) + ε > u(γ(−t), σ(t)).
Note that the above inequality is equivalent to the condition that there is an η ∈
C([−σ(t), 0]; γ(−t)) such that
u∞(γ(−t)) + ε >
∫ 0
−σ(t)
L(η(s), η˙(s)) ds+ u0(η(−σ(t)).
In our next theorem, condition (C3) is used together with one of the conditions
(A7)± on H, which are certain strict convexity requirements on H. We set Q :=
{(x, p) ∈ R2n |H(x, p) = 0} and
S := {(x, ξ) ∈ R2n | (x, p) ∈ Q, ξ ∈ D−2 H(x, p) for some p ∈ Rn},
5
where D−2 H(x, p) stands for the subdifferential of H with respect to the p-variable.
(A7)+ (resp. (A7)−) There exists a modulus ω satisfying ω(r) > 0 for r > 0 such
that for all (x, p) ∈ Q, ξ ∈ D−2 H(x, p) and q ∈ Rn,
H(x, p+ q) ≥ ξ · q + ω((ξ · q)+) (resp. ≥ ξ · q + ω((ξ · q)−)), (9)
where r± := max{±r, 0} for r ∈ R.
We remark here that condition (A7)+ has already been used in [2, 12] to replace the
strict convexity of H(x, ·) in order to get the convergence (6) and also that condition
(A7)− has been discussed in [13] when n = 1.
Theorem 5. Assume that (C3) and either of (A7)+ or (A7)− are satisfied. Then (6)
holds.
We will apply the criteria above, to obtain more specific conditions on (H, u0) for
the convergence (5).
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Asymptotic solutions of Hamilton-Jacobi equations with convex Hamiltonians

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EPrint Series of Department of Mathematics, Hokkaido University