Remarks on Schauder estimates and existence of classical solutions for a class of uniformly parabolic Hamilton-Jacobi-Bellman integro-pde

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1 Remarks on Schauder estimates and existence of classical solutions for a class of uniformly parabolic Hamilton-Jacobi-Bellman integro-pde Chenchen Mou Department of Mathematics, UCLA Los Angeles, CA 995, U.S.A. muchenchen@math.ucla.edu Abstract We prove Schauder estimates and obtain existence of classical solutions of Dirichlet initial boundary value problems for a class of uniformly parabolic non-local Hamilton-Jacobi- Bellman equations. Keywords: integro-pde, viscosity solution, Hamilton-Jacobi-Bellman equation, Schauder estimates. 1 Mathematics Subject Classification: 35R9, 35D4, 35K61, 45K5, 47G, 93E. 1 Introduction In this paper, we study existence of classical solutions and Schauder estimates for Dirichlet initial boundary value problems for uniformly parabolic non-local Hamilton-Jacobi-Bellman (HJB integro-pde {1 u t + inf Tr ( σ a (t, xσa T (t, xd u + I a [t, x, u] (1.1 +b a (t, x Du c a (t, xu + f a (t, x } =, in Q T := [, T Ω, where A is an index set, Ω is a bounded open subset of R n, σ a : Q T R n m, b a : Q T R n, c a : Q T R, f a : Q T R are continuous functions and I a [t, x, u] is an integro-differential operator. The nonlocal operator I a is of Lévy-Itô form, i.e. I a [t, x, u] := [u(t, x + j a (t, x, z u(t, x Du(t, x j a (t, x, z] µ(dz, (1. where j a : Q T R n is a function, continuous in (t, x and Borel measurable in z, that determines the size of the jumps for the diffusion related to the operator I a and µ is a Lévy measure. The Lévy measure µ is a Borel measure on \ {} satisfying ( z 1µ(dz < +. \{} 1

2 We will make an additional assumption about µ later. We extend µ to a measure on by setting µ({} =. Existence of W,p loc solutions of Dirichlet boundary value problems for linear uniformly elliptic versions of (1.1 has been obtained first in [17, 4] even though the arguments in [4] are not entirely correct. The same problem for uniformly elliptic HJB integro-pde have also been studied in [17], where existence of W, loc solutions was proved under an additional assumption about the non-local part. The equation studied in [17] was written in a slightly different form from (1.. The non-local operators in [17] were of the form I a [x, u] := [u(x + z u(t, x Du(x z] k a (x, zµ(dz and the additional condition there required that for every a, z and x Ω, k a (x, z = if x + z Ω. For the associated optimal control problem this corresponds to the requirement that the controlled diffusions never exit Ω and, thus, the parabolic boundary is different from that in this paper. Existence of strong solutions for the Neumann boundary value problem was investigated in [5]. Cauchy problems for integro-differential operators and HJB integro-pde have been studied in Hölder and Sobolev spaces by Mikulyavichyus and Pragarauskas in [6, 7]. Existence of classical solutions in Hölder spaces of Dirichlet initial boundary value problems for uniformly parabolic integro-pde of type (1.1 was proved in [8] under a condition which is similar to the one used in [17] (discussed above. In [31] the authors established existence of unique solutions in weighted Sobolev spaces of a Cauchy-Dirichlet problem for a linear uniformly parabolic integro-pde in a smooth bounded domain. A theory of Green functions for initial boundary value problems for parabolic non-local integro-differential operators can be found in [16]. Generalized and viscosity solutions of initial boundary value problems for non-local HJB equations have been studied in [9, 3]. Existence of viscosity solutions for a Dirichlet boundary value problem for a time dependent non-local equation was also investigated in [1] and existence of viscosity solutions for elliptic integro-differential equations was proved in []. Finally we mention that there are many recent regularity results for purely non-local equations, see e.g. [5, 4, 6, 7, 8, 9, 1, 11, 1,, 35, 34], where regularity is derived as a consequence of ellipticity/parabolicity and regularity of the non-local part. In this paper we study the case where the regularity of the solution is a consequence of the uniform parabolicity of the differential operators. The motivation of studying such C 1+α/,+α regularity results comes from the stochastic representation for the solution to a degenerate nonlocal HJB equation (1.1. Indeed, to obtain the stochastic representation in the degenerate case, we first need to derive it for the equation, by adding ɛ u to the nonlocal HJB equation, which is a uniformly parabolic equation where the uniform parabolicity comes from the second order term. The C 1+α/,+α regularity for the uniformly parabolic nonlocal HJB equation are crucial for the application of the Itô formula for Lévy process to derive the stochastic representation formula in the uniform parabolic case. Then, by the comparison principle, the analysis of the exit times and an approximation argument, we can obtain the stochstic representation formulas for the degenerate nonlocal HJB equation. The focus of this paper is to obtain interior Schauder estimates and establishing the existence, in the parabolic Hölder spaces C 1+α/,+α, of classical solutions of Dirichlet initial boundary value problems for a class of HJB equations (1.1 (i.e. classical solutions of problems (4.1 without any restrictions on the jump functions j a besides their Lipschitz and Hölder regularity. Thus we generalize the results of [8], however we have to assume here that the index set A is finite. Our method, in its initial part, is partially reminiscent of these of [17, 4]. We first truncate the Lévy measure µ and consider truncated problems for

3 which classical solutions exist. To prove uniform estimates for the solutions of the truncated problems, we employ a bootstrap technique. We begin with Lipschitz estimates which can be proved by continuous dependence estimates for viscosity solutions. Since the truncated problems can be rewritten as PDE, we then use localization and the W 1,,p estimates for fully nonlinear uniformly parabolic HJB equations to obtain uniform interior W 1,,p estimates independent of the truncation. Unfortunately the use of the W 1,,p estimates seems to be the weak link of the argument since the nature of these estimates for fully nonlinear equations allows us to be successful here only if A is finite, thus the restriction. Having the W 1,,p estimates and using Sobolev type embeddings one can then convert the truncated PDE to PDE with Hölder continuous coefficients and finally use C 1+α/,+α estimated for fully nonlinear uniformly parabolic HJB equations to produce uniform C 1+α/,+α estimates. The plan of the paper is the following. In Section we collect the definitions, assumptions and some preliminary results. In Section 3, following [19], we prove continuous dependence estimates for viscosity solutions of Dirichlet initial boundary value problems for (1.1, from which we deduce Lipschitz in x and Hölder in t continuity estimates for viscosity solutions. In Section 4 we construct Lipschitz in x and Hölder in t viscosity solutions of Dirichlet initial boundary value problems for (1.1 by Perron s method. We adapt the barrier function technique of [17] to construct barrier functions for our problem. Finally Section 5 is devoted to the proof of the interior C 1+α/,+α estimates which we outlined above. The main result of the paper is stated in Theorem 5.3. Definitions and assumptions The Euclidean distance between x and y in R n is denoted by x y and the inner product of x and y by x y. The standard parabolic boundary of Q T = [, T Ω is denoted by p Q T : p Q T := ({T } Ω ([, T Ω. The nonlocal parabolic boundary of Q T = [, T Ω is denoted by pn Q T : pn Q T := ({T } R n ([, T Ω c, (.1 where Ω c = R n \ Ω. The parabolic distance between (t, x and (s, y in R R n is For Q Q T we define d p ((t, x, (s, y := ( t s + x y 1. d p (Q, p Q T := inf { d p ((t, x, (s, y : (t, x Q, (s, y p Q T } and for x R n, A R n d(x, A := inf { x y : y A}. We set for δ > Ω δ = {x Ω : d(x, Ω > δ}. The notation Q Q T means Q Q T and d p (Q, p Q T >. 3

4 Let Q be any open subset of R R n. We will use the function spaces C(Q, L p (Q, C α/,α (Q, C 1/+α/,1+α (Q, C 1+α/,+α (Q and W 1,,p (Q, together with their local versions. The spaces C( Q and L p (Q, 1 p <, are equipped respectively with the norms ( u L (Q and u L p (Q := u(t, x p dtdx Q C α/,α (Q, < α < 1, is the space of functions u : Q R such that u C α/,α (Q := u L (Q+[u] C α/,α (Q := u L (Q+ 1 p. (t,x,(s,y Q,(t,x (s,y u(t, x u(s, y d α p ((t, x, (s, y <. C 1/+α/,1+α (Q, < α < 1, is the space of functions u : Q R whose spatial gradient Du(t, x exists classically for any (t, x Q and for which u C 1/+α/,1+α (Q := u L (Q + Du L (Q u(t, x u(s, y Du(s, y (x y + (t,x,(s,y Q,(t,x (s,y d 1+α p ((t, x, (s, y <. C 1+α/,+α (Q, < α < 1, is the space of functions u : Q R such that u t (t, x, Du(t, x and D u(t, x exist classically for any (t, x Q and such that u C 1+α/,+α (Q := u L (Q + Du L (Q + D u L (Q + u t L (Q + (t,x,(s,y Q,(t,x (s,y u(t, x u(s, y u t (s, y(t s Du(s, y (x y 1 (x yt D u(s, y(x y d +α <. p ((t, x, (s, y It can be seen that if u C 1/+α/,1+α (Q then Du C α/,α (Q, and if u C 1+α/,+α (Q then Du C 1/+α/,1+α (Q and u t, D u C α/,α (Q. W 1,,p (Q, 1 p <, is the space of functions u : Q R for which u, Du, D u, u t L p (Q in the sense of distributions and is equipped with the norm ( u W 1,,p (Q = u p L p (Q + u t p L p (Q + Du p L p (Q + D u p 1 p L p (Q. We will write USC([, T R n (respectively, LSC([, T R n for the space of upper (respectively, lower semi-continuous functions on [, T R n. We denote by R n m the space of real n m matrices and by S n the space of real n n symmetric matrices. We will be using the following calculus lemma and the well known Minkowski s inequality for integrals, see e.g. [15], Theorem Lemma.1. Let δ >. For any x R n, z B δ ( and f C (B δ (x, we have f(x + z f(x Df(x z = 1 1 D f(x + tszz ztdsdt. Lemma.. Let (X, M, ν 1 and (Y, N, ν be σ-finite measure spaces and let 1 p <. If f is an M N measurable function on X Y, then ( Y ( X p 1 p f(x, y ν 1 (dx ν (dy 4 X ( Y 1 f(x, y p p ν (dy ν1 (dx.

5 We recall two equivalent definitions of a viscosity solution of (1.1. In order to do it, we introduce two associated operators Ia 1,δ and Ia,δ, Ia 1,δ [t, x, p, u] = [u(t, x + j a (t, x, z u(t, x p j a (t, x, z] µ(dz, I,δ a [t, x, p, u] = z δ [u(t, x + j a (t, x, z u(t, x p j a (t, x, z] µ(dz. Definition.1. A bounded function u USC([, T R n is a viscosity subsolution of (1.1 if whenever u ϕ has a maximum over [, T R n at (t, x Q T for some bounded test function ϕ C 1, ([, T R n, then {1 ϕ t (t, x + inf Tr ( σ a (t, xσa T (t, xd ϕ(t, x + I a [t, x, ϕ] +b a (t, x Dϕ(t, x c a (t, xu(t, x + f a (t, x }. A bounded function u LSC([, T R n is a viscosity ersolution of (1.1 if whenever u ϕ has a minimum over [, T R n at (t, x Q T for a bounded test function ϕ C 1, ([, T R n, then {1 ϕ t (t, x + inf Tr ( σ a (t, xσa T (t, xd ϕ(t, x + I a [t, x, ϕ] +b a (t, x Dϕ(t, x c a (t, xu(t, x + f a (t, x }. A function u is a viscosity solution of (1.1 if it is both a viscosity subsolution and viscosity ersolution of (1.1. Definition.. A bounded function u USC([, T R n is a viscosity subsolution of (1.1 if whenever u ϕ has a maximum over [, T R n at (t, x Q T for a bounded test function ϕ C 1, ([, T R n, then, for any < δ < 1, {1 ϕ t (t, x + inf Tr ( σ a (t, xσa T (t, xd ϕ(t, x + Ia 1,δ [t, x, Dϕ(t, x, ϕ] + Ia,δ [t, x, Dϕ(t, x, u] +b a (t, x Dϕ(t, x c a (t, xu(t, x + f a (t, x }. A bounded function u LSC([, T R n is a viscosity ersolution of (1.1 if whenever u ϕ has a minimum over [, T R n at (t, x Q T for a bounded test function ϕ C 1, ([, T R n, then, for any < δ < 1, {1 ϕ t (t, x + inf Tr ( σ a (t, xσa T (t, xd ϕ(t, x + Ia 1,δ [t, x, Dϕ(t, x, ϕ] + Ia,δ [t, x, Dϕ(t, x, u] +b a (t, x Dϕ(t, x c a (t, xu(t, x + f a (t, x }. A function u is a viscosity solution of (1.1 if it is both a viscosity subsolution and viscosity ersolution of (1.1. The equivalence of Definition.1 and Definition. is standard. We remark that the definitions above may be slightly different from typical definitions of viscosity sub/ersolutions which also incorporate the boundary and initial conditions into the definitions. We make the following assumptions. 5

6 (H ρ(z µ(dz < +, (. where ρ : \ {} R + is a Borel measurable, locally bounded function satisfying lim z ρ(z = and inf z r ρ(z > for any r >. (H1 c a in Q T for every a A. (H For every a A, z and (t, x, (s, y Q T, there exist constants < α < 1 and L > such that f a (t, x f a (s, y + c a (t, x c a (s, y + σ a (t, x σ a (s, y + b a (t, x b a (s, y L( t s α + x y (H3 and and j a (t, x, z j a (s, y, z Lρ(z( t s α + x y. f a L (Q T + c a L (Q T + j a (,, z L (Q T Lρ(z. b a L (Q T + σ a L (Q T L Remark.1. Assumption (H1 is not essential since one can perform a standard change of dependent variable to guarantee that the new function satisfies an equation for which (H1 holds. 3 Hölder regularity In this section we prove that viscosity solutions of (1.1 are Hölder continuous in time and Lipschitz continuous in the space variable. We first introduce a nonlocal version of the Jensen- Ishii lemma which we borrow from [19], Theorem.. The reader can consult [3, ] for more information about the Jensen-Ishii lemma for integro-differential equations. Lemma 3.1. Let u, ū be respectively a bounded viscosity subsolution of (1.1 and a bounded viscosity ersolution of where {1 ū t + inf Tr ( σ a (t, x σ a T (t, xd ū + Īa[t, x, ū] (3.1 + b a (t, x Dū c a (t, xū + f a (t, x } = in Q T, Ī a [t, x, ū] := [ū (t, x + j a (t, x, z ū(t, x Dū(t, x j a (t, x, z] µ(dz. Assume that (H holds and (H1-(H3 are satisfied for the coefficients of (1.1 and (3.1. Let ϕ C 1, ([, T R n and (t, x, y [, T R n be such that (t, x, y u(t, x ū(t, y ϕ(t, x, y 6

7 has a maximum at (t, x, y Q T over [, T R n. Furthermore, assume that in a neighborhood of (t, x, y there are continuous functions g : [, T ] R n R, g 1, g : [, T ] R n S n with g (t, x, y >, satisfying ( ( D I I g1 (t, x ϕ(t, x, y g (t, x, y +. I I g (t, y Then, for any < δ < 1 and ɛ >, there are a, b R and X, Y S n satisfying a b = ϕ t (t, x, y and ( ( ( X g1 (t, x I I (1 + ɛ Y g (t, y g (t, x, y, I I such that {1 a + inf Tr ( σ a (t, x σa T (t, x X + b a (t, x D x ϕ(t, x, y c a (t, x u(t, x + f a (t, x +Ia 1,δ [t, x, D x ϕ(t, x, y, ϕ(t,, y ] + Ia,δ [t, x, D x ϕ(t, x, y, u(t, ] }, (3. {1 b + inf Tr ( σ a (t, y σ a T (t, y Y b a (t, y D y ϕ(t, x, y c a (t, y ū(t, y + f a (t, y +Ī1,δ a [t, y, D y ϕ(t, x, y, ϕ(t, x, ] + Ī,δ a [t, y, D y ϕ(t, x, y, ū(t, ] }. (3.3 Remark 3.1. Lemma 3.1 is a weaker version of Theorem. in [19]. The following theorem provides continuous dependence estimates for viscosity solutions of (1.1, which allow us to obtain regularity estimates for viscosity solutions in time and space. It is an analog of Theorem 3.1 in [19] for an initial boundary value problem. Theorem 3.1. Assume that (H holds and (H1-(H3 are satisfied for the coefficients of (1.1 and (3.1. For any η >, let u, ū be respectively a bounded viscosity subsolution of (1.1 and a bounded viscosity ersolution of (3.1 such that, for some C 1, η, and u(t, x u(t, y + ū(t, x ū(t, y C 1 x y for any x, y R n (3.4 u(t, x u(t, y + ū(t, x ū(t, y C 1 x y + η for any t [, T, x Ω c, y R n. (3.5 Then there exists a constant C, depending on L, C1, n, u L ([,T R n, ū L ([,T R n and ρ(z µ(dz such that u(t, x ū(t, x η + {u(t, x ū(t, x} + (t,x pq T + CT ( f a f a L (Q T + c a c a L (Q T ( + CT 1 b a b a L (Q T + σ a σ a L (Q T +,(t,x Q T 7 ( 1 j a (t, x, z j a (t, x, z µ(dz.

8 Proof. Let R > be such that Ω B R (. Throughout the proof, C max( u L ([,T R n, ū L ([,T R n will be a generic constant which will also depend on L, C 1, n and ρ(z µ(dz. Let ϕ(t, x, y := e γ(t t β x y + ɛ ( x + y and ψ(t, x, y := u(t, x ū(t, y ϕ(t, x, y κσ T (T t ɛ t, where κ, ɛ, ɛ (, 1, γ, β >, { σ := u(t, x ū(t, y ϕ(t, x, y ɛ } σ t and σ := (t,x,y [,T R n { u(t, x ū(t, y ϕ(t, x, y ɛ } +. (t,x,y {T } R n ((,T (Ω Ω c t Without loss of generality we assume that σ >. Since u, ū USC([, T R n, the boundedness of u, ū and the growth properties of ϕ and ɛ t guarantee that there exists (t, x, y [, T R n such that (t,x,y [,T R n ψ(t, x, y = ψ(t, x, y. Thus, ψ(t, x, y σ + σ κσ > σ. If t = T, then ψ(t, x, y = u(t, x ū(t, y ϕ(t, x, y ɛ T σ. If (t, x, y [, T (Ω Ω c, then ψ(t, x, y = u(t, x ū(t, y ϕ(t, x, y κσ T (T t ɛ t σ. Therefore, we can conclude that (t, x, y [, T Ω Ω. By Lemma 3.1, we have, for any δ (, 1, there are a, b R and X, Y S n satisfying a b = ϕ t (t, x, y κσ T ɛ t, ( ( ( X e γ(t t I I I β + ɛ, Y I I I and (3.-(3.3 hold. Then we have { 1 b a Tr ( σ a (t, x σa T (t, x X 1 Tr ( σ a (t, y σ a T (t, y Y +b a (t, x (e γ(t t β(x y + ɛx b a (t, y (e γ(t t β(x y ɛy + c a (t, y ū(t, y c a (t, x u(t, x + f a (t, x f a (t, y +Ia 1,δ [t, x, D x ϕ(t, x, y, ϕ(t,, y ] Ī1,δ a [t, y, D y ϕ(t, x, y, ϕ(t, x, ] } +Ia,δ [t, x, D x ϕ(t, x, y, u(t, ] Ī,δ a [t, y, D y ϕ(t, x, y, ū(t, ]. (3.6 Now let us estimate the right hand side of (3.6 term by term. By (H, we have f a (t, x f a (t, y f a (t, x f a (t, x + f a (t, x f a (t, y f a f a L (Q T +C x y. (3.7 Since (H1-(H hold and σ > implies u(t, x > ū(t, y, we have c a (t, y ū(t, y c a (t, x u(t, x = c a (t, y (ū(t, y u(t, x + u(t, x ( c a (t, y c a (t, x C ( c a c a L (Q T + x y. (3.8 8

9 By (H-(H3 and the boundedness of Ω, we have b a (t, x (e γ(t t β(x y + ɛx ba (t, y (e γ(t t β(x y ɛy e γ(t t β(x y ( b a b a L (Q T + C x y + Cɛ e γ(t t β b a b a L (Q T + Ceγ(T t β x y + Cɛ (3.9 and 1 Tr ( σ a (t, x σa T (t, x X 1 Tr ( σ a (t, y σ a T (t, y Y 1 [( Tr σa (t, x σa T (t, x σ a (t, x σ a T ( ] (t, y X σ a (t, y σa T (t, x σ a (t, y σ a T (t, y Y [( σa (t Tr, x σa T (t, x σ a (t, x σ a T ( (t, y σ a (t, y σa T (t, x σ a (t, y σ a T e γ(t t β (t, y ( I I + ɛ I I e γ(t t βtr [ (σ a (t, x σ a (t, y (σ a (t, x σ a (t, y T ] + Cɛ e γ(t t β( σ a σ a L (Q T + C x y + Cɛ ( ] I I e γ(t t β σ σ L (Q T + Ceγ(T t β x y + Cɛ. (3.1 Moreover = = Ia 1,δ [t, x, D x ϕ(t, x, y, ϕ(t,, y ] Ī1,δ a [t, y, D y ϕ(t, x, y, ϕ(t, x, ] +Ia,δ [t, x, D x ϕ(t, x, y, u(t, ] Ī,δ a [t, y, D y ϕ(t, x, y, ū(t, ] [ϕ(t, x + j a (t, x, z, y ϕ(t, x, y D x ϕ(t, x, y j a (t, x, z] µ(dz + + z δ z δ [ϕ(t, x, y + j a (t, y, z ϕ(t, x, y D y ϕ(t, x, y j a (t, y, z] µ(dz [u(t, x + j a (t, x, z u(t, x D x ϕ(t, x, y j a (t, x, z] µ(dz [ū(t, y + j a (t, y, z ū(t, y + D y ϕ(t, x, y j a (t, y, z] µ(dz [ e γ(t t β x + j a (t, x, z y + ɛ ( x + j a (t, x, z + y e γ(t t β x y ɛ ( x + y ] (e γ(t t β(x y + ɛx j a (t, x, z µ(dz [ + e γ(t t β y + j a (t, y, z x + ɛ ( x + y + j a (t, y, z + z δ e γ(t t β y x ɛ ( x + y ] (e γ(t t β(y x + ɛy j a (t, y, z µ(dz [ u(t, x + j a (t, x, z u(t, x ū(t, y + j a (t, y, z + ū(t, y ] (e γ(t t β(x y + ɛx, e γ(t t β(y x + ɛy (j a (t, x, z, j a (t, y, z µ(dz. 9

10 Since ψ(t, x, y = (t,x,y [,T R n ψ(t, x, y, we have Therefore, u(t, x + j a (t, x, z u(t, x ū(t, y + j a (t, y, z + ū(t, y e γ(t t β x + j a (t, x, z y j a (t, y, z e γ(t t β x y + ɛ ( x + j a (t, x, z + y + j a (t, y, z ɛ ( x + y. Ia 1,δ [t, x, D x ϕ(t, x, y, ϕ(t,, y ] Ī1,δ a [t, y, D y ϕ(t, x, y, ϕ(t, x, ] +Ia,δ [t, x, D x ϕ(t, x, y, u(t, ] Ī,δ a [t, y, D y ϕ(t, x, y, ū(t, ] + e γ(t t β + ɛ j a (t, x, z e γ(t t β + ɛ µ(dz + j a (t, y, z µ(dz [ e γ(t t β x + j a (t, x, z y j a (t, y, z e γ(t t β x y z δ ] e γ(t t β(x y (j a (t, x, z j a (t, y, z µ(dz [ ɛ + z δ ( x + j a (t, x, z + y + j a (t, y, z ɛ ( x + y ] ɛx j a (t, x, z ɛy j a (t, y, z µ(dz w β,ɛ,γ (δ + e γ(t t β j a (t, x, z j a (t, y, z µ(dz z δ ɛ + z δ ( j a(t, x, z + j a (t, y, z µ(dz w β,ɛ,γ (δ + e γ(t t β j a (t, x, z j a (t, x, z µ(dz + Ce γ(t t β x y +Cɛ, z δ where lim δ w β,ɛ,γ (δ =. By (3.6-(3.11, we now have where b a βη 1 + η + C x y + Ce γ(t t β x y + Cɛ + w β,ɛ,γ (δ, ( η 1 := e γ(t t σ a σ a L (Q T + b a b a L (Q T + j a (t, x, z j a (t, x, z µ(dz,(t,x Q T (3.11 and Thus, η := C( c a c a L (Q T + f a f a L (Q T. κσ T + γeγ(t t β x y βη 1 + η + C x y + Ce γ(t t β x y + Cɛ + w β,ɛ,γ (δ. 1

11 We set γ = (C + 1 and let δ, κ 1 to obtain σ T βη 1 + η + C x y e γ(t t β x y + Cɛ. (3.1 Maximizing the right hand side of (3.1 with respect to x y yields σ T βη 1 + η + Cβ 1 + Cɛ. (3.13 We now need an estimate for σ. Using (3.4, (3.5 and arguing similarly as above, we obtain σ Combining (3.13 and (3.14 gives us σ + σ (t,x pq T {u(t, x ū(t, x} + + Cβ 1 + η. (3.14 (t,x pq T {u(t, x ū(t, x} + + βt η 1 + T η + Cβ 1 + Cɛ + η. (3.15 Since (3.15 holds for all β >, we minimize the right hand side of (3.15 with respect to β to obtain σ + σ {u(t, x ū(t, x} + + CT 1 1 η 1 + T η + Cɛ + η. (3.16 (t,x pq T To complete the proof, we use the definition of σ to see that u(t, x ū(t, x ɛ x ɛ t σ + σ, for any (t, x [, T R n. (3.17 By (3.16, (3.17 and sending ɛ and ɛ, it thus follows u(t, x ū(t, x {u(t, x ū(t, x} + + CT 1 1 η 1 + T η + η (t,x pq T η + {u(t, x ū(t, x} + (t,x pq T +CT ( ( +CT 1 c a c a L (Q T + f a f a L (Q T σ a σ a L (Q T + b a b a L (Q T +,(t,x Q T ( 1 j a (t, x, z j a (t, x, z µ(dz. Theorem 3.. Assume that (H-(H3 hold. Let u be a bounded viscosity solution of (1.1, where u(t, x u(t, y C 1 x y for any x, y R n (3.18 and u(t, x u(t, y C 1 x y for any t [, T, x Ω c, y R n. (3.19 Then there exists a constant C, depending on L, C 1, n, u L ([,T R n and ρ(z µ(dz such that u(t, x u(t, y C x y for any t [, T, x, y R n. (3. 11

12 Proof. Since u is a viscosity solution of (1.1, ū(t, x := u(t, x + h is a viscosity solution of {1 ū t + inf Tr ( σ a (t, x + hσa T (t, x + hd ū + Ia h [t, x, ū] + b a (t, x + h Dū c a (t, x + hū + f a (t, x + h } =, in Q h T := [, T Ω h, where h is a sufficiently small constant, Ω h := Ω h and Ia h [t, x, ū] := [ū(t, x + j a (t, x + h, z ū(t, x Dū(t, x j a (t, x + h, z] µ(dz. We will apply Theorem 3.1 for u and ū in [, T (Ω Ω h. By (3.18, it is obvious that (3.4 holds. For any x (Ω Ω h c and y R n, we have either x Ω c or x (Ω h c. If x (Ω h c, then we have x + h Ω c. Thus, for any t [, T, u(t, x u(t, y u(t, x u(t, x+h + u(t, x+h u(t, y C 1 h +C 1 x+h y C 1 h +C 1 x y and If x Ω c, then, for any t [, T, and ū(t, x ū(t, y = u(x + h u(y + h C 1 x y. u(t, x u(t, y C 1 x y ū(t, x ū(t, y u(t, x+h u(t, x + u(t, x u(t, y+h C 1 h +C 1 x y h C 1 h +C 1 x y. Therefore (3.5 holds for η := C 1 h. Applying Theorem 3.1 and using (3.18, (3.19, (H, we thus have u(t, x u(t, x + h C 1 h + for some C. (t,x {T } R n ((,T (Ω Ω h c {u(t, x u(t, x + h} + + CT ( c a (, + h c a (, L ([,T (Ω Ω h + f a (, + h f a (, L ([,T (Ω Ω h + CT ( 1 σ a (, + h σ a (, L ([,T (Ω Ω h C h + b a (, + h b a (, L ([,T (Ω Ω h ( + j a (t, x, z j a (t, x + h, z µ(dz 1,(t,x [,T (Ω Ω h Theorem 3.3. Assume that (H-(H3 hold. Let u be a bounded viscosity solution of (1.1 which satisfies (3.18 and (3.19. Assume moreover that u(t, x u(t, x C 1 T t α for any t [, T, x R n (3.1 1

13 and u(t, x u(s, x C 1 t s α for any t, s [, T, x Ω c. (3. Then there exists a constant C, depending on L, C 1, n, u L ([,T R n and ρ(z µ(dz such that u(t, x u(s, y C( t s α + x y for any t, s [, T, x, y R n. (3.3 Proof. By Theorem 3., we have (3.. Since u is a viscosity solution of (1.1, ū(t, x := u(t h, x is a viscosity solution of {1 ū t + inf Tr ( σ a (t h, xσa T (t h, xd ū + Īh a [t, x, ū] + b a (t h, x Dū c a (t h, xū + f a (t h, x } = in [h, T Ω, where h is a sufficiently small constant and Īa h [t, x, ū] := [ū(t, x + j a (t h, x, z ū(t, x Dū(t, x j a (t h, x, z] µ(dz. We will apply Theorem 3.1 for u and ū in [h, T Ω. By (3., estimate (3.4 holds and (3.5 is true with η =. Thus, applying Theorem 3.1 and using (3.1, (3., (H, we obtain u(t, x u(t h, x for some C. 4 Existence (t,x {T } R n ((h,t Ω c {u(t, x u(t h, x} + + CT ( c a ( h, c a (, L ([h,t Ω + f a ( h, f a (, L ([h,t Ω ( + CT 1 σ a ( h, σ a (, L ([h,t Ω + b a ( h, b a (, L ([h,t Ω ( 1 + j a (t h, x, z j a (t, x, z µ(dz,(t,x [h,t Ω C h α For any q = 1,,..., we define the truncated measures µ q ( := µ(b1/q c ( and nonlocal operators I a,q [t, x, u] := [u(t, x + j a (t, x, z u(t, x Du(t, x j a (t, x, z] µ q (dz. We investigate the existence of viscosity solutions of the initial boundary value problems u t + inf { 1 Tr ( σ a (t, xσa T (t, xd u + I a [t, x, u] +b a (t, x Du c a (t, xu + f a (t, x} = in Q T, (4.1 u(t, x = g(t, x on pn Q T 13

14 and (u q t + inf { 1 Tr ( σ a (t, xσa T (t, xd u q + Ia,q [t, x, u q ] +b a (t, x Du q c a (t, xu q + f a (t, x} = in Q T, u q (t, x = g(t, x on pn Q T, where g C 1+α/,+α ([, T ] R n. We are interested in viscosity solutions which satisfy (3.3 uniformly in q. The main issue here is to construct a viscosity sub/ersolution ψ of (4.1 and (4. (which is independent of q such that ψ = g on pn Q T and and (4. ψ(t, x ψ(t, y C x y for any t [, T, x Ω c, y Ω (4.3 ψ(t, x ψ(t, x C T t α for any t [, T, x R n (4.4 for some constant C. We first construct a barrier function assuming that Ω is a bounded domain with smooth boundary Ω. We define d Ω (x := d(x, Ω c. The construction is essentially taken from [17], Theorem II.1. Lemma 4.1. Let Ω be a bounded domain with smooth boundary Ω. Assume that (H, (H1, (H3 hold and λi σ a σ T a in Q T for < λ. Then there exist < δ < 1, κ > and a nonnegative Lipschitz function ψ in R n such that ψ = in Ω c, ψ κ in Ω δ, ψ C (Ω \ Ω δ and, for any a A, (t, x [, T (Ω \ Ω δ, 1 Tr ( σ a (t, xσ T a (t, xd ψ(x + I a,q [t, x, ψ] + b a (t, x Dψ(x c a (t, xψ(x + f a (t, x κ. Proof. Since Ω has a smooth boundary, let δ 1 > be such that d Ω ( C ({x Ω : d Ω (x < δ 1 }. We set β(s := ρ(zµ(dz and define Ψ(s := s Lρ(z s e Ml M l β(τdτ dl s, where M > will be determined later. It is obvious that there exists a constant s(m > such that, for any < s < s(m, Ψ (s 1. We now define { Ψ(dΩ (x, if d ψ(x = Ω (x < δ := 1 min{s(m, δ 1}, (4.5 Ψ(δ, if d Ω (x δ and set δ := δ 4. It is easy to see that ψ = on Ω c, ψ κ in Ω δ, ψ C (Ω \ Ω δ for some κ < 1 and ψ is a Lipschitz function in R n with Lipschitz constant 1. For any x Ω \ Ω δ, we have Dψ(x = Ψ (d Ω (xdd Ω (x, D ψ(x = Ψ (d Ω (xd d Ω (x + Ψ (d Ω (xdd Ω (x Dd Ω (x. Since d Ω ( C ({x Ω : d Ω (x < δ 1 }, D d Ω (x L ({x Ω:d Ω (x δ 1 } C. Thus, for any a A and (t, x [, T (Ω \ Ω δ, we have 1 Tr ( σ a (t, xσ T a (t, xd ψ(x C + Ψ (d Ω (xλ, 14

15 b a (t, x Dψ(x C for some C independent of a and I a,q [t, x, ψ] = [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] µ q (dz +. j a(t,x,z d Ω (x j a(t,x,z >d Ω (x Since Ψ (d Ω ( in Ω \ Ω δ, we have [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] µ q (dz C j a(t,x,z d Ω (x j a(t,x,z d Ω (x j a(t,x,z d Ω (x D ψ(x + sκj a (t, x, zj a (t, x, z j a (t, x, zs dκdsµ q (dz j a (t, x, z µ q (dz C ρ(z µ(dz C. Since ψ is a Lipschitz function in R n, we have [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] µ q (dz C j a(t,x,z >d Ω (x Lρ(z>d Ω (x Lρ(z>d Ω (x ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z µ q (dz ρ(zµ(dz Cβ(d Ω (x. Therefore, for any a A, (t, x [, T (Ω \ Ω δ, we have 1 Tr ( σ a (t, xσ T a (t, xd ψ(x + I a,q [t, x, ψ] + b a (t, x Dψ(x c a (t, xψ(x + f a (t, x Ψ (d Ω (xλ + C(β(d Ω (x + 1 Mλ(β(d Ω (x + 1e Md Ω(x M d Ω (x β(sds + C(β(d Ω (x + 1 Mλ(β(d Ω (x C(β(d Ω (x + 1 (β(d Ω (x if we set M := C+1 λ. We now assume that Ω satisfies the uniform exterior ball condition, i.e., there is a constant r Ω > such that, for any x Ω, there exists y x Ω c satisfying B rω (y x Ω = {x}. Without loss of generality, we can assume that r Ω < 1. Lemma 4.. Let Ω be a bounded domain satisfying the uniform exterior ball condition. Assume that (H-(H3 hold, λi σ a σa T in Q T for < λ and g C 1+α/,+α ([, T ] R n. Then there exists a viscosity ersolution ψ (which is independent of q of (4.1 and (4. such that ψ = g in pn Q T and (4.3-(4.4 hold. 15

16 Proof. We first assume that g in [, T ] R n. We extend our non-local parabolic equation by u t + λ u = on [, T Ω c. By the boundedness of Ω, there exists a sufficiently large constant R such that, for any x Ω, we have Ω B R 1(y x \ B rω (y x. By Lemma 4.1, applied in B R (y x \ B rω (y x, there are δ >, κ > and a non-negative Lipschitz function ψ x in R n with Lipschitz constant 1 such that ψ x = in B R (y x c B rω (y x, ψ x κ in Ω \ B rω +δ (y x, ψ x C ( Ω B rω +δ (y x and, for any (t, y [, T (Ω B rω +δ (y x, (ψ x t (y + inf {1 Tr ( σ a (t, yσ T a (t, yd ψ x (y + I a,q [t, y, ψ x ] +b a (t, y Dψ x (y c a (t, yψ x (y + f a (t, y} κ and the same is true if I a,q [t, y, ψ x ] is replaced above by I a [t, y, ψ x ]. It follows from the construction that the constants δ, κ are independent of x Ω and q. We take a sufficiently large constant C 1 such that C 1 κ T ( f a L (Q T + 1. (4.6 It is obvious that C 1 ψ x is a viscosity ersolution of (4.1 and (4. in [, T (Ω B rω +δ (y x and ( f a L (Q T + 1 (T t is a viscosity ersolution of (4.1 and (4. in Q T. We define ψ x (t, y := min {( f a L (Q T + 1 (T t, C 1 ψ x (y }. Then ψ x (t, x = for any t [, T, ψ x (T, y = for any y R n, ψ x in [, T R n and x Ω { D ψx L ([,T R n + ( ψ } x t L ([,T R n < +. It is easy to see that ψ x is a viscosity ersolution of (4.1 and (4. in Q T. We define ψ(t, y := inf x Ω ψx (t, y. Then ψ is a non-negative viscosity ersolution of (4.1 and (4. in Q T, ψ(t, y = for any (t, y [, T Ω, ψ(t, y = for any y R n and (4.3-(4.4 hold for ψ. Therefore { ψ(t, y, if (t, y QT, ψ(t, y :=, if (t, y pn Q T is a viscosity ersolution of (4.1 and (4. in Q T, ψ = in p Q T and (4.3-(4.4 hold for ψ. We now consider the case where g is an arbitrary C 1+α/,+α ([, T ] R n function. Suppose that u and u q are viscosity solutions of (4.1 and (4. respectively. Define v = u g, v q := u q g. Then v q is a viscosity solution of { (v q t (t, x + inf 1 Tr ( σ a (t, xσa T (t, xd v q (t, x + I a,q [t, x, v q ] +b a (t, x Dv q (t, x c a (t, xv q (t, x + f a (t, x } =, in Q T, (4.7 v q (t, x =, in p Q T, where f a,q (t, x := f a (t, x + g t (t, x + 1 Tr ( σ a (t, xσ T a (t, xd g(t, x + I a,q [t, x, g] +b a (t, x Dg(t, x c a (t, xg(t, x and v is a viscosity solution of (4.7 if I a,q [t, x, v q ] is replaced by I a [t, x, v] above. Since g C 1+α/,+α ([, T ] R n, it follows (by similar computations to these in (5.11 that I a,q [,, g] C α/,α (Q T C(1 + g C 1+α/,+α ([,T ] R n. 16

17 Thus, we have f a,q C α/,α (Q T C(1 + g C 1+α/,+α ([,T ] R n. By the first part of the proof, we know that there is a ersolution ψ, which is independent of q, of (4.7 in Q T, ψ = in pn Q T and (4.3-(4.4 hold for ψ. We define ψ := ψ + g. Then ψ is a viscosity ersolution of (4. such that ψ = g in pn Q T and (4.3-(4.4 hold. Applying the same construction to v we obtain a required viscosity ersolution of (4.1. Theorem 4.1. Let Ω be a bounded domain satisfying the uniform exterior ball condition. Assume that (H-(H3 hold, λi σ a σa T in Q T for < λ and g C 1+α/,+α ([, T ] R n. Then there exist unique viscosity solutions u and u q of (4.1 and (4. respectively which satisfy (3.3 uniformly in q. Moreover u q converge uniformly to u on [, T ] R n. Proof. By Lemma 4., there exists a viscosity ersolution ψ 1 of (4.1 and (4. (for every q such that ψ 1 = g in pn Q T and (4.3-(4.4 hold for ψ 1. Similarly, we can construct a viscosity subsolution ψ of (4.1 and (4. (for every q such that ψ = g in pn Q T and (4.3-(4.4 hold for ψ. Using Perron s method, we can thus construct viscosity solutions u and u q of (4.1 and (4. which are uniformly bounded and satisfy (3.18, (3.19, (3.1 and (3. uniformly in q. The reader can consult [1,, 3, 33] for Perron s method for integro-differential equations. We remark that the uniqueness of viscosity solutions of the initial boundary value problems (4.1 and (4. is standard, see [3, ]. It now follows from Theorem 3.3 that (3.3 holds uniformly in q. The uniform convergence of u q to u is a consequence of (3.3, stability properties of viscosity solutions and the uniqueness of viscosity solutions of ( Schauder regularity In this section we prove a result about C 1+α/,+α regularity of viscosity solutions for a slightly restricted class of (1.1. We first recall the W 1,,p and C 1+α/,+α estimates for uniformly parabolic equations from [14] and [37]. The first result is a combination of Lemma.9, Corollary.1 and Theorem 9.1 of [14] (see also [36] and is adjusted to our purposes. We notice that in Theorem 5.1 below all coefficients are uniformly continuous, uniformly in a A, so the notion of L p -viscosity solution is equivalent to the usual notion of viscosity solution, called C-viscosity solution in [14] (see Lemma.9 of [14], which we use here. Theorem 5.1. Let u be a viscosity solution of {1 u t + inf Tr ( σ a (t, xσa T (t, xd u + b a (t, x Du c a (t, xu + f a (t, x } = in Q T, where Ω is a bounded domain in R n. Suppose that σ a, b a, c a, f a are uniformly continuous in Q T, uniformly in a A, and assume that λi σ a σa T ΛI in Q T for some < λ Λ for all a A, b a L (Q T < +, g := f a L p (Q T for some n + 1 p < +, c a L (Q T < +, c a for all a A. Then u W 1,,p loc (Q T and for any Q Q T, there exists C 1 > such that u W 1,,p (Q C 1 ( u L ( pq T + g L p (Q T, where C 1 depends on n, p, b a L (Q T, c a L (Q T, λ, Λ, diam(ω, T, d p (Q, p Q T and the uniform modulus of continuity of the σ a. 17

18 Theorem 5. below follows from Theorem 1.1 of [37]. Theorem 5.. Let u be a viscosity solution of {1 u t + inf Tr ( σ a (t, xσa T (t, xd u + f a (t, x } = in Q T, where Ω is a bounded domain in R n. Suppose that λi σ a σa T ΛI in Q T for some < λ Λ for all a A. There exists < α < 1 such that if [σ a σa T ] C β/,β (Q T < +, f a C β/,β (Q T < + and < β < α, then u C 1+β/,+β loc (Q T. Moreover, for any Q Q T, there exists C > such that ( u C 1+β/,+β (Q C u L ( pq T + f a C β/,β (Q T, where C depends on n, β, α, [σ a σ T a ] C β/,β (Q T, λ, Λ, diam(ω, T, d p(q, p Q T. Theorem 5.3. Let A be finite and let Ω be a bounded domain satisfying the uniform exterior ball condition. Assume that (H-(H3 hold, λi σ a σ T a in Q T for < λ and g C 1+α/,+α ([, T ] R n. Assume moreover that for any a A, z and (t, x, (s, y Q T, j a (t, x, z j a (s, y, z Cρ(z( t s 1 + x y. (5.1 Let < α < 1 be from Theorem 5. and let α in (H3 satisfy α < α. Let u be the unique viscosity solution of (4.1 (i.e. the one constructed in Theorem 4.1. Then u C 1+α/,+α loc (Q T and for every Q Q T, there exists a constant C > such that u C 1+α/,+α (Q C, (5. where C depends on n, α, α, λ, L, T, ρ, diam(ω, d p (Q, p Q T, g C 1+α/,+α ([,T ] R n number of elements of A. and the Proof. Let u q be the viscosity solution of (4. constructed in Theorem 4.1. We know that u q satisfy (3.3 uniformly in q, q u q L ([,T ] R n C for some constant C and the functions u q converge uniformly to u. Thus it is enough to show that (5. is satisfied for every u q, uniformly in q. We can rewrite (4. in the following form { 1 (u q t + min Tr ( σ a (t, xσa T (t, xd ( u q + b a (t, x j a (t, x, zµ q (dz Du q R } l c a (t, xu q + g a (t, x =, (5.3 where g a (t, x := f a (t, x + R [u l q (t, x + j a (t, x, z u q (t, x] µ q (dz. We want to prove that u q is a viscosity solution of (5.3. Suppose that ϕ C 1, ([, T R n is a bounded function such that u ϕ has a maximum over [, T R n at (t, x Q T. Since u q is a viscosity solution of (4., we have, for any < δ < 1, { 1 ϕ t (t, x + min Tr ( σ a (t, xσa T (t, xd ϕ(t, x ( + b a (t, x j a (t, x, zµ q (dz Dϕ(t, x R } l c a (t, xu q (t, x + ḡ a (t, x, (5.4 18

19 where ḡ a (t, x := f a (t, x + z δ [u q (t, x + j a (t, x, z u q (t, x] µ q (dz + [ϕ(t, x + j a (t, x, z ϕ(t, x] µ q (dz. Since µ q is a finite measure, letting δ in (5.4, we obtain { 1 ϕ t (t, x + min Tr ( σ a (t, xσa T (t, xd ϕ(t, x ( + b a (t, x j a (t, x, zµ q (dz Dϕ(t, x R } l c a (t, xu q (t, x + g a (t, x. Thus u q is a viscosity subsolution of (5.3. The proof of the ersolution property is the same. We notice here that (5.3 is a partial differential equation. For any (t, x Q T, we have j a (t, x, z µ q (dz L ρ(zµ q (dz C(q, R l u q (t, x + j a (t, x, z u q (t, x µ q (dz CL ρ(zµ q (dz < C(q, where C(q > is a constant depending on q. By Theorem 5.1, we thus have u q W 1,,p loc (Q T for every p n+1. If p > n+, the space W 1,,p loc (Q T can be embedded into C 1/+α(p/,1+α(p loc (Q T for α(p := 1 n+ p (see e.g. [3], Lemma 3.3, p. 8. Thus, we choose p sufficiently large such that u q C 1/+α/,1+α loc (Q T. For any (t, x, (s, y Q T, we have j a (t, x, z j a (s, y, z µ q (dz L ρ(zµ q (dz( t s α + x y and C(q( t s α + x y, [u q (t, x + j a (t, x, z u q (t, x u q (s, y + j a (s, y, z + u q (s, y] µ q (dz [ u q (t, x + j a (t, x, z u q (s, y + j a (s, y, z + u q (t, x u q (s, y ] µ q (dz R l C (1 + Lρ(zµ q (dz( t s α + x y C(q( t s α + x y. Therefore, by Therorem 5., u q C 1+α/,+α loc (Q T and is a classical solution of (4. and (5.5. It remains to prove the uniform C 1+α/,+α loc estimate. Let η C ([, T and ψ C (Ω satisfy η, ψ 1, p(η [, T δ, p(ψ Ω δ, [, T δ ] {η = 1} and Ω δ {ψ = 1} for some sufficiently small δ >. Then ηψu q is a classical solution of the equation { 1 (ηψu q t (t, x + min Tr ( σ a (t, xσa T (t, xd (ηψu q (t, x } +b a (t, x D(ηψu q (t, x c a (t, x(ηψu q (t, x + g a (t, x =, (5.5 19

20 where g a (t, x := η t (tψ(xu q (t, x η(t Tr ( σ a (t, xσa T (t, xdψ(x Du q (t, x η(t u q(t, xtr ( σ a (t, xσa T (t, xd ψ(x + I a,q [t, x, ηψu q ] η(t [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] u q (t, x + j a (t, x, zµ q (dz η(t [u q (t, x + j a (t, x, z u q (t, x] Dψ(x j a (t, x, zµ q (dz η(tu q (t, xb a (t, x Dψ(x + η(tψ(xf a (t, x. It is easy to see that the functions g a are continuous in Q T for every q. By (3.3, we have, for any (t, x (, T δ Ω δ, and η(t Tr(σ a(t, xσa T (t, xdψ(x Du q (t, x C R η(t [u q (t, x + j a (t, x, z u q (t, x] Dψ(x j a (t, x, zµ q (dz l C j a (t, x, z µ q (dz C for some absolute constant C independent of q. We recall that lim z ρ(z =. Hence, by (H3, there exists δ 1 > such that j a (,, z L (Q T δ 4 for any z < δ 1. Using Lemma.1 and (3.3, we have, for any (t, x (, T δ Ω δ, 1 1 I a,q [t, x, ηψu q ] D (ηψu q (t, x + κrj a (t, x, zj a (t, x, z j a (t, x, zrdκdrµ q (dz + η(t [(ψu q (t, x + j a (t, x, z (ψu q (t, x D(ψu q (t, x j a (t, x, z] µ q (dz z δ 1 1 D (ηψu q (t, x + κrj a (t, x, zj a (t, x, z j a (t, x, zr dκdrµq (dz + C(δ, where C(δ is a constant depending on δ. By Lemma., we have I a,q [,, ηψu q ] L p ([,T δ ] Ω δ ( T δ ( 1 1 pdxdt 1 D p (ηψu q (t, x + κrj a (t, x, zj a (t, x, z j a (t, x, zr dκdrµ q (dz Ω δ +C(δ +C(δ 1 1 ( T δ Ω δ D (ηψu q (t, x + κrj a (t, x, zj a (t, x, z j a (t, x, zr p dxdt 1 p dκdrµ q (dz

21 C +C(δ. 1 1 ( T δ Ω δ D (ηψu q (t, x + κrj a (t, x, z p dxdt 1 p dκdrρ(z µ q (dz (5.6 There exists < δ < δ 1 such that, if δ < δ, we have κrj a (t, x, z κrj a (t, y, z 1 4 x y for any t [, T, x, y Ω, κ, r [, 1] and z δ. Then there exists a sequence of continuously differentiable functions {f m } + m=1 such that f m κrj a (t,, z uniformly in Ω and x Ω Df m (x 1, where is the -norm for matrices. Therefore x f m (x is one-to-one in Ω, x Ω det(i + Df m (x (1 1 n = 1. Since f n m κrj a (t,, z uniformly in Ω, we have 1 1 = lim m + 1 C ( T δ Ω δ ( T δ ( T δ D (ηψu q (t, x + κrj a (t, x, z p dxdt Ω δ 1 p D (ηψu q (t, x + f m (x p dxdt Ω δ D (ηψu q (t, y p dydt 1 p 1 p dκdrρ(z µ q (dz dκdrρ(z µ q (dz dκdrρ(z µ q (dz (5.7 (recall that p(ψ Ω δ and p(η [, T δ. By (., (5.6 and (5.7, we choose δ 3 < δ sufficiently small such that if δ < δ 3 then I a,q [,, ηψu q ] L p ([,T δ ] Ω δ 1 C 1 K ηψu q W 1,,p ([,T δ ] Ω δ + C, where C 1 is given by Theorem 5.1 and K is the number of elements of A. Using Theorem 5.1, we now have ηψu q W 1,,p ([,T δ ] Ω δ C 1 (C + g a L p ([,T δ ] Ω δ C 1 (C + K C 1 K ηψu q W 1,,p ([,T δ ] Ω δ, which implies ηψu q W 1,,p ([,T δ ] Ω δ C for some constant C independent of q. By the definition of η and ψ, and the embedding theorem, it then follows u q C 1/+α/,1+α ([,T δ ] Ω δ C ηψu q W 1,,p ([,T δ ] Ω δ C. (5.8 We can now use a bootstrap argument to improve the regularity to C 1+α/,+α. We now let η C ([, T and ψ C (Ω satisfy η, ψ 1, p(η [, T 5δ, p(ψ Ω 5δ, [, T 6δ ] {η = 1} and Ω 6δ {ψ = 1}. By (3.3, (5.1, (5.8, if δ < δ, we have, for any 1

22 (t, x, (s, y [, T 4δ ] Ω 4δ, η(t [u q (t, x + j a (t, x, z u q (t, x] Dψ(x j a (t, x, zµ q (dz R l η(s [u q (s, y + j a (s, y, z u q (s, y] Dψ(y j a (s, y, zµ q (dz R l [ 1 η(tdu q (t, x + κj a (t, x, z j a (t, x, zdκdψ(x j a (t, x, z ] η(sdu q (s, y + κj a (s, y, z j a (s, y, zdκdψ(y j a (s, y, z µ q (dz [ + η(t [u q (t, x + j a (t, x, z u q (t, x] Dψ(x j a (t, x, z z δ ] η(s [u q (s, y + j a (s, y, z u q (s, y] Dψ(y j a (s, y, z µ q (dz C(1 + u q C 1/+α/,1+α ([,T δ ] Ω δ ( t s α + x y α + C( t s α + x y C( t s α + x y α. (5.9 By Lemma.1 and (3.3, we have η(t [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] u q (t, x + j a (t, x, zµ q (dz R l η(s [ψ(y + j a (s, y, z ψ(y Dψ(y j a (s, y, z] u q (s, y + j a (s, y, zµ q (dz R l 1 1 [ η(t [ D ψ(x + κrj a (t, x, z j a (t, x, zj a (t, x, zr ] u q (t, x + j a (t, x, z z <1 η(s [ D ψ(y + κrj a (s, y, zj a (s, y, z j a (s, y, zr ] ] u q (s, y + j a (s, y, z dκdrµ q (dz [ + η(t [ψ(x + j a (t, x, z ψ(x Dψ(x j a (t, x, z] u q (t, x + j a (t, x, z z 1 ] η(s [ψ(y + j a (s, y, z ψ(y Dψ(y j a (s, y, z] u q (s, y + j a (s, y, z µ q (dz C( t s α + x y α. (5.1 Using (3.3, (5.1, u q C 1/+α/,1+α ([,T δ ] Ω δ C and Corollary.1, we can choose δ < δ sufficiently small such that I a,q [t, x, ηψu q ] I a,q [s, y, ηψu q ] [ (ηψu q (t, x + j a (t, x, z (ηψu q (t, x D(ηψu q (t, x j a (t, x, z ] (ηψu q (s, y + j a (s, y, z (ηψu q (s, y D(ηψu q (s, y j a (s, y, z µ q (dz 1 1 [ D (ηψu q (t, x + κrj a (t, x, zj a (t, x, zj a (t, x, zr ] D (ηψu q (s, y + κrj a (s, y, zj a (s, y, zj a (s, y, zr dκdrµ q (dz [ + (ηψu q (t, x + j a (t, x, z (ηψu q (t, x D(ηψu q (t, x j a (t, x, z z δ

23 ] (ηψu q (s, y + j a (s, y, z (ηψu q (s, y D(ηψu q (s, y j a (s, y, z µ q (dz 1 C ηψu q C 1+α/,+α ([,T δ ] Ω δ ( t s α + x y α + C( t s α + x y α, (5.11 where C is the constant from Theorem 5.. Therefore, by Theorem 5., (5.8-(5.11 and the properties of the functions η, ψ, ηψu q C 1+α/,+α ([,T 5δ ] Ω 5δ C (C + b a D(ηψu q c a ηψu q + g a C α/,α ([,T 4δ ] Ω 4δ C (C + 1 ηψu q C C 1+α/,+α ([,T δ ] Ω δ It thus follows C + 1 ηψu q C 1+α/,+α ([,T 5δ ] Ω 5δ. u q C 1+α/,+α ([,T 6δ ] Ω 6δ ηψu q C 1+α/,+α ([,T 5δ ] Ω 5δ C. This gives the required uniform estimate which will also be satisfied by u. References [1] O. Alvarez and A. Tourin, Viscosity solutions of nonlinear integro-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996, no.3, [] G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J. 57 (8, no. 1, [3] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (8, no. 3, [4] L. A. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 6 (9, no. 5, [5] L. A. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. (11, no. 1, [6] L. A. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math. ( 174 (11, no., [7] H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal parabolic equations, Calc. Var. Partial Differential Equations 49 (14, no. 1, [8] H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal parabolic equations II, J. Differential Equations 56 (14, no. 1, [9] H. Chang Lara and G. Dávila, Hölder estimates for non-local parabolic equations with critical drift, J. Differential Equations 6 (16, no. 5, [1] H. Chang Lara and G. Dávila, C σ+α estimates for concave, nonlocal parabolic equations with critical drift, to appear in J. Integral Equations Applications. 3

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