Existence, uniqueness and regularity for nonlinear parabolic equations with nonlocal terms

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1 Existence, uniqueness and regularity for nonlinear parabolic equations with nonlocal terms Nathaël Alibaud 04/08/006 Université Montpellier II, Département de mathématiques, CC 051, Place E. Bataillon, Montpellier cedex 5, France Abstract. We study evolution equations that are fully nonlinear, degenerate parabolic, nonlocal and nonmonotone. The major difficulty lies in nonmonotonicity, i.e. in the fact that no comparison principle can be obtained. This implies that the classical method used to prove existence in the context of fully nonlinear degenerate equations, namely Perron s one, does not apply. We thus need to use a fixed point argument and to get suitable a priori estimates, we need a refined version of classical continuous dependance estimates. This technical result is of independent interest. We also obtain results such as uniform or Hölder continuity of the solution. Keywords: nonlinear degenerate parabolic integro-partial differential equation; nonmonotone equation; viscosity solution; continuous dependance estimate; Hölder estimate. Mathematics Subject Classification: 35A05, 35B30, 35B65, 35G0, 35K65, 47G0 1 Introduction In recent years there has been an interest in developing viscosity solutions theory for parabolic integro-pdes. Particularly equations that occur in the theory of optimal control of jumpdiffusion Lévy processes [7, 14, 16,, 11, 13]. The use of viscosity solutions is appropriately chosen because most of these equations are degenerate or fully nonlinear. In general, existence is proved by Perron s method with the help of a comparison principle. But in few other many applications, like signal [4], there is no such a comparison principle. The equations are then said to be nonmonotone. The notion of viscosity solution can still be used but existence must be proved by classical fixed point methods. Our purpose is to find a general framework to treat these difficulties that are degeneracy, nonlinearity, presence of nonlocal terms and nonmonotonicity. For other recent works on nonmonotone equations, we refer the reader to [1, 3, 1] which study equations that are involved in the theory of dislocation. Let us present our mathematical framework. We are interested in existence, uniqueness and regularity of viscosity solution of fully nonlinear degenerate parabolic integro-pdes of the form t u + Ft, x, u, Du, D u, g[u] = 0 in Q T, 1.1 u0,. = u 0 on R N, 1. where Q T :=]0, T[ R N, F : [0, T] R N R R N S N R M R is a given functional nonincreasing with respect to w.r.t. for short the D u-variable and g[u] is a nonlocal term. Here S N denotes the space of symmetric N N real valued matrices. We investigate the case 1

2 where no monotonicity assumption w.r.t. the nonlocality is assumed see Remark 3.1 for more details; this led us to consider a class of nonlocal term of order 0 satisfying a Lipschitz condition of the form sup [0,t] R N g[u] g[v] τ, x C sup [0,t] R N u v τ, x, 1.3 where C does not depend on t, u and v. Examples of nonlocal terms satisfying 1.3 are nonlinear integral operators of the form R N Mt, x, z, ut, x, ut, x + zdµ t,x z, 1.4 where sup QT µ t,x R N < + and M is Lipschitz w.r.t. its two last arguments. Further examples are given by integral operators in both time and space variables such as g[u]t = Stv 0 + t 0 St τfuτdτ, 1.5 where S. is a semigroup generated by a linear operator A, f : R R is Lipschitz and v 0 is a given initial condition. Then, the Cauchy problem is equivalent to the following system: t u + Ft, x, u, Du, D u, v = 0 in Q T, u0,. = u 0 on R N, d v + Av = fu in ]0, T[, dt v0 = v 0. When A is the Laplacian operator, for example, the associated semigroup S. is defined by Stv := Gt v, where designs the convolution product in R N and Gtx is the Green Kernel. As far as the Hamiltonian F is concerned, we first study the case where it is Lipschitz w.r.t. the g[u]-variable and next the case where there is a coupling between g[u] and the derivatives of u. The last one case can be seen as a generalization of [4] which treats the nonmonotone equation t u fdg ut,.tr ADuD u = 0 in Q T, for f > 0 Lipschitz-continuous, G which is a Gaussian function and A 0 bounded continuous on R N {0}. In fact, our technics combined with these ones used to treat the mean curvature flow by the level set method could allow to treat discontinuous Hamiltonians at Du = 0 and such that F t, x, r,0, 0, λ = F t, x, r,0, 0, λ. But, for the sake of clarity we have chosen to present only the continuous case. An illustrating example of what kind of coupling can be considered is the following quasilinear Hamiltonian Ft, x, r, p, X, λ = Ht, x, r, p, λ tr t σt, x, p, λ σt, x, p, λx, 1.6 where H and σ are Lipschitz w.r.t. x, λ respectively locally and globally in p.

3 Another interest of this paper is a so-called continuous dependence estimate see Theorem 4.1 for local parabolic equations which allow to obtain lots of needed a priori estimates. This can be seen as a generalization of results of Souganidis [15, Proposition 1.4] for first-order equations and of Jakobsen and Karlsen [10, Theorem 3.1] for second-order equations. Let us recall that some of their applications are a priori Lipschitz and Hölder estimates. In our setting, their results are not sufficient, in particular they do not permit to prove that the function is uniformly continuous. This improved version seems to us of independent interest. The rest of this paper is organized as follows: in Section, we introduce the definitions and notations that will be used throughout this paper. In Section 3, we state our results and prove them in Section 4. The last one section also contains our continuous dependance estimate for local equations. Preliminaries Throughout the paper, we will use the notations that follow. For a, b R, we let a b denote the real max{a, b}. We let a + denote the real a 0. Let k be an integer. For x R k, we let x denote the Euclidean norm of x. We let Q T denote the cylinder ]0, T[ R N. We let M N denote the space of N N real valued matrices and S N the space of such matrices which are symmetric. For every X, Y S N, we say that X Y when Xξ, ξ Y ξ, ξ for all ξ R N. The notation.,. is the Euclidean scalar product of R N. Let E be a metric space. The closed ball of E centered at x and of radius R is denoted by B E x, R. Let us now introduce some functional vector spaces. Consider µ ]0, 1] and u : Q T R k. Define u := sup t [0,T],x R N ut, x, ut, x ut, y [u] µ := sup t [0,T],x,y R N,x y x y µ, u µ := u + [u] µ. We let C b Q T, R k and C 0,µ b Q T, R k denote the spaces of continuous functions u : Q T R k such that u < + and u µ < +, respectively. We let BUCQ T, R k denote the space of bounded uniformly continuous functions u. When k = 1, we let C b Q T, C 0,µ b Q T and BUCQ T denote the preceding spaces. Consider a function h : O R k R and a nonnegative real α. We let ω α h denote the modulus of continuity of size α of h. That is to say, ω α h = sup hx hy. x,y O, x y α We call modulus a function m : R + R + such that m is continuous, nondecreasing, m0 = 0 and such that mα 1 + α mα 1 + mα for all nonnegative reals α 1 and α. Following [9], we now recall the notion of viscosity solution. This last one notion can be defined for discontinuous locally bounded functions and Hamiltonians. Here, we only need the continuous case. Consider the following general equation: t u + Gt, x, u, Du, D u = 0 in Q T,.1 3

4 where the Hamiltonian G : [0, T] R N R R N S N R is continuous and nonincreasing w.r.t. its last argument. For each bounded continuous function u : Q T R and each subset O of Q T, we let P,+ O ut, x denote the second-order parabolic superjet subjet of u at t, x O relatively to O. Let us recall that a, p, X P,+ O ut, x if and only if iff for short us, y ut, x + as + p, y x + 1 Xy x, y x + o s t + y x as O s, y t, x. We let simply P,+ ut, x denote the semijet P,+ Q T ut, x. Definition.1. Let u belong to CQ T. 1. The function u is a viscosity subsolution of.1 iff for every t, x Q T and a, p, X P,+ ut, x, a + Gt, x, u, p, X 0.. The function is a u viscosity supersolution of.1 iff for every t, x Q T and a, p, X P, ut, x, a + Gt, x, u, p, X The function is a u viscosity solution of.1 iff it is both a viscosity sub- and supersolution of.1. Remark.1. Define the closure P,+ O ut, x as the set of a, p, X R R N S N such that, there are t n, x n O and a n, p n, X n P,+ ut n, x n such that ut n, x n ut, x and O t n, x n, a n, p n, X n t, x, a, p, X. In fact, penalization technics used in [9] allow to prove that the definitions above are still true by replacing Q T by O :=]0, T] R N and P,+ ut, x by P,+ O ut, x. In a similar way, we can define the notion of continuous viscosity semisolution of the following differential equation: f + Gf = 0 in ]0, T[, where f : [0, T] R is the unknown function, and G C[0, T] is given. Then, the notion of parabolic semijet is replaced by the notion of first-order Fréchet semidifferential: for t ]0, T[ recall that a 1,+ ft iff fs ft + as + o s t, as s t. Let us now define the notion of viscosity solution of 1.1 which is used in our paper. Consider a continuous Hamiltonian F and a nonlocal term g[.] : C b Q T CQ T, R M. Let F u : [0, T] R N R R N S N R denote the functional defined by F u t, x, r, p, X = Ft, x, r, p, X, g[u]. Note that F u is continuous when u is bounded continuous. Consider the following equation in w: t w + F u t, x, w, Dw, D w = 0 in Q T.. Definition.. A function u C b Q T is a viscosity solution of 1.1 iff it is a viscosity solution of.. 4

5 3 Main results Let us state our main results. We first consider nonlocal term uncoupled with the derivatives of u and next we study the case where there is a coupling. All the constants appearing in this section are noted C F resp. C g when depending on the Hamiltonian F resp. the nonlocal term g[.] and CR F when also depending on a real number R. 3.1 Uncoupled nonlocal term Let us consider a class of nonlocal terms g[.] that satisfy the following conditions: H1 The operator g[.] is well-defined from C b Q T into C b Q T, R M in particular, g[0] C g < +. H If u is uniformly continuous w.r.t. x independently of t, then so is g[u]. H3 There exists a constant C g 0 such that for every u, v C b Q T and t [0, T], sup g[u]τ,. g[v]τ,. C g sup uτ,. vτ,.. As far as F is concerned, let us assume the following conditions: H4 The Hamiltonian F is continuous and for each R 0, F is uniformly continuous w.r.t. the Du, D u-variable, independently of the others, on [0, T] R N [ R, R] B R N0, R B S N0, R B R M0, R. H5 The Hamiltonian F is nondecreasing w.r.t. the u-variable. There is a modulus m R., depending on a real number R, such that for every > 0, t [0, T], x, y R N, r [ R, R], X, Y S N and λ B R M0, R, H6 if then F t, y, r, x y, Y, λ 3 X 0 0 Y F 3 I I I I t, x, r, x y, X, λ m R x y + x y. 3.1 H7 The Hamiltonian F is Lipschitz w.r.t. the g[u]-variable, independently of the others, with a Lipschitz constant that is denoted by C F. H8 sup QT Ft, x,0, 0, 0, 0 C F < +. Remark 3.1. Assumptions H4-H6 and H8 are classical when studying local equations see [9]. Monotonicity assumptions w.r.t. the nonlocality see [14, 16,, 11] and remarks in Example 3.1 below are replaced here by H3 and H7. Assumption H is necessary to solve after having frozen the nonlocal part. If we strengthen H4, then we can actually omit H to solve in the space of bounded continuous functions to see this, one could combine technics used in [] with our technics. 5

6 Then we have the following result. Theorem 3.1 Existence, uniqueness and BUC-regularity. Assume H1-H8. Let u 0 belong to BUCR N. Then there exists a unique u BUCQ T such that u is a viscosity solution of Moreover, for each t [0, T] where C = C F + C F C g and γ 0 = C F C g. ut,. u 0 + Cte γ 0t, 3. Remark 3.. This theorem is also a regularity result, since we get the uniform continuity of the solution. Moreover, the assumptions in particular H and H6 seem to us quite general to get this. Example 3.1. Suitable assumptions under which nonlocal terms of the general form 1.4 or 1.5 satisfy H1-H3 can easily be determinate by the readers. Some simple illustrating examples of such nonlocal terms are: convolution operators K ut,., where K L 1 R N ; Lévy operators of the form R N ut, x + z ut, xdµz, where µ is a bounded Borel measure under suitable assumptions to ensure H, µ can also depend on t, x; Volterra operators of the form t 0 Bt sus, xds, where B L1 0, T. Note that if the measure and the kernels above are nonnegative and if F is nondecreasing w.r.t. the g[u]-variable, then 1.1 is monotone see [16,, 11]; but, in the general case 1.1 can become nonmonotone. We are now interested in Hölder regularity of u w.r.t. x. For µ ]0, 1], let us consider the following condition: H There exists a constant C g 0 such that for every u C 0,µ b Q T and t [0, T], sup [g[u]τ,.] µ C g 1 + sup uτ,. µ. We let also H6 denote Assumption H6 for m R x y + x y = CR F x y µ + x y, where C F R is a nonnegative constant that depends on R. Let us state our regularity result. Theorem 3. Hölder regularity. Assume that H1, H, H3-H5, H6, H7 and H8 hold true for any µ ]0, 1]. Let u be the unique viscosity solution of Define R = u and R g = C g 1 + R. Then for each t [0, T], [ut,.] µ 4 [u 0 ] µ e µ γt + C R e γ1tt, 3.3 where γ = C F R R g + 1, γ 1t = 4C F C g e µ γt µ and C R = C F C g 1 C F R R g + C F R g. 6

7 3. Nonlocal term coupled with the derivatives Now, we strengthen H by: H There exists a constant C g 0 such that for every u C b Q T and t [0, T], Assumptions H6 and H7 on F are relaxed by: sup [g[u]τ,.] 1 C g 1 + sup uτ,.. H9 There exists a constant C F 0 such that for each R 0 there exists a constant C F R 0 such that for every > 0, t [0, T], x, y R N, r R, X, Y S N and λ, µ R, if 3.1 holds true then F t, y, r, x y, Y, λ F t, x, r, x y CR x F x y y +, X, µ + C F x y λ µ + λ µ + λ µ. Remark 3.3. Let us comment assumption H9. Under H6 with µ = 1 and H7, we have F t, y, r, x y, Y, λ F t, x, r, x y, X, µ C F x y R x y + + C F λ µ. The role of the new term C F λ µ x y + C F λ µ the following simple example of Hamiltonian: that appears in H9 can be illustrate by where A = t σσ and σ is Lipschitz. Let us state our last result. Ft, x, r, p, X, λ = λ p traλx, Theorem 3.3 Existence, uniqueness and Lipschitz regularity. Let us assume H1, H, H3-H5, H8 and H9. Then for each u 0 W 1, R N there exists a unique u C b Q T such that u is a viscosity solution of 1.1 and 1.. Moreover, defining R = u and R g = C g 1 + R, Estimate 3. still holds true and for each t [0, T] [ut,.] 1 [u 0 ] 1 + C R te γ t, 3.4 where γ = C F R R g + CF C g 1 + C g + R + C g R + 1 and C R = C F R R g + CF R g. Example 3.. Simple assumptions under which quasilinear Hamiltonians of the form 1.6 satisfy H9 are the following: there are nonnegative constants C i i = 1, such that for each R 0 there exists C R 0 such that: 1. For every t [0, T], x R N, r R, p R N, λ R M, λ Ht, x, r, p, λ C p and λ σt, x, p, λ C.. If r, λ R, then x Ht, x, r, p, λ C R 1 + p and x σt, x, p, λ C R. 7

8 4 Proofs of the results In this section, we prove the preceding results. This section is organized as follows: in Subsection 4.1, we state a technical result Theorem 4.1. This result is proved in Subsection 4.4. Subsection 4. is devoted to the proofs of Theorems 3.1 and 3.. Theorem 3.3 is proved in Subsection Continuous dependence estimate for local parabolic equations To state our technical result, we have to introduce some notations. Let us consider equations of the form t u i + G i t, x, u i, Du i, D u i = 0 in Q T, 4.1 where G i : [0, T] R N R R N S N R is a given functional i = 1,. We make the following assumptions on G i : The Hamiltonian G i is continuous and for each R 0, G i is uniformly continuous w.r.t. the Du i, D u i -variables, independently of the others, on [0, T] R N [ R, R] B R N0, R B S N0, R. 4. There exists γ 0 such that for every t [0, T], x R N, r, s R, p R N and X S N, if r s then G i t, x, r, p, X G i t, x, s, p, X + γr s. 4.3 For each R 0, there exists a modulus m R. such that for every > 0, t [0, T], x, y R N, r [ R, R] and X, Y S N, if 3.1 holds true then G i t, y, r, x y, Y G i t, x, r, x y x y, X m R x y Finally, we assume that for each R 0, sup G t, x, r,0, 0 G 1 t, x, r,0, 0 < t,x,r Q T [ R,R] Let u i be a bounded continuous sub- and supersolution of 4.1 for respectively i = 1 and i =, γ be a nonnegative constant that will be appropriately chosen when using Theorem 4.1 in Subsection 4. and 4.3 and be a positive real. For t [0, T], define m t := σ t := sup x,y R N R N inf sup n N d n t + γt x y u 1 t, x u t, y e, G t, y, r, e γt p, e γt Y G 1 t, x, r, e γt p, e γt X γe γt x y +, where we let d n t denote the set of x, y, r, p, X, Y such that, r u 1 t,. u t,., p = x y, Condition 3.1 holds true, x y 4 m t m t + 1 n. 8

9 We also define Σ t := sup D t G τ, y, r, e γτ p, e γt Y G 1 τ, x, r, e γt p, e γτ X γe where we let D t denote the set of τ, x, y, r, p, X, Y such that τ [0, t], r u 1 τ,. u τ,., p = x y, Condition 3.1 holds true. Then, we have the following result. γτ x y Theorem 4.1. Assume that G i satisfies i = 1,. Assume that u i C b Q T is a sub- and a supersolution of 4.1 for i = 1 and i =, respectively. Then: i For each > 0 the function σ. is measurable and for each t [0, T], m t m 0 + t 0 σ τdτ. ii If moreover γ > 0, then for each t [0, T], m t m γ Σ t. +, The proof of this result is given in Subsection Case of an uncoupled nonlocal term Proof of Theorem 3.1. We use a contracting fixed point theorem. We leave it to the reader to verify that for u BUCQ T the Hamiltonian F u satisfies Under these assumptions, it is well-known that there is a comparison principle between semicontinuous semisolutions of.. Moreover, sup QT F u t, x,0, 0, 0 < + and for u 0 BUCR N there exists a unique bounded continuous solution of. such that 1. is satisfied; we let Θu denote this solution. For a proof of these results we refer the reader to [9]. Let us first prove that Θ maps BUCQ T into itself. We let m., σ. and d n. denote the functions introduced in Subsection 4.1 for u i = u, G i = F u i = 1, and γ = 0. Let m u R. denote the modulus deriving from 4.4. For > 0, t [0, T] and x, y, r, p, X, Y d n t, F u t, y, r, p, Y F u t, x, r, p, X m u R x y + where R = Θu. By the definition of d n t, we see that x y 4 x y x y + m t m t + 1n + 4 x y, m t m t + 1 n. Thus m t m t + 1 n Let α n t denote the right hand side of this inequality and define α t := inf n N α n t. We get F u t, y, r, p, Y F u t, x, r, p, X m u R αn t. 9.

10 Taking the supremum w.r.t. x, y, r, p, X, Y d n t, we see that for all > 0 and all t [0, T], σ t inf n N mu R αn t m u R α t. By the item i of Theorem 4.1, we find that for all > 0 and all t [0, T], T m t m 0 + m u R α τ dτ We let I denote the integral term of this inequality. Let us recall that for each τ [0, T], m τ is nonnegative and nondecreasing w.r.t.. The limit lim > 0 m τ then exists and lim > 0 m τ m τ = 0. Moreover, we know that m τ Θu and we infer that the family of functions m u R α. is uniformly bounded and converges to 0 pointwise as >0 > 0. By Lebesgue s convergence Theorem, we deduce that lim > 0 I = 0. Since 4.6 implies that x y Θut, x Θut, y m 0 + I +, for all t [0, T], all x, y R N and all > 0, we have proved that Θu is uniformly continuous w.r.t. x independently of t. The uniform continuity in both time and space variable is now a consequence of the following result. Proposition 4.1. For each ν > 0, there exists C ν 0 that only depends on the Hamiltonian F, u, Θu and the modulus of continuity w.r.t. the x-variable of Θu and that is such that, for each α 0, sup ω α Θu., x inf {ν + C να}. 4.7 x R N ν>0 The proof of Proposition 4.1 is well-known for local equations see [6, Lemma 9.1] and can easily be adapted to nonlocal equations; for the reader s convenience, a sketch of the proof of this result is given in Appendix A. The proof of the fact that Θ maps BUCQ T into itself is complete. Define now the space E := {u BUCQ T : u0,. = u 0 } that we endow with the distance d E u, v = u 1 u γ, where γ := C F C g and w γ t, x := e γt wt, x. 4.8 We know that ΘE E. Let us prove that Θ : E E is a contraction. Let u i belong to E i = 1,. Define v := Θu + eγt d Eu 1, u. Let a, p, X belong to P, vt, x. Using successively H5, H7 and H3, we show that a + F u 1 t, x, v, p, X a + Ft, x,θu, p, X, g[u 1 ], a + Ft, x,θu, p, X, g[u ] C F g[u 1 ]t, x g[u ]t, x, a + Ft, x,θu, p, X, g[u ] C F C g sup u 1 τ,. u τ,., a + F u t, x,θu, p, X, C F C g e γt d E u 1, u. Since a C F C g e γt d E u 1, u, p, X P, Θu t, x, the viscosity inequalities applied to Θu supersolution of. with u = u imply that v is a supersolution of. with u = u 1. By 10

11 the comparison principle, we infer that Θu 1 v and Θu 1 Θu γ 1 d Eu 1, u. We can argue similarly to get the other inequality and we conclude that Θ is a contraction. Since the metric space E, d E is complete, Banach fixed point Theorem implies that the Cauchy problem admits a unique viscosity solution in BUCQ T. What is left to prove is Estimate 3.. It will be a consequence of the following result. Proposition 4.. If u satisfies 3., then Θu also satisfies 3.. Proof. Let t [0, T[ and h > 0 be such that t + h T. A simple computation shows that the function s vs := sup Θuτ,. + s t sup F u τ, x,0, 0, 0 τ,x Q t+h is a supersolution of. in the domain ]t, t + h[ R N. Since Θut,. vt, the comparison principle implies that sup Θu sup Θuτ,. + h sup Fτ, x,0, 0, 0, g[u]. Q t+h τ,x Q t+h Using successively H7, H3 and H1 and H8, we deduce that for all τ, x Q t+h, Fτ, x,0, 0, 0, g[u] Fτ, x,0, 0, 0, 0 + C F g[u]τ, x, Fτ, x,0, 0, 0, 0 + C g[0] F + C g sup uτ,., τ [0,τ] C F + C F C g 1 + sup uτ,., τ [0,t+h] = C + γ 0 sup uτ,., τ [0,t+h] where C and γ 0 are defined as in 3.. We get sup Θu sup Θuτ,. + h Q t+h C + γ 0 sup τ [0,t+h] uτ,. We can argue similarly to get the other inequality and we deduce that for all 0 t t + h T, sup Θuτ,. sup Θuτ,. + h C + γ 0 sup uτ,.. τ [0,t+h] τ [0,t+h] Since Θu is uniformly continuous, the function t [0, T] ft := sup Θuτ,. is continuous. Lemma 4.1 then implies that if u satisfies 3., then f is a continuous viscosity subsolution of the following equation: f = C + γ 0 g in ]0, T[, where gt is equal to the right hand side of 3.. The comparison principle then completes the proof of Proposition

12 Let us return to the proof of 3.. Since {u E : 3. holds true} is a nonempty closed subspace of E which is stable by Θ, we see that the unique fixed point of Θ belongs to this subspace. This completes the proof of 3. and a fortiori the proof of Theorem 3.1. Proof of Theorem 3.. Consider the space E µ := {u E : 3. and 3.3 hold true}, where R is any L bound on E µ. Let us prove that ΘE µ E µ. Proposition 4. implies that for all u E µ, Θu satisfies 3. and Θu R. To prove the Hölder continuity of Θu, we have to introduce some notations. For γ > 0, define F γ t, x, r, p, X, λ := e γt Ft, x, e γt r, e γt p, e γt X, λ, F γ u t, x, r, p, X := F γ t, x, r, p, X, g[u]. The function Θu γ, defined as in 4.8, is a viscosity solution of the following equation in w: t w + γw + F u γ t, x, w, Dw, D w = 0 in Q T. Following [15] and [10], we derive an Hölder estimate on Θu γ. Theorem 4.1 item ii, applied to G i t, x, r, p, X = γr + F u γ t, x, r, p, X and u i = Θu γ i = 1,, implies that for all > 0, all t [0, T] and all x, y R N, We have Θu γ t, x Θu γ t, y m γ Σ γt x y t + e. 4.9 m 0 = sup u 0 x u 0 y x,y R N R N + x y, sup x,y R N R N [u 0 ] µ µ µ [u 0 ] µ x y µ + x y, µ Let us derive an upper bound on Σ t. For τ, x, y, r, p, X, Y D t, define I := F γ u τ, y, r, e γτ p, e γτ Y F γ u τ, x, r, e γτ p, e γτ X, 4.11 = F γ τ, y, r, e γτ x y, e γτ Y, g[u]τ, y F γ τ, x, r, e γτ x y, e γτ X, g[u]τ, x. Condition H7 implies that I F γ τ, y, r, e γτ x y, e γτ Y, g[u]τ, x F γ τ, x, r, e γτ x y, e γτ X, g[u]τ, x + e γτ C F g[u]τ, y g[u]τ, x. 1

13 By H1 and H3, we see that g[u]τ, x R g where R g is defined as in Theorem 3.. By the definition of D t, we see that e γτ r e γτ Θu γ τ,. Θu R. Moreover, Condition 3.1 holds true and H6 then implies that I e γτ CR R F g x y µ + e CR R F g x y µ + e Using H, we get γ+γτ x y γτ x y + e γτ C F g[u]τ, y g[u]τ, x, + C F e γτ [g[u]τ,.] µ x y µ. I C 1 x y µ γτ x y + C e, 4.1 where C 1 = CR R F g + CF R g + C g sup s [0,t] [u γ s,.] µ and C = CR R F g. Taking the supremum w.r.t. τ, x, y, r, p, X, Y D t, we get Σ t sup, x,y R N + C 1 x y µ γτ x y γτ x y + C e γe. If we take γ = C + 1, then for all t [0, T] and all > 0, Σ t sup r>0 C 1 r µ r + C µ 1 µ µ Inequalities 4.9 and 4.10 then imply that for all t [0, T] and all x, y R N, µ Θu γ t, x Θu γ t, y inf [u >0 0 ] µ + C µ 1 µ µ γt x y + e γ, that is to say, [Θut,. γ ] µ [u 0 ] µ + C 1 γ µ µ [u 0 ] µ + C µ 1 γ e µ γt [u 0 ] µ + C 1 γ µ [u 0 ] µ + CF R R g + CF R g γ µ µ e µ γt x y µ ; e µ γt x y µ, e µ γt + CF C g γ µ e µ γt sup [u γ τ,.] µ. 13

14 Recalling that γ > 0 is arbitrary, we can take γ = γ 1 t > 0 in this inequality, where γ 1 t is such that CF C g e µ γt = 1. We then have proved that for all u Eµ and all t [0, T], γ 1 t µ [ ] Θut,. γ1 t µ [u 0 ] µ + CF R R g + CF R g γ 1 t µ 1 e µ γt + 1 sup [u γ1 tτ,.], 4[u 0 ] µ e µ γt + C R + 1 sup [u γ1 tτ,.], 4.14 where R is any L bound on E µ and γ 1 t and C R are defined as in Theorem 3.. A simple computation now completes the proof of the stability of E µ by Θ. Since E µ is a nonempty closed subspace of E, we deduce that the unique fixed point of Θ belongs to E µ. Thus, the unique viscosity solution u of is Hölder continuous and satisfies 3.3 note that we have also proved 4.14 for R = max{ u, Θu }. The proof of Theorem 3. is now complete. 4.3 Case of a coupled nonlocal term Proof of Theorem 3.3. The proof is based on a contracting fixed point theorem. Note that for each u C b Q T, the Hamiltonian F u satisfies and sup QT F u t, x,0, 0, 0 < + ; hence, there still exists a unique bounded continuous viscosity solution Θu of. which satisfies 1.. First step: a needed gradient estimate. We begin by the proof of the following property: for all u C b Q T and all t [0, T] [Θut,.] 1 [u 0 ] 1 + C R te γ t, 4.15 where R = max{ u, Θu } and γ and C R are defined as in Theorem 3.3. The item i of Theorem 4.1 for G i = F u and u i = Θu i = 1, implies that Θut, x Θut, y γt x y m 0 + t sup σ τ + e, γt x y m 0 + t Σ t + e Condition H9 gives the following estimate on I I being defined as in 4.11 with γ = 0: for all > 0, t [0, T] and all τ, x, y, r, p, X, Y D t, I CR R F g x y γτ x y 1 + e + C F γτ x y γτ g[u]τ, y g[u]τ, x g[u]τ, y g[u]τ, x 1 + e + e, CR R F g x y γτ x y 1 + e + C F γτ x y [g[u]] 1 x y 1 + e + e γτ [g[u]] 1 x y, CR R F g + CF [g[u]] 1 x y + C F R R g + C F [g[u]] 1 + C F [g[u]] 1 e γτ 14 x y,

15 where R := max{ u, Θu } and R g is defined as in Theorem 3.3. By H, we get 4.1 for µ = 1 and C 1 which is now equal to C F R R g + CF R g. Inequalities 4.16, 4.10, 4.13 and an optimization w.r.t. as in the precedent proof then complete the proof of Second step: local-in-time solvability of For t ]0, T], define the space E t := { u C b Q t : u satisfies 1. and 3. } endowed with the distance of the uniform convergence. We claim that Proposition 4. still holds true under the assumptions of Theorem 3.3. This implies that Θ maps E t into itself. Let us prove that Θ : E t E t is a contraction for t sufficiently small. In what follows, we let R denote a L bound on E t and we let R denote a bound on ΘE t for the [.] 1 - seminorm. Such a number R exists, thanks to Consider u i E t i = 1,. Define N := sup t [0,t ]{Θu 1 t,. Θu t,.} and M := sup t [0,t ] u 1 t,. u t,.. The item i of Theorem 4.1, applied to the functions Θu i, the Hamiltonians F ui and γ = 0, implies that for all > 0, N m 0 + t sup t [0,t ] σ t [u 0] 1 + t sup σ t R t [0,t ] + t sup σ t t [0,t ] Let us derive an upper bound on sup t [0,t ] σ t. Consider t [0, t ] and x, y, r, p, X, Y d n t. Condition H9 implies that F u t, y, r, p, Y F u1 t, x, r, p, X CR R F g x y 1 + x y where λ = g[u 1 ]t, y and µ = g[u ]t, x. By H and H3, Using 4.18 and 4.19, + C F x y λ µ λ µ [g[u 1 ]t,.] 1 x y + g[u 1 ]t, x g[u ]t, x, F u t, y, r, p, Y F u 1 t, x, r, p, X C g 1 + sup u 1 τ,. x y + C g M, τ [0,t ] λ µ, 4.18 R g x y + C g M C 1 x y 1 + x y x y + C M M where C i only depends on C F, C g, CR R F g and R i = 1,. Let us prove that x y, 4.0 m t m t + 1 n R + 1 n. 4.1 If m t = 0, then 4.1 is immediate since x = y. In the other case, that is to say if m t > 0, we can deduce from 4.36 that the approximate supremum m,η t defined in 4.5 is positive for all η sufficiently small. If m,η t is achieved at some x,y by 4.4, 15

16 such a maximal point always exists, then we see that x y 1,+ Θu t,.y and it follows that x y R. By a simple computation, we get m,η t m,η t x y 4 R. The limit as η > 0 in this inequality then gives m t m t R. Inequality 4.1 is now a immediate consequence of the definition of d n t. By 4.0, it follows that sup σ t C 3 + C 4 M t [0,t ] 1 + M where C i only depends on C F, C g, CR R F g, R and R i = 3, 4. Inequality 4.17 implies that for all > 0, N R + t C 3 + C 4 M 1 + M R = t C 4 M + + t C 3 + t M C 4. By taking the infimum w.r.t., there exists a universal constant C 0 such that sup {Θu 1 t,. Θu t,.} t [0,t ] R t C 4 + C + t C 3 By exchanging the role of Θu 1 and Θu, we conclude that sup Θu 1 t,. Θu t,. t [0,t ] R t C 4 + C + t C 3, 1 t C 4 1 t C 4 M. sup u 1 t,. u t,.. t [0,t ] Since C i only depends on C F, C g, C F R R g, R and R i = 3, 4, we have proved that Θ is a contraction for t > 0 sufficiently small. This established the local-in-time solvability of the Cauchy problem Third step: global solvability. Define now I := { t ]0, T] :! u t C b Q t such that u t satisfies 3. and u t is solution of in Q t } and t max = supi. By the preceding step, I. Let us prove that t max = T. Let us assume the contrary and let us seek a contradiction. By construction, there exists a unique u tmax C b Q tmax such that u tmax satisfies 3. and u tmax is solution of in Q tmax. By 4.15 and Proposition 4.1, u tmax satisfies 3.4 and 4.7. We leave it to the reader to verify that Proposition 4.1 still holds true under the assumptions of Theorem 3.3. The family u tmax s,. s [0,tmax[ thus satisfies the Cauchy property for s < t max and it follows that u tmax C 0,1 b Q tmax. Consider the following Cauchy problem: t u + Ft, x, u, Du, D u, g[u] = 0 in ]t max, T[ R N, 4. ut max,. = u tmax t max,. on R N

17 where g[u] := g[w] with w which is defined as follows: w := u tmax on Q tmax and w = u on [t max, t] R N. Arguing as in the second step, we can prove that there exists t ]t max, T] such that there exists a unique u C b [t max, t] R N which satisfies 3. and which is solution of in ]t max, t[ R N. For each u C b [t max, t] R N which satisfies 4.3, define ũ C b Q t as follows: ũ = u tmax on Q tmax and ũ = u on [t max, t] R N. Since P,+ ũt max, x P,+ ]0,t max] R N u tmax t max, x, the function u is solution of 4. in ]t max, t[ R N iff ũ is solution of 1.1 in Q t. We deduce that t I. Since t > t max, we get a contradiction and necessarily t max = T. The proof of Theorem 3.3 is now complete Proof of Theorem 4.1 Before proving our continuous dependence estimate, we have to recall two classical lemmas. The first lemma establishes a relation between the Fréchet subdifferential and the Clarke generalized derivative of a function of the real variable and the second lemma is a semicontinuity result on marginal functions. Here is the first result. Lemma 4.1. Let f : [0, T] R be a locally bounded function. For each s ]0, T[, sup 1,+ fs lim sup t s h > 0 ft + h ft. h Proof. Let ξ belong to 1,+ fs. For each τ sufficiently small, fs + τ fs + ξτ + oτ. If τ < 0, then fs + τ fs ξ + oτ ft ft + h = + oh τ τ h h, where t + h := s and t := s + τ. Since t s and h > 0 when τ < 0, the proof of Lemma 4.1 is complete. To state the second lemma, we have to introduce some notations and recall some definitions on multiapplications. Consider E and F two metric spaces, k an integer, and f a given function from E F into R. For x E, define gx := sup y dx fx, y where we let dx denote any subset of F depending on x. Definition 4.1. The multiapplication d : E F is said to be nonempty valued and compact valued if for each x E, dx is a nonempty compact subset of F. Moreover, we say that: 1. The multiapplication d is upper semicontinuous on E u.s.c. for short iff for all x E and all neighborhood U of dx, there exists η > 0 such that for all x B E x, η, dx U; 1 In fact, the uniqueness of the solution has been proved amongst the bounded continuous functions that satisfy 3.. But, we claim that the ideas of the proof of Proposition 4. allow to show that a priori solutions satisfies 3.; hence, the solution is unique in C b Q T. 17

18 . The multiapplication d is lower semicontinuous on E l.s.c. for short iff for all x E, all sequence x m x and all y dx, there exists a sequence y m dx m such that y m y; 3. The multiapplication d is continuous on E iff it is both u.s.c. and l.s.c. on E. Here is the second lemma. Lemma 4.. Assume that f is continuous and that d is nonempty valued, compact valued and u.s.c. on E. Then, g : E R is well-defined and upper semicontinuous. If moreover d is l.s.c. on E, then g is continuous on E. For a proof of this result we refer the reader to [5, Theorem 7.3.1]. Let us now return to the proof of our continuous dependence estimate. Proof of Theorem 4.1. Let us introduce some notations that will be needed. Define and γt x y ψ t, x, y := u 1 t, x u t, y e, Ψt, x, y, r 1, r, p, X, Y := G t, y, r, e γt p, e γt Y G 1 t, x, r1, e γt p, e γt X + γt x y γe. Let us perturb the functions m. and σ. the following way: let φ C R N be nonnegative and such that φ0 = 0, C φ := Dφ + D φ < + and for η ]0, 1] and t [0, T], define lim φx = + ; 4.4 x + m,η t := σ,η t := sup ψ t, x, y ηφx + φy +, 4.5 x,y R N R N sup Ψt, x, y, r, r, p, X, Y, d,ηt where we let d,η t denote the set of x, y, r, p, X, Y such that, φx u 1 u η, r max i=1, sup { φ u 1 u η p = x y, Condition 3.1 holds true, x y 4 m,η t m,η t. } u i t,., Let us give some properties on these functions that we admit until the end of the proof of Theorem 4.1. Proposition 4.3. The functions m,η. and σ,η. are well-defined from [0, T] into R + and are continuous. 18

19 The strategy of the proof of Theorem 4.1 is the following: first, we will prove that there exists a constant R 0 such that for each η ]0, 1], m,η. is a continuous viscosity subsolution of the following differential equation in f: f = σ,η γf + + ω R ηc φ in ]0, T[, 4.6 where we let ω R. denote the modulus deriving from 4.; this will give us some approximate continuous dependence estimates, thanks to the comparison principle, and we will conclude by taking the limit as η > 0 in these estimates. First step: proof of 4.6. By Proposition 4.3 and Lemma 4.1, it remains to prove that for each s ]0, T[, m,η t + h m,η t lim sup σ,η s γm,η s + + ω R ηc φ. 4.7 h t s h > 0 Let us perturb again the functions m,η. and σ,η. the following way: define and for every t, h E and λ R +, define E := {t, h R : 0 t t + h T } M t, h, λ := Σt, h, λ := sup τ [t,t+h], x,y R N R N ψ τ, x, y ηφx + φy λτ t +, sup Ψτ, x, y, r 1, r, p, X, Y, Dt,h,λ where we let Dt, h, λ denote the set of τ, x, y, r 1, r, p, X, Y such that, τ [t, t + h], φx u 1 u η, r 1 r M t, h, λ, r i max j=1, sup [t,t+h] {φ u 1 u η p = x y, Condition 3.1 holds true, x y 4 M t, h, λ M t, h, λ. } u j, i = 1, The new domain, on which the supremum M t, h, λ is computed, is introduced in order to use Equation 4.1. Note the presence of the penalization terms, ηφx + φy and x y, which are classically used when working with viscosity solutions see [9]. Let us recall that there are respectively used to treat unbounded domain and to split in two the space variables and use Ishii Lemma. Here are some properties on the functions above that we admit until the end of the proof of Theorem 4.1. Proposition 4.4. The functions M.,.,. and Σ.,.,. are well-defined from E R + into R + and are continuous. 19

20 Let us take the parameter λ = λt, h large enough in order that M t, h, λt, h M t, 0, λt, h = m,η t. 4.8 Roughly speaking, λt, h will be an upper bound on the velocity of m,η. on [t, t+h] and 4.7 will be obtained by taking the limit as t > s and h > 0. Take { u1 u e R = max u 1 u, γt For every t, h E, define where we let dt, h denote the set of λ R + such that We have the following result: + C φ, 6eγT + C φ }. 4.9 λt, h := supdt, h, 4.30 λ Σt, h, λ ω R ηc φ Proposition 4.5. The function λ.,. is well-defined from E into R + and is u.s.c.. Proof. It is easy to see that Σt, h, λ is nonnegative and bounded independently of t, h, λ E R +. Then, d.,. is nonempty valued and compact valued thanks to the continuity of Σ.,.,.. Moreover, t,h E dt, h is relatively compact and the continuity of Σ.,.,. also ensures the upper semicontinuity of d.,. on E. By Lemma 4., the proof of Proposition 4.5 is complete. Let us prove that for all t, h E, 4.8 holds true. When M t, h, λt, h = 0, 4.8 is immediate since M t, h, λ is nonnegative. Assume that M t, h, λt, h > 0. The continuity of M.,.,. implies that for all λ > λt, h sufficiently close to λt, h, M t, h, λ > 0. Let λ be such a number. By 4.4, there exists τ, x,y at which M t, h, λ is achieved. Necessarily, { φx u 1 u η, 4.3 e γτ x y 4 M t, h, λ M t, h, λ. Let us prove that τ = t. Assume the contrary and let us seek a contradiction. We let O denote the cylinder ]t, t + h] R N. Ishii Lemma see [8] or [9, Theorem 8.3] implies that there are a, e γτ p + ηdφx, e γτ X + ηd φx P,+ O u 1 τ, x, b, e γτ p ηdφy, e γτ Y ηd φy P, O u τ, y, with p = x y, a b = λ + γe γτ x y, Condition 3.1 holds true. Indeed, multiapplications of the form dx = {y F : hx, y 0}, where E, F are metric spaces, h : E F R k is continuous and x Edx is relatively compact, are u.s.c. on E. 0

21 Using the viscosity inequalities, we get a b G τ, y, u, e γτ p ηdφy, e γτ Y ηd φy G 1 τ, x, u1, e γτ p + ηdφx, e γτ X + ηd φx. By 4.3 and 3.1, we know that e γτ u p 1 u e γτ and X, Y 6. Recalling that γ is nonnegative, the definition of R see 4.9 and 4. imply that We then deduce that a b G τ, y, u, e γτ p, e γτ Y G 1 τ, x, u1, e γτ p, e γτ X + ω R ηc φ. λ G τ, y, u, e γτ p, e γτ Y G 1 τ, x, u1, e γτ p, e γτ X γτ x y γe + ω R ηc φ, Ψτ, x,y, r 1, r, p, X, Y + ω R ηc φ, 4.33 where r 1 = u 1 τ, x and r = u τ, y. Recalling that M t, h, λ > 0, we get As a result of this, ψ τ, x,y ηφx + φy λτ t ψ τ, x,y ηφx + φy λτ t + > 0. = ψ τ, x,y ηφx + φy λτ t + = M t, h, λ. The nonnegativity of e γτ x y + ηφx + φy + λτ t implies that r 1 r M t, h, λ. According to 4.3 and 3.1, τ, x,y, r 1, r, p, X, Y Dt, h, λ and 4.33 implies that λ Σt, h, λ + ω R ηc φ. By 4.30, 4.31 and the fact that λ > λt, h, we get a contradiction. Consequently, τ = t and M t, h, λ M t, 0, λ. By the continuity of M.,.,., we can pass to the limit in λ > λt, h to complete the proof of 4.8. Now, a simple computation shows that for all t, h E, m,η t + h M t, h, λt, h + λt, hh M t, 0, λt, h + λt, hh = m,η t + λt, hh. It follows that for all s ]0, T[, lim sup t s h > 0 m,η t + h m,η t h lim sup λt, h λs,0, t s h 0 > thanks to the upper semicontinuity of λ.,.. Moreover, the continuity of Σ.,.,. and the definition of λ.,. imply that λs,0 = Σs,0, λs,0 + ω R ηc φ in fact, this holds true not only for t, h = s,0 but for all t, h E. Since 4.3 implies that Σs,0, λs,0 σ,η s γm s,0, λs,0 + = σ,η s γm,η s +, 1

22 we deduce that λs,0 σ,η s γm,η s + + ω R ηc φ. Inequality 4.7 is now immediate and this completes the proof of 4.6. Second step: taking limit as η > 0. By the comparison principle, we deduce from 4.6 that for each t [0, T], m,η t m,η 0 + and if γ > 0, then for each t [0, T] m,η t t 0 σ,η τdτ + t ω R ηc φ 4.34 m,η γ sup σ,η τdτ + t ω R ηc φ Indeed, the right hand side of 4.34 resp is a classical solution resp. a continuous supersolution of 4.6 as function of the t-variable. Let us pass to the limit as η > 0 in these inequalities. One can see that for all t [0, T], lim m i,η t = m i t i = 1, and lim supσ,η t σ t η 0 > The first limit is classical and its verification is left to the reader. Let us prove the second limit. Let n be a nonnegative integer and let t belong to [0, T]. For all positive η sufficiently small, m,η t m,η t m t m t + 1 n. Consequently, d,ηt d n t and σ,η t σ n t. Taking the infimum w.r.t. n N implies that lim sup η > 0 σ,η t σ t. This finishes the proof of Note now that the set d,η t is bounded independently of t and η. We deduce, by 4.4 and 4.5, that the family of functions σ,η η>0 is uniformly bounded on [0, T]. By 4.36 and Fatou s Lemma, the limit as η > 0 in 4.34 completes the proof of the item i of Theorem 4.1 for a proof of the measurability of σ., see Appendix B. Moreover, when γ > 0 it is immediate that {τ} d,η τ D t for all t [0, T] and all η > 0. The item ii of Theorem 4.1 can thus easily be deduced from 4.35 and this completes the proof of Theorem 4.1. Proof of Propositions 4.3 and 4.4. We only show that Σ.,.,. is a well-defined real valued continuous function since the other points can be proved by the same ideas. Defining r as in 4.38, we see that t, 0, 0, r, r, 0, 0, 0 Dt, h, λ for all t, h, λ E R +. It follows that D.,.,. is nonempty valued. Moreover, it is clear that D.,.,. is compact valued and since t,h,λ E R +Dt, h, λ is relatively compact, D.,.,. is u.s.c. on E R + see Footnote on page 0. According to Lemma 4., we are then reduce to proving that D.,.,. is l.s.c.. Let t, h, λ E R +, t m, h m, λ m E R + be a sequence which converges to t, h, λ and τ, x, y, r 1, r, p, X, Y Dt, h, λ. For m N, define and r m := max sup { j=1, [t m,t m+h m] M m := M t m, h m, λ m η > 0 φ u 1 u η τ m := max{t m, min{τ, t m + h m }}, r 1,m := max{ r m + M m, min{r 1, r m }}, r,m := max{ r m,min{r, r 1,m M m }}. } u j,

23 For any ν [0, 1], define y ν := 1 νx + νy. We get x yν 4 = ν x y 4. Define { } νm 4 := min 1, x y M t m, h m, λ m M t m, h m, λ m and y m = y νm and p m = x ym. Observe that for all reals a b and c a max{a,min{c, b}} b and if a c b, then max{a,min{c, b}} = c. Since M m r m, r m + M m r m. By 4.37 and the nonnegativity of M m, r m r m + M m r 1,m r m. Using now that r m r 1,m M m, 4.37 implies that r m r,m r 1,m M m r m M m r m. We then have proved that r i,m r m i = 1, and that r 1,m r,m M m. We deduce that τ m, x, y m, r 1,m, r,m, p m, X, Y Dt m, h m, λ m, 4.37 the detailed verification of the conditions on τ m, y m and p m being left to the reader. Moreover, we have the following properties: rm r := max sup { j=1, [t,t+h] M m M := M t, h, λ. φ u 1 u η } u j, 4.38 This can be seen by using Lemma 4. to prove the continuity of M.,.,. and the continuity of r as function of the t, h-variable. Then, r 1,m max{ r + M, min{r 1, r }}, { } r,m max r, min{r, limr 1,m M}. m Since r i r and r 1 r M, r + M r + M r 1 r and 4.37 implies that lim m r 1,m = r 1 and lim m r,m = r. We easily deduce that τ m, x, y m, r 1,m, r,m, p m, X, Y τ, x, y, r 1, r, p, X, Y and this finishes the proof of the lower semicontinuity of D.,.,.. The proof of Propositions 4.3 and 4.4 is now complete. A Sketch of the proof of Proposition 4.1 Let C ν := sup β>0 sup t [0,T] ω β Θut,. ν β, R M := C ν, C ν := sup Ft, x, r, p, X, λ, K 3

24 where R = u Θu and K denote the set of t, x, r, p, X, λ such that, t [0, T], r R, p C νm, X C ν, λ C g 1 + R. If Θu is uniformly continuous w.r.t. x independently of t, then C ν is finite. By H4, H7 and H8, it follows that C ν is finite. Let s and y be fixed. A simple computation shows that the function vt, x := Θus, y + ν + C ν x y + C ν t s is a viscosity supersolution of. in ]s, T[ B R Ny, M. Moreover, for each x Θus, x Θus, y ω x y Θus,. ν + C ν x y. It follows that Θus,. vs,.. Another simple computation shows that Θu v on the domain ]s, T[ B R Ny, M. Using the comparison principle with a Dirichlet condition see [9] and choosing x = y, we deduce that Θut, x Θus, x + ν + C ν s t. We can argue similarly to obtain the other inequality and prove that Θut, x Θus, x ν +C ν s t for all t, s, x. The proof of Proposition 4.1 is complete. B Proof of the measurability of σ. For t [0, T], define σ n t := sup d n t Ψt, x, y, r, r, p, X, Y. Let us prove that σ n. is measurable. For a set A of a metric space E, define { 0 if x A, I A x := if not. We let A denote the set of X, Y S N such that 3.1 holds true. For t [0, T], we let Bt resp. Ct denote the set of r R resp. x, y R N such that r u 1 t,. u 1 t,. resp. x y 4 m t m t + 1 n. We see that σ n t = sup x,y,r,x,y R N R A For each t [0, T], σ n t = { Ψ t, x, y, r, r, x y, X, Y Ct = Ct and either Bt = {0} or { sup Ψ Q N Q D t, x, y, r, r, x y, X, Y } + I Bt r + I Ct x, y. Bt = Bt; this implies that } + I Bt r + I Ct x, y, where D is a countable dense subset of A. Moreover, t u 1 t,. and m i. are measurable as l.s.c. functions i = 1,. Consequently, for each x, y, r, X, Y Q N Q D, the function t Ψ t, x, y, r, r, x y, X, Y + I Bt r + I Ct x, y 4

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