Uniqueness for $L_{p}$-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms

In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be H\"older continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.


Introduction
For a real-valued measurable function H(u, t, x), where S is the set of symmetric d × d matrices, and sufficiently regular functions v(t, x) we set If R ∈ (0, ∞) and (t, x) ∈ R d+1 , then We also take a bounded domain Ω ⊂ R d of class C 1,1 and set Π = [0, T ) × Ω, ∂ ′ Π =Π \ {0} ×Ω Remark 1.1. We assumed that Ω ∈ C 1,1 just to be able to refer to the results available at this moment, but actually much less is needed for our Theorems 2.1, 4.1, 4.2, 4.3, 4.4, and 5.1 to hold. For instance the exterior cone condition would suffice.
We will be dealing with viscosity solutions of (1.1) in Π. The following definition is taken from [3] and has the same spirit as in [1]. Definition 1.1. For each choice of "regularity" class R = C or R = L p we say that u is an R-viscosity subsolution of (1.1) in Π provided that u is continuous in Π and, for anyC r (t 0 , x 0 ) ⊂ Π and any function φ, that is continuous in C r (t 0 , x 0 ) and whose generalized derivatives satisfy ∂ t φ, Dφ, D 2 φ ∈ R C r (t 0 , x 0 ) , and is such that u − φ attains its maximum over C r (t 0 , x 0 ) at (t 0 , x 0 ), we have lim ρ↓0 ess sup Cρ(t 0 ,x 0 ) ∂ t φ(t, x) + H u(t, x), Dφ(t, x), D 2 φ(t, x), t, x ≥ 0. (1.2) In a natural way one defines R-viscosity supersolutions and calls a function an R-viscosity solution if it is an R-viscosity supersolution and an R-viscosity subsolution.
Note that C r (t 0 , x 0 ) contains (t, x) : t = t 0 , |x − x 0 | < r , which is part of its boundary. Therefore, the conditions like D 2 φ ∈ C C r (t 0 , x 0 ) mean that the second-order derivatives of φ are continuous up to this part of the boundary.
In Section 2 we discuss uniqueness of C-viscosity solutions for general equations when H is Lipschitz continuous with respect to u. The result we obtain is crucial for proving uniqueness of L p -viscosity solutions in Section 5 for the Isaacs equations with measurable lower order terms. The proof of the main result in Section 2 hinges on Lemma 2.3, whose rather long proof is given in Section 3. Section 4 concentrates on the extremal L p -viscosity solutions, their existence and stability. Precisely the stability of L p -viscosity minimal and maximal solutions is used in Section 5.
Assumption 2.1. (i) The function H(u, t, x) is a continuous function of (u, t, x) and is Lipschitz continuous with respect to u with Lipschitz constant K 0 .
(ii) At all points of differentiability of H with respect to u ′′ we have D u ′′ H ∈ S δ .
Theorem 2.1. Under the above assumptions there exists a unique v ∈ C(Π) which is a C-viscosity solution of (1.1) in Π with boundary condition v = g on ∂ ′ Π. Furthermore, there exists a constant N ∈ (0, ∞) such that for any Remark 2.1. The assumptions of Theorem 2.1 are almost identical to the assumptions made in the elliptic case in [15], that, to the best of our knowledge, provides the most general result to date concerning the uniqueness of C-viscosity solutions for the uniformly elliptic case (see Remark 3.1 there). Our Theorem 2.1 is a parabolic counterpart of Trudinger's result from [15]. In the parabolic case the uniqueness of L p -viscosity solutions is proved in Lemma 6.2 of [3] when H is independent of (t, x). In the case of the Isaacs equations, under the assumptions on the coefficients guaranteeing that our assumptions are satisfied as well, the statement about the uniqueness of Cviscosity solutions is found in Theorem 9.3 of [3]. However, this statement is not provided with a proof with the excuse that its proof is similar to the one known in the elliptic case.
One of the features of our proof is that it also allows one to establish an algebraic rate of convergence of numerical approximations (see [10]).
The proof of Theorem 2.1 is based on a few auxiliary results. Denote It is easy to see that if v(t, x) is a C-viscosity subsolution of (1.1) in Π, then, for any constant c, the functions w(t, x) := e ct v(t, x) is a C-viscosity subsolution of The function H c (u, t, x) has the same Lipschitz constant with respect to [u ′ ], u ′′ and its derivative with respect to u ′ 0 , wherever it exists, is If we take c = K 0 + 1 and redefine H c for t ∈ [0, T ] as its value at the closest end point of [0, T ], then H c will satisfy all assumptions of Theorem 2.1 with K 0 + 1 in place of K 0 and additionally satisfy D u ′ 0 H c ≤ −1. That is why without loss of generality we suppose that not only Assumption 2.1 is satisfied but also for all values of arguments wherever the left-hand side exists.
Below we suppose that the above assumptions are satisfied, take the convex positive homogeneous of degree one function P (u ′′ ) on S from Theorem 4.1, and set P [u](t, x) = P D 2 u(t, x) . Recall that κ is introduced before Assumption 2.1.
in Π (a.e.) with boundary condition u ±K = g on ∂ ′ Π has a unique solution u ±K ∈ W 1,2 p,loc (Π ρ ) ∩ C(Π) for any p ≥ 1 and ρ > 0. (ii ) We have u −K ≤ u K and, as K increases, u −K increase and u K decrease.
(iv ) There exists a constant N ∈ (0, ∞) such that for any ρ > 0 and The existence part in assertion (i) for the sign + follows from Theorem 4.1 which holds under more general assumptions than the ones imposed here. For the sign − it suffices to replace H(u, t, x) with −H(−u, t, x). Uniqueness and assertion (ii) are direct consequences of the maximum principle.
Assertion (iii) for u K follows from the linear theory and the observation that Indeed, for any K, there exist Sδ-valued a, R d -valued b, and real-valued c ≥ 0 and f such that and |b| ≤ K 0 , c ≤ K 0 , |f | ≤H. For u −K the argument is similar. Assertion (iv) follows from Theorem 2.1 of [11].
Under the assumptions of this section, for K → ∞ we have This lemma is proved in Section 3.
Proof of Theorem 2.1. First we prove uniqueness. Introduce ψ ∈ C 2 (R d ) as a global barrier for Ω, that is, in Ω we have ψ ≥ 1 and Such a ψ can be found in the form cosh µR − cosh µ|x| for sufficiently large µ and R.
Then we take and fix a radially symmetric with respect to x function ζ = ζ(t, x) of class C ∞ 0 (R d+1 ) with support in (−1, 0)× B 1 and unit integral.
Let Ω n , n = 2, 3, ..., be a sequence of strictly expanding smooth domains whose union is Ω and set Π n = [0, T (1−1/n))×Ω n . Then for any n 0 = 3, 4, ... and all sufficiently small ε > 0 Since the second-order derivatives with respect to x and the first derivative with respect to t of u K are in L p (Π ρ ) for any p and ρ, we have ξ ε,K → 0 as ε ↓ 0 in any L p (Π n 0 ) for any K and any p > 1. Furthermore, ξ ε,K are continuous because H(u, t, x) is continuous. Therefore, there exist smooth functions ζ ε,K onΠ n 0 such that −ε ≤ ξ ε,K + ζ ε,K ≤ 0 in Π n 0 for all small ε > 0.
Since Ω n 0 is smooth, by Theorem 1.1 of [4] there exists a unique w ε,K ∈ p>1 W 1,2 p (Π n 0 ) satisfying ∂ t w ε,K + sup in Π n 0 (a.e.) and such that w ε,K = 0 on ∂ ′ Π n 0 . By the maximum principle such w ε,K is unique. Then owing to the continuity of ζ ε,K , for any ε and K, for all sufficiently small β > 0, we have This and the definition of C-viscosity solutions imply that, if v is a continuous inΠ, C-viscosity solution of ∂ t v + H[v] = 0 with boundary data g, then the minimum of u (ε) K + w β ε,K + βψ − v inΠ n 0 −1 is either nonnegative or is attained on the parabolic boundary of Π n 0 −1 . The same conclusion holds after letting first β ↓ 0 and then ε ↓ 0. Combining this with the Aleksandrov estimates showing that w ε,K → 0 as ε ↓ 0 uniformly onΠ n 0 , we get that in Π which after letting n 0 → ∞ and then K → ∞ yields v ≤ u, where u is the common limit of u K , u −K , which exists by Lemma 2.3. By comparing v with u −K , we get v ≥ u, and hence uniqueness. After that estimate (2.2) follows immediately from (2.6), and the theorem is proved.

Proof of Lemma 2.3
To prove Lemma 2.3 we need an auxiliary result. In the following theorem Ω can be just any bounded domain. Below by C 1,2 loc (Π) we mean, as usual, the space of functions u = u(t, x) which are continuous in Π along with their derivatives ∂ t u, D ij u, D i u. We recall that κ is introduced before Assumption 2.1 and fix a constant τ ∈ (0, 1).
in Π and v ≥ u on ∂ ′ Π. Also assume that, for a constant M ∈ [1, ∞), Then there exist a constant N ∈ (0, ∞), depending only on τ , the diameter of Ω, d, K 0 ,H, and δ, and a constant η > 0, depending only on τ , d, and δ, such that, if K ≥ N M η and Remark 3.1. The purpose of introducing τ is that for ω = t τ estimate (3.4) becomes u − v ≤ N K −(κ−1)/4 + N K −τ , which was used in [10] to estimate the rate of convergence of finite-difference approximations for (1.1).
The statement of this theorem is almost identical to that of Theorem 3.1 of [10] although that theorem is about the Isaacs equations and our equations are more general. However, the most part of the proof follows that of Theorem 3.1 of [10] and, as there, we are going to adapt to our situation an argument from Section 5.A of [1]. For that we need a construction and two lemmas. From the start throughout the section we will only concentrate on K satisfying (3.3).
We take and fix a function ζ = ζ(t, x) as before (2.7) and use the notation u (ε) introduced in (2.7). Recall some standard properties of parabolic mollifiers in which no regularity properties of Ω are required: where the constants N depend only on d and κ.
Proof. Since ε 0 < 1, κ ≤ 2, and M ≥ 1, the left-hand side of (3.6) is less than One easily checks that and this proves the lemma.
where and below by N with indices or without them we denote various constants depending only on d, K 0 ,H, δ, τ , and the diameter of Ω, unless specifically stated otherwise. By the way, recall that κ and, hence, γ depend only on d and δ.
Proof. The first inequality in (3.9) follows from (3.5) and the fact that the first derivatives of W with respect to x vanish atx, that is, Also the matrix of second-order derivatives of W with respect to (x, y) is nonpositive at (t,x,ȳ) as well as its (at least one sided ift = 0) derivative with respect to t, which yields (ii).
By taking η = 0 in (3.10) and using the fact that Similarly, Also it follows from (3.11) and (3.5) that and at (t,ȳ) Here M ε κ−2 ≤ νK, which is equivalent to (3.8) with N = 1. Hence which is impossible if we choose and fix ν such that and this proves (3.12).
which again is impossible with the above choice of ν for K satisfying (3.8). This yields (3.13). Moreover, not only M ε κ−2 ≤ νK for K satisfying (3.8), but we also have N M ε κ−2 ≤ νK, where N is taken from (3.9), if we increase N in (3.8). This yields the second inequality in (3.9).
The lemma is proved.
Everywhere below in this section ν is the constant from Lemma 3.3.
Proof. It follows from (3.7) that (recall that ν is already fixed) Hence we have from (3.16) that and (3.17) follows provided that In any case in light of (3.17), (3.7), and (3.19) 4εM which is less than (1/2)K −(κ−1)/4 in light of (3.19) and Lemma 3.2, again after perhaps further adjusting the constant in (3.20). This is impossible due to (3.16). Hence,t < T −ε 2 and this finishes the proof of the lemma. We also need a simple result based on solving quadratic inequalities.
Proof of Theorem 3.1. Fix a (large) constant µ > 0 to be specified later as a constant, depending only on d, K 0 ,H, δ, τ , and the diameter of Ω, recall that ν is found in Lemma 3.3 and first assume that for (t, x), (t, y) ∈Π ε . Observe that for any point (t, x) ∈Π one can find a point (s, y) ∈Π ε with |x − y| ≤ ε and |t − s| ≤ ε 2 and then In that case, as follows from Lemma 3.2, (3.4) holds for K satisfying (3.20) with any N ≥ 1 in (3.20).
It is clear now that, to prove the theorem, it suffices to find µ such that the inequality (3.16) is impossible if K ≥ N M η with N and η as in the statement of the theorem and at least not smaller than those in (3.20). Of course, we will argue by contradiction and suppose that (3.16) holds, despite (3.20) is valid and (3.8) is satisfied with ν fixed in Lemma 3.3 and ε 0 ∈ (0, 1), which is yet to be specified.
Then (3.17) holds, and, in particular, Also by Lemma 3.4 the pointsx,ȳ are in Ω ε (even in Ω 2ε ) andt < T − ε 2 , so that we can use the conclusions of Lemma 3.3. Denote and interpret matrices as linear operators in a usual way and constants as operators of multiplications by these constants. Observe that (3.10) implies that the operator B + 2νK is nonnegative and, hence, B + 4νK is strictly positive. Then for Hence A ≤ 4νKB(B + 4νK) −1 and We now use (3.9) to get that |x −ȳ| ≤ N M K −1 ε κ−1 , and in light of (3.5) that and as is easy to see Thus far, we have and, therefore, This along with (3.13) and Assumption 2.1 (iii) yields (recall that M ≥ 1) Upon combining this with (3.17) we arrive at Now we choose µ = N 1 and observe that (see (3.9)) where Then for such K we infer from (3.25) that Here, owing to (3.5) and (3.9), where θ 2 = (7κ−19)/(14−6κ), and as is easy to see θ 2 ≤ −3/2 for κ ∈ [1,2], so that if, for instance, In what concerns the last term in (3.27), note that We conclude that, for K satisfying (3.28) and (3.29), relation (3.27) yields Next, observe that, by Lemma 3.5 applied after we diagonalize B and set B = Kα implies that the right-hand side of (3.30) is where the inequality holds owing to Cauchy's inequality. We can certainly assume that N 5 ≥ 1 and then we can choose ε 0 ∈ (0, 1) so that Since M ≥ 1, for η ′ defined as the sum of the above powers of M and N ′ defined as the sum of the above N 's, the inequality (3.16) is impossible for K ≥ N ′ M η ′ and this brings the proof of the theorem to an end.
The function H is a nonincreasing function of u ′ 0 , is continuous with respect to u ′ 0 , uniformly with respect to other variables [u ′ ], (t, x) ∈ R d+1 , u ′′ ∈ S, is measurable with respect to (t, x) for any u, and is Lipschitz continuous in [u ′ ] with Lipschitz constant independent of u ′ 0 , u ′′ , (t, x).
(ii) For any u ′ , (t, x) ∈ R d+1 , the function H(u, t, x) is Lipschitz continuous with respect to u ′′ and at all points of differentiability of H(u, t, x) with respect to u ′′ , we have Assumption 4.2. We are given a function g ∈ C(∂ ′ Π).
We are going to use the following local version of Theorem 1.14 of [12], proved there for g ∈ W 1,2 p (R d+1 ) with the solution in global rather than local spaces W 1,2 p,loc . This local version is easier to prove because no boundary estimates are needed and we will provide the proof elsewhere.
By the maximum principle the solutions v = v K are unique and decrease as K → ∞.
Theorem 4.2. Under the above assumptions, as K → ∞, v K converges uniformly onΠ to a continuous function v which is an L d+1 -viscosity solutions of (1.1) with boundary condition v = g on ∂ ′ Π. Furthermore, v is the maximal L d+1 -viscosity subsolution of (1.1) of class C(Π) with given boundary condition. This yields the following result.
denote a unique solution of (4.2) (a.e.) in Π with boundary data v −K = g on ∂ ′ Π. Then, as K → ∞, v −K converges uniformly onΠ to a continuous function w which is an L d+1 -viscosity solutions of (1.1) with boundary condition w = g on ∂ ′ Π. Furthermore, w is the minimal L d+1 -viscosity supersolution of (1.1) of class C(Π) with given boundary condition.
Remark 4.2. The existence of extremal C-viscosity solutions is proved in the elliptic and parabolic cases in [2] when H is a continuous function. Our function H(u, t, x) is just measurable in (t, x) and we are dealing with L d+1viscosity solutions.
Also note that the existence of the extremal L p -viscosity solution for the elliptic case was proved in [5] with no continuity assumption on H with respect to x. We provide a method which in principle allows one to find it.
Here is a stability result for the extremal L d+1 -viscosity solutions. In the following assumption there are two objects: κ 1 = κ(d, δ) ∈ (1, 2) (close to 1), and θ = θ(κ, d, δ) ∈ (0, 1] (close to 0), κ ∈ (1, κ 1 ). The values of κ 1 and θ are specified in the proof of Lemma 5.3 of [9].  (iii) The function F is positive homogeneous of degree one with respect to u ′′ , is Lipschitz continuous with respect to u ′′ , and at all points of differentiability of F with respect to u ′′ we have D u ′′ F ∈ S δ .
Assume that for any M > 0 in L d+1 (Π) as n → ∞. Also assume that for all values of the arguments and n H n (u, t, x) − H 0 (u, t, x) ≤H(t, x) 1 + |u ′ | . Then v n → v 0 in C(Π) as n → ∞. The same holds true if v n are minimal L d+1 -viscosity solutions of class C(Π).
Proof. According to Theorem 4.2, it suffices to show that sup K≥1 v n,K − v 0,K → 0, (4.5) in C(Π), where v n,K are the solutions of in Π (a.e.) with boundary condition v n,K = g n on ∂ ′ Π.
Observe that for certain measurable R d -valued b, and measurable real-valued τ and c, such that |b|, |c| ≤ K 0 , and |τ | ≤ 1. Also max F n [0], P [0] − K = 0. Therefore, by the mean-value theorem we have ∂ t v n,K + a ij D ij v n,K + b i D i v n,K + cv n,K + τH = 0 (4.6) (a.e.) in Π for certain measurable Sδ-valued (a ij ), and perhaps different measurable R d -valued b, and measurable real-valued τ and c, such that |b|, |c| ≤ K 0 , and |τ | ≤ 1. By the parabolic Aleksandrov estimates (4.6) implies that |v n,K | are uniformly bounded inΠ and by the linear theory of parabolic equations we conclude that the family {v n,K : K ≥ 1, n ≥ 0} is precompact in C(Π). Next, fix a ρ > 0 such that Π ρ = ∅ and observe that, as we know from [4], [7], there is a number γ = γ(d, δ, K 0 ) ∈ (0, 1) such that there is a constant N , depending only on ρ, d, δ, and K 0 , such that for any cylinder C ρ (t 0 , x 0 ) ⊂ Π we have due to (4.6) that for all n, K. Here the right-hand side is dominated by a constant independent of n, K, and it follows, by Chebyshev's inequality that there is a constant N (perhaps depending on ρ) for which for all n ≥ 0, K ≥ 1, M > 0.
To finish with preparations, set H n,K = max(H n , P − K), F n,K = max(F n , P ), G n,K = H n,K − F n,K , where F n is taken from Assumption 4.3 written for H n . Then F n,K and G n,K satisfy Assumption 4.3 (i) and (ii) with the same K 0 andH. Assumption 4.3 (iii) also is satisfied withδ in place of δ. Finally, easy manipulations, using the fact that in the assumptions of the theorem we suppose that Assumption 4.3 (iv) is satisfied for F n with θ(κ, d,δ)/2 in place of θ, show that Assumption 4.3 (iv) is satisfied for F n,K with θ = θ(κ, d,δ). Thus, all the assumptions of Theorem 2.1 of [9] are satisfied apart from g ∈ W 1,2 ∞ (R d+1 ) and Ω ∈ C 2 . We will show in a separate publication that these assumptions can be replaced with the current ones. Now since v n,K is a classical solution of (4.1), we obtain from that theorem, for any small ρ > 0, the estimates of the C 1+α (Π ρ )-norms of v n,K uniform with respect to n and K. Therefore, by interpolation theorems we get sup Π ρ |Dv n,K | ≤ N, (4.8) where and below by N we denote various constants independent of K and n, perhaps depending on ρ. Now set w n,K = v 0,K − v n,K , and observe that 0 = ∂ t w n,K + I 1 + I 2 + I 3 , where and (a ij ) is an Sδ-valued function. By assumption where b and c are bounded uniformly with respect to n, K. Upon observing that by assumption and (4.8) in Π ρ for any M > 0 we have |I 3 | ≤HN I |D 2 v n,K |≥M + ∆ n,M +N I |D 2 v n,K |≤M and using the parabolic Aleksandrov estimates in Π ρ we conclude that there exists a constant N such that for all n, K, M in Π (4.9) Here sup as M → ∞, sinceH ∈ L d+1 (Π) and (4.7) holds. Therefore, by first taking the sup's with respect to K ≥ 1 in (4.9), then sending n → ∞, using assumption (4.3), and then sending M → ∞, we infer from (4.9) that, for any small ρ > 0 After that it only remains to set ρ ↓ 0 and use the equicontinuity of v n,K and the fact that g n → g 0 uniformly in ∂ ′ Π. The theorem is proved.
Remark 4.3. It follows from the above proof thatH in (4.4) can be replaced withH n , provided that the family |H n | d+1 is uniformly integrable over Π.
An obvious consequence of this theorem is the stability of uniqueness. Coming back to Theorem 4.2, observe that, as we have mentioned above, by the maximum principle v K decreases as K increases. The precompactness of {v K , K ≥ 1} in C(Π) is proved in the same way as in the above proof after (4.6) using the fact that It follows that v K converges uniformly onΠ as K → ∞ to a function v ∈ C(Π). To prove that v is an L d+1 -viscosity solution we need the following.
Proof. Observe that in C r (t, x) (a.e.) where the constant N is of the type described in the statement of the present lemma. We obtain (4.10) from (4.12) by letting K → ∞. In the same way (4.11) is established. The lemma is proved.
Proof of Theorem 4.2. First we prove that v is an L d+1 -viscosity solution. Let (t 0 , x 0 ) ∈ Π and φ ∈ W 1,2 d+1,loc (Π) be such that v − φ attains a local maximum at (t 0 , x 0 ). Then for ε > 0 and all small r > 0 for Hence, by Lemma 4.6 L d+1 (Cr(t 0 ,x 0 )) . By letting r ↓ 0 and using the continuity of H(u, t, x) in u ′ 0 , which is assumed to be uniform with respect to other variables and also using the continuity of φ (embedding theorems) and v, we obtain N lim r↓0 ess sup where φ ε = φ+ε |x−x 0 | 2 +t−t 0 . Finally, observe that H(u, t, x) is Lipschitz continuous with respect to [u ′ ], u ′′ with Lipschitz constant independent of u ′ 0 , (t, x) by assumption. Then letting ε ↓ 0 in (4.13) proves that v is an L d+1viscosity subsolution. The fact that it is also an L d+1 -viscosity supersolution is proved similarly on the basis of (4.11).
Finally, we prove that v is the maximal continuous L d+1 -viscosity subsolution. Let w be an L d+1 -viscosity subsolution of (1.1) of class C(Π) with boundary data g. To prove that v ≥ w, it suffices to show that for any ε > 0 and K > 1 we have u K + ε(T − t) ≥ w inΠ.
Assume the contrary and observe that, since and Since H is a decreasing function of u ′ 0 , in light of (4.14), −ε+lim r↓0 ess sup This is however impossible since ∂ t u K + H[u K ] ≤ 0 in Π (a.e.). This contradiction finishes proving the theorem.

Uniqueness of L p -viscosity solutions for parabolic Isaacs equations
Fix some constants δ ∈ (0, 1], K 0 , T ∈ (0, ∞), p > d + 2. Assume that we are given countable sets A and B, and, for each α ∈ A and β ∈ B, we are given an S δ -valued function a αβ on R d+1 = (t, x) : t ∈ R, x ∈ R d , a realvalued function b αβ (u ′ , t, x) on R d+1 × R d+1 , and a real-valuedH(t, x) ≥ 0 on R d+1 . (ii ) The function a αβ (t, x) is uniformly continuous with respect to (t, x) uniformly with respect to α, β and, with γ introduced before Assumption 2.1, for all values of indices and arguments a αβ (t, x) − a αβ (t, y) ≤ K 0 |x − y| γ .
(iii ) The function b αβ (u ′ , t, x) is nonincreasing with respect to u ′ 0 , is Lipschitz continuous with respect to u ′ with Lipschitz constant K 0 , and for all values of indices and arguments (where as everywhere the summation convention is enforced and the summations are done inside the brackets). For sufficiently smooth functions u = u(t, x) define As is usual in this article we take an open bounded subset Ω of R d of class C 1,1 and set Π = [0, T ) × Ω. Here is the main results of this section.
Theorem 5.1. Under the above assumption for any g ∈ C(∂ ′ Π) there exists a unique continuous inΠ, L d+1 -viscosity solution g of the Isaacs equation in Π with boundary condition u = g on ∂ ′ Π.
with uniformly continuous (in (t, x) uniformly in α, β) and uniformly bounded coefficients and the free terms, the uniqueness of C-viscosity solutions is stated without proof in Theorem 9.3 in [3]. This case is covered by Theorem 2.1.
For general H, not necessarily related to Isaacs equations, uniqueness is claimed for ∂ t u + H(D 2 u) = 0 in Lemma 4.7 of [16]. It is proved for L pviscosity solutions in Lemma 6.2 of [3] in case H is independent of (t, x) with no reference to Wang's Lemma 4.7 of [16].
In the elliptic case Jensen andŚwiȩch [5] proved the uniqueness of continuous L p -viscosity solutions for Isaacs equations, assuming that b αβ i , c αβ are bounded, sup α,β |f αβ | ∈ L p and Assumptions 5.1 (i), (ii) are satisfied. Their proof uses a remarkable Corollary 1.6 ofŚwiȩch [14] of which the parabolic counterpart is given in [3].
An important difference with [5] here is that we consider lower-order terms in a more general form, but in [5] the summability assumption is weaker (some p < d are allowed).
Here are a few properties of H n .
Lemma 5.2. (i ) For each n ≥ 1, the function H n (ξ, u ′ , t, x) is continuous, is Lipschitz continuous in u ′ with Lipschitz constant K 0 , and the function H n (0, 0, t, x) is bounded.
(ii ) For each n ≥ 1, the function H n (ξ, u ′ , t, x) is infinitely differentiable with respect to x (and t), and there exists a constant N (depending on n) such that |D x H n | ≤ N 1 + |u ′ | for all values of arguments.