Prueba de hipótesis

≤ A^ε_n 0 0 −B_n^ε

!

≤ 3n Id −Id

−Id Id

!

. (6.24)

Using (6.16), (6.21), (6.22), (6.23), (6.24) and (i) of (2.2), we then deduce that 0 ≤ −m(κ + δ( ˆρ^ε_n)) + C n|x^ε_n− y_n^ε|² − ε(κ + δ(ˆρ^ε_n)) {(f + G^ε)(t^ε_n, x^ε_n) + f (y_n^ε)}

+ εL^ρˆεn(f + G^ε)(t^ε_n, x^ε_n) + L^ρˆεnf (y^ε_n)

for some C > 0 independent of n. Sending n to ∞, it follows from the compactness of K˜_η and (6.22) that

0 ≤ −m(κ + δ( ˆρ^ε)) + 2εL^ρˆεf (x_ε) − (κ + δ( ˆρ^ε))f (x_ε)

for some ˆρ^ε ∈ ˜K_η. Sending ε to 0 and using (6.20) finally leads to a contradiction since

κ, m > 0 and δ ≥ 0 by (2.9). 2

We now conclude the proof of Theorem 3.1.

Proof of (ii) and (iii) of Theorem 3.1. Observe that a supersolution u of B_dϕ = 0 on ¯D^∗ is also a supersolution of (6.14) on D, and, by Proposition 6.8, satisfies u ≥ w_η on ∂_xD^∗ ∪ ∂_TD^∗ for all η ≥ 1. In view of Proposition 6.9 and (6.13), it follows that u ≥ lim_η→∞↑ w_η = v on D whenever u satisfies (3.10). In particular, since v∗ is a supersolution of (3.8) satisfying (3.10), see Proposition 3.1, we have v_∗ ≥ v so that

v_∗ = v ≤ u on D and v_∗ ≤ u_∗ on ¯D^∗. 2

7 A uniqueness result

We now proceed with the proof of Theorem 3.2. It is an immediate consequence of Proposition 3.1, Theorem 3.1 and the following comparison result.

Theorem 7.1 Assume that the conditions of Theorem 3.2 hold. Let u be an upper-semicontinuous viscosity subsolution of Bϕ = 0 on ¯D^∗. Assume furthermore that u satisfies the growth condition (3.10). Then, u ≤ v_∗ on ¯D^∗.

Remark 7.1 1. It will be clear from the proof that the above Theorem can be stated as follows. Let u and w be respectively sub- and supersolution of Bϕ = 0 and B_dϕ = 0 on D ∪ ∂_xD^∗ satisfying the growth condition (3.10). Assume further that w satisfies C: ∀ (t, x) ∈ D ∪ ∂_xD^∗ and ρ ∈ ˜K₁(x, ¯O), ∃ λ₀ > 0 s.t. λ ∈ [0, λ₀] 7→ w(t, xe^λρ)e^−λδ(ρ) is non-increasing.

Then, u ≤ w on ∂_TD^∗ implies u ≤ w on ¯D^∗.

2. One can actually show that any supersolution of H_dϕ = 0 satisfies the condition C.

Since it is not useful for our main result, we do not provide the proof which is rather long.

3. Combining the above assertions provides a general comparison result for super- and subsolutions of, respectively, Bdϕ = 0 and Bϕ = 0 on D ∪ ∂xD^∗.

In order to prove Theorem 7.1, we need the following intermediate Lemma.

Lemma 7.1 Assume that HO holds. Fix x₀ ∈ ∂O^∗. If ρ ∈ ˜K₁ satisfies Dd(x₀)⁰diag [x₀] ρ > 0 ,

then there exists some positive parameters r₀ and λ₀ such that xe^{λ ρ} ∈ O for all x ∈ B(x₀, r₀) ∩ ¯O and λ ∈ (0, λ₀).

Proof. Recall from HO that the function d is C² on a neighbourhood of x₀. Thus, Dd(x₀)⁰diag [x₀] ρ > 0 implies that for some δ₀, r₀ > 0

Dd(¯x)⁰diag [x] ρ ≥ δ₀ for all ¯x, x ∈ B(x₀, r₀) . (7.1) Given that xe^λρ − x = λdiag [x] ρ + o(λ), we can fix some λ₀ > 0 such that, for all x ∈ B(x₀, r₀/2) and λ ∈ (0, λ₀)

[x, xe^λρ] ⊂ B(x₀, r₀) and |xe^λρ− x − λdiag [x] ρ| < λδ₀/2 1 + max

x∈ ¯B(x0,r0)

|Dd(x)| . (7.2) Let x be in B(x₀, r₀/2) ∩ ¯O, so that d(x) ≥ 0. Since d is C¹, for each λ ∈ (0, λ₀) there exists ¯x ∈ [x, xe^{λ ρ}] ⊂ B(x₀, r₀) such that

d(xe^{λ ρ}) = d(x) + Dd(¯x)⁰ xe^{λ ρ}− x

= d(x) + λDd(¯x)⁰diag [x] ρ + Dd(¯x)⁰xe^{λ ρ}− x − λdiag [x] ρ

≥ d(x) + λδ₀

2 > 0 ,

where the last inequality follows from (7.1) and (7.2). This shows that xe^{λ ρ} ∈ O. 2 Proof of Theorem 7.1: In order to avoid too many complications, we make the proof under the assumption

H⁰⁰ : (i) ∃ % > 1 s.t. %¯γ ∈ K ∩ (0, ∞)^d, (ii) 0 ∈ int(K),

(iii) ∀ x ∈ ∂O^∗ ∃ ρ ∈ ˜K1 s.t. Dd(x)⁰diag [x] ρ > 0 ,

in place of H⁰. We shall explain in the last step how to adapt this proof when ¯O is bounded but (i) of H⁰⁰ does not hold, or 0 /∈ int(K) but ¯O ∩ ∂R^d+= ∅ and int(K) 6= ∅.

1. Given some positive parameter κ, we introduce the functions ˜u(t, x) := e^κtu(t, x),

˜

v(t, x) := e^κtv∗(t, x) and ˜g(t, x) := e^κtˆg(t, x). One easily checks that the function ˜u (resp. ˜v) is a viscosity subsolution (resp. supersolution) of ˜Bϕ = 0 (resp. ˜B_dϕ = 0), where for a smooth function ϕ

Bϕ =˜











minn ˜Lϕ , Hϕo

on D min {ϕ − ˜g , Hϕ} on ∂_xD^∗

ϕ − ˜g on ∂_TD^∗

B˜_dϕ =

( Bϕ˜ on D ∪ ∂_TD^∗

min {ϕ − ˜g , H_dϕ} on ∂_xD^∗ and

Lϕ := κϕ − Lϕ .˜ Let % ∈ R be as in H⁰⁰, i.e.

γ := %¯γ ∈ K ∩ (0, ∞)^d and % > 1 . (7.3) Since 0 ∈ int(K) by H⁰⁰, it follows from (2.9) and Remark 2.4 that the map defined by β(t, x) = e^{τ (T −t)}(1 + x^γ) on ¯D^∗ satisfies

H (β(t, x), diag [x] Dβ(t, x)) ≥ cK > 0 for all (t, x) ∈ ¯D^∗ . (7.4) Moreover, by the same computations as in the proof of Proposition 6.9, one easily checks that we can choose τ large enough so that

Lβ ≥ 0˜ on ¯D^∗ . (7.5)

2. We argue by contradiction. We assume that sup

D¯^∗

(u − v∗) > 0 and work towards a contradiction.

2.1. In this step, we follow the same construction as in the proof of Proposition 6.9.

By the growth condition on ˜u, ˜v and (7.3), we deduce that 0 < 2m := sup

D¯^∗

(˜u − ˜v − 2αβ) < ∞ (7.6) for α > 0 small enough. Fix ε > 0 and let f be defined as in (6.18). Arguing as in the proof of Proposition 6.9, we obtain that

Φ^ε:= ˜u − ˜v − 2(αβ + εf )

admits a maximum (t_ε, x_ε) on ¯D^∗, which, for ε > 0 small enough, satisfies

Φ^ε(t_ε, x_ε) ≥ m > 0 . (7.7) Moreover, using the same arguments as in 2.1 of the proof of Proposition 6.9, we obtain that

lim sup

ε→0

sup

ρ∈ ˜K1

ε (|f (x_ε)| + |diag [x_ε] Df (x_ε)| + |Lf (x_ε)|) = 0 . (7.8)

Finally, since β, f ≥ 0 and v∗(T, ·) ≥ u(T, ·), (7.7) implies that t_ε< T , i.e.

(t_ε, x_ε) ∈ [0, T ) × ¯O^∗ . (7.9) 2.2. In the following, we fix ρ_ε ∈ ˜K₁ such that

ρ_ε := 0 if x_ε ∈ O

Dd(x_ε)⁰diag [x_ε] ρ_ε> 0 if x_ε∈ ∂O^∗ , (7.10) see (iii) of H⁰⁰. By Lemma 7.1 and (3.11), we can fix r_ε, λ_ε > 0, such that

xe^{λ ρε} ∈ O and e^{−λδ(ρε)}v(t, xe˜ ^λρε) ≤ ˜v(t, x) (7.11) for all t ∈ (t_ε− r_ε, t_ε+ r_ε) ∩ [0, T ), x ∈ B_ε := B(x_ε, r_ε) ∩ ¯O and λ ∈ (0, λ_ε) .

For n ≥ 1 and ζ ∈ (0, 1), we then define the function Ψ^ε_n,ζ on [0, T ] × ( ¯O^∗)² by Ψ^ε_n,ζ(t, x, y) := Θ(t, x, y) − ε(f (x) + f (y)) − ζ(|x − x_ε|²+ |t − t_ε|²) − n²|xe^nζρε− y|² , where

Θ(t, x, y) := u(t, x) − ˜˜ v(t, y) − α (β(t, x) + β(t, y)) .

It follows from (7.3) and the growth condition (3.10) satisfied by ˜v and ˜u that Ψ^ε_n,ζ attains its maximum at some (t^ε_n, x^ε_n, y_n^ε) ∈ [0, T ]×( ¯O^∗)². The inequality Ψ^ε_n,ζ(t^ε_n, x^ε_n, y^ε_n)

≥ Ψ^ε_n,ζ(t_ε, x_ε, x_εe^nζρε) implies that

Θ(t^ε_n, x^ε_n, y^ε_n) ≥ Θ(t_ε, x_ε, x_εe^nζρε) − ε

f (x_ε) + f (x_εe^nζρε)

+ n²|x^ε_ne^ζnρε − y_n^ε|²+ ζ |x^ε_n− x^ε|²+ |t^ε_n− t_ε|²
+ ε (f (x^ε_n) + f (y_n^ε)) , which combined with the growth condition (3.10) and (7.3) shows that n²|x^ε_ne^ζnρε − y_n^ε|²+ f (x^ε_n) is bounded in n so that, up to a subsequence,

(i) x^ε_ne^nζρε, x^ε_n, y_n^ε −−−→

n→∞ x¯^ε ∈ ¯O^∗ and t^ε_n−−−→

n→∞

t¯^ε∈ [0, T ] .

Let n be large enough so that _n^ζ < λ_ε. Recall from (7.11) that this implies that

˜

v(tε, xεe^nζρε) ≤ ˜v(tε, xε)e^ζnδ(ρε), which combined with the previous inequality yields Θ(t^ε_n, x^ε_n, y^ε_n) ≥ u(t˜ ε, xε) − ˜v(tε, xε)e^nζδ(ρε)− α

β(tε, xε) + β(tε, xεe^nζρε)



− ε

f (x_ε) + f (x_εe^nζρε)

+ n²|x^ε_ne^ζnρε − y_n^ε|²+ ζ |x^ε_n− x_ε|²+ |t^ε_n− t_ε|²
+ ε (f (x^ε_n) + f (y_n^ε)) . Sending n → ∞ and using the maximum property of (t_ε, x_ε), we get

0 ≥ Φ^ε(¯t^ε, ¯x^ε) − Φ^ε(t_ε, x_ε)

≥ lim sup

n→∞



n²|x^ε_ne^nζρε − y_n^ε|²+ ζ |x^ε_n− x_ε|²+ |t^ε_n− t_ε|²
 . Recalling (7.7) and (7.9), this shows that

(ii) n²|x^ε_ne^ζnρε − y_n^ε|²+ ζ |x^ε_n− xε|²+ |t^ε_n− tε|²
−−−→

n→∞ 0 , (iii) u(t˜ ^ε_n, x^ε_n) − ˜v(t^ε_n, y^ε_n) −−−→

n→∞ (˜u − ˜v) (t_ε, x_ε) ≥ m + 2αβ(t_ε, x_ε) + 2εf (x_ε) , (iv) (t^ε_n, x^ε_n) ∈ [0, T ) × ¯O^∗ for n large enough.

3. From Theorem 8.3 in [4], we deduce that, for each η > 0, there are real coefficients b^ε_1,n, b^ε_2,n and symmetric matrices X_n^ε,η and Y_n^ε,η such that

b^ε_1,n, p^ε_n, X_n^ε,η

∈ ¯P_O⁺_¯u(t˜ ^ε_n, x^ε_n) and −b^ε_2,n, q_n^ε, Y_n^ε,η

∈ ¯P_O⁻_¯˜v(t^ε_n, y^ε_n) , see [4] for the standard notations ¯P_O⁺_¯ and ¯P_O⁻_¯, where

p^ε_n := 2n²(x^ε_ne^nζρε − y^ε_n)e^ζnρε + 2ζ(x^ε_n− x_ε) + αDβ(t^ε_n, x^ε_n) + εDf (x^ε_n) q^ε_n := 2n²(x^ε_ne^nζρε − y^ε_n) − αDβ(t^ε_n, y_n^ε) − εDf (y_n^ε) ,

and b^ε_1,n, b^ε_2,n, X_n^ε,η and Y_n^ε,η satisfy







b^ε_1,n+ b^ε_2,n = 2ζ(t^ε_n− tε) − ατ (β(t^ε_n, x^ε_n) + β(t^ε_n, y_n^ε)) X_n^ε,η 0

0 −Y_n^ε,η

!

≤ (A^ε_n+ B_n^ε) + η(A^ε_n+ B_n^ε)² (7.12)

with

A^ε_n := 2n² diag[e^2ζnρε] + 2ζI_d −2n²diag[e^nζρε]

−2n²diag[e^ζnρε] 2n²I_d

!

B_n^ε := αD²β(t^ε_n, x^ε_n) + εD²f (x^ε_n) 0

0 αD²β(t^ε_n, y_n^ε) + εD²f (y_n^ε)

! .

3.1. We now show that, up to a subsequence,

y^ε_n∈ O . (7.13)

In view of (ii), this is clearly true when x_ε ∈ O. In the case x_ε ∈ ∂O, we deduce from (ii) that

y_n^ε = x^ε_ne^nζρε+ o(n⁻¹) = x^ε_n+ ζ

ndiag [x^ε_n] ρε+ o(n⁻¹) . This implies that, for some _n → 0,

d(y^ε_n) = d(x^ε_n) + ζ

n (Dd(x^ε_n)⁰diag [x^ε_n] ρ_ε+ _n) ,

so that (7.13) is a consequence of (7.10), the continuity of Dd and (ii).

3.2. In this step, we show that there is a subsequence of (t^ε_n, x^ε_n, y_n^ε) such that

x^ε_n ∈ O and κ˜u(t^ε_n, x^ε_n) − b^ε_1,n− ¹₂Tr [a(t^ε_n, x^ε_n)X_n^ε,η] ≤ 0 . (7.14)

First observe that we can not have x^ε_n ∈ ∂O^∗ and ˜u(t^ε_n, x^ε_n) ≤ ˜g(t^ε_n, x^ε_n) for all n. In view of (ii), this is obvious if x_ε ∈ O. If x_ε ∈ ∂O^∗, it follows from (7.9) and the viscosity property of ˜v that ˜v(t_ε, x_ε) ≥ ˜g(t_ε, x_ε). Since ˜g is upper-semicontinuous, see H_g, this would imply that ˜u(t^ε_n, x^ε_n) ≤ ˜v(t^ε_n, y_n^ε) + m/2 for n large enough, see (ii), thus leading to a contradiction to (iii) since β, f ≥ 0. By (iv) and the viscosity subsolution property of ˜u, we then deduce that either (7.14) holds or

H (˜u(t^ε_n, x^ε_n), diag [x^ε_n] p^ε_n) ≤ 0 . (7.15) Thus, it remains to prove that the above inequality leads to a contradiction. Using the supersolution property of ˜v, (7.13), (ii)-(iii) and (2.9), we observe that (7.15) implies

0 ≥ H (˜u(t^ε_n, x^ε_n), diag [x^ε_n] p^ε_n) − H (˜v(t^ε_n, y_n^ε), diag [y_n^ε] q_n^ε)

≥ α {H (β(t^ε_n, x^ε_n), diag [x^ε_n] Dβ(t^ε_n, x^ε_n)) + H (β(t^ε_n, y_n^ε), diag [y_n^ε] Dβ(t^ε_n, y^ε_n))}

+ ε {H (f (x^ε_n), diag [x^ε_n] Df (x^ε_n)) + H (f (y_n^ε), diag [y_n^ε] Df (y^ε_n))}

+ inf

ρ∈ ˜K1

δ(ρ) [Θ(t^ε_n, x^ε_n, y^ε_n) − ε (f (x^ε_n) + f (y_n^ε))]

− sup

ρ∈ ˜K1

h

2n²ρ⁰diagh

x^ε_ne^ζnρε − y_n^εi 

x^ε_ne^nζρε − y^ε_n

+ 2ζρ⁰diag [x^ε_n] (x^ε_n− x_ε)i

≥ inf

ρ∈ ˜K1

δ(ρ)(m/2) + 2αH(β(tε, xε), diag [xε] Dβ(tε, xε))

+ _n+ ε {H (f (x_ε), diag [x_ε] Df (x_ε)) + H (f (y_ε), diag [y_ε] Df (y_ε))}

where _n → 0 when n → ∞, but depend on ε. Recalling (7.4) and (7.8), we get a contradiction for ε small and n large enough. This concludes the proof of (7.14).

3.3. We can now provide the required contradiction and conclude the proof. Let ˜σ be defined on ¯D by ˜σ(t, x) = diag [x] σ(t, x). By the viscosity supersolution property of ˜v, (7.13), (7.14) and (7.12), (t^ε_n, x^ε_n, y_n^ε) must satisfy

κ (˜u(t^ε_n, x^ε_n) − ˜v(t^ε_n, y^ε_n)) ≤ b^ε_1,n+ b^ε_2,n+1

2Tr [a(t^ε_n, x^ε_n)X_n^ε,η− a(t^ε_n, y_n^ε)Y_n^ε,η]

≤ 2ζ(t^ε_n− t_ε) − ατ (β(t^ε_n, x^ε_n) + β(t^ε_n, y^ε_n)) + 1

2TrΞ(t^ε_n, x^ε_n, y^ε_n) A^ε_n+ B_n^ε+ η(A^ε_n+ B_n^ε)²


where

Ξ(t^ε_n, x^ε_n, y^ε_n) := σ(t˜ ^ε_n, x^ε_n)˜σ⁰(t^ε_n, x^ε_n) ˜σ(t^ε_n, y^ε_n)˜σ⁰(t^ε_n, x^ε_n)

˜

σ(t^ε_n, x^ε_n)˜σ⁰(t^ε_n, y_n^ε) σ(t˜ ^ε_n, y_n^ε)˜σ⁰(t^ε_n, y^ε_n)

!

is a non-negative symmetric matrix. Using (ii)-(iii), (7.5) and (7.8), it follows that for ε small and n large enough

κ m/2 ≤ κ (˜u(t^ε_n, x^ε_n) − ˜v(t^ε_n, y_n^ε) − (αβ + εf )(t^ε_n, x^ε_n) − (αβ + εf )(t^ε_n, y_n^ε))

≤ 2ζ(t^ε_n− tε) + 1

2TrΞ(t^ε_n, x^ε_n, y^ε_n) A^ε_n+ η(A^ε_n+ B_n^ε)²
 + θ(ε, n) where θ(ε, n) is independent of (η, ζ) and satisfies

lim sup

ε→0

lim sup

n→∞

|θ(ε, n)| = 0 . (7.16)

Sending η → 0 in the previous inequality provides κ m/2 ≤ 2ζ(t^ε_n− t_ε) + 1

where C_ε > 0 denotes a generic constant independent of n and ζ. Plugging this in the previous inequality implies that there is some C_ε > 0 independent of n and ζ for which κ m/2 ≤ 2ζ(t^ε_n− t_ε) + ζTr [˜σ⁰(t^ε_n, x^ε_n)˜σ(t^ε_n, x^ε_n)] + C_ε

ζ + n²|x^ε_ne^nζρε − y^ε_n|²

+ θ(ε, n) . Finally, using (ii) and sending n to ∞ and then ζ to 0 in the last inequality implies

κ m/2 ≤ lim sup

n→∞

θ(ε, n) ,

which by (7.16) provides the required contradiction and concludes the proof.

4. We now explain how to adapt this proof to the alternative assumptions of H⁰. 4.1. Observe that the penalty function β is introduced in order to obtain a finite supremum for ˜u− ˜v −2αβ and existence of an optimum for Φ^εand Ψ^ε_n,ζ. If ¯O is bounded, the introduction of such a penalty function is not required and we can reproduce the same proof with β ≡ 0 whenever 0 ∈ int(K). Indeed, by Remark 2.4, inf_{ρ∈ ˜K}

1δ(ρ) > 0 so that we still obtain a contradiction at the end of 3.2. The arguments of 3.3 also work with β ≡ 0. The case where 0 /∈ int(K) is discussed below.

4.2. Similarly, the map f is introduced only to prevent the different maxima to take values outside ¯O^∗. If ¯O ∩ ∂R^d+ = ∅, this penalty function is useless and can be fixed to f ≡ 0. In this case, one can also fix some γ ∈ int(K), if non-empty, and add the term e^{τ (T −t)}x^γ in the definition of β. Thus, β becomes e^{τ (T −t)}(1 + x^γ+ x^γ) or e^{τ (T −t)}(1 + x^γ) depending whether ¯O is bounded or not, see 4.1. For fixed ε > 0, we deduce from Remark 2.4 and the fact that γ ∈ int(K) that H(β(t, x), diag [x] Dβ(t, x))

> 0. Since f = 0, there is no ε to send to 0 at the end of 3.2 and 3.3, and we obtain the same contradictions by simply sending n to ∞ and ζ to 0.

2

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