Perron s method for nonlocal fully nonlinear equations

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1 Peon s method fo nonlocal fully nonlinea equations Chenchen Mou Depatment of Mathematics, UCLA Los Angeles, CA 90095, U.S.A. muchenchen@math.ucla.edu Abstact This pape is concened with existence of viscosity solutions of non-tanslation invaiant nonlocal fully nonlinea equations. We constuct a discontinuous viscosity solution of such nonlocal equation by Peon s method. If the equation is unifomly elliptic, we pove the discontinuous viscosity solution is Hölde continuous and thus it is a viscosity solution. Keywods: viscosity solution; intego-pde; Hamilton-Jacobi-Bellman-Isaacs equation; Peon s method; weak Hanack inequality. 010 Mathematics Subject Classification: 35D40, 35J60, 35R09, 45K05, 47G0, 49N70. 1 Intoduction In this pape, we investigate existence of a viscosity solution of { I(x, u(x), u( )) = 0, in Ω, u = g, in Ω c, (1.1) whee Ω is a bounded domain in R n, I is a non-tanslation invaiant nonlocal opeato and g is a bounded continuous function in R n. An impotant example of (1.1) is the Diichlet poblem fo nonlocal Bellman-Isaacs equations, i.e., { supa A inf b B { I ab [x, u] + b ab (x) u(x) + c ab (x)u(x) + f ab (x)} = 0, in Ω, u = g, in Ω c (1.), whee A, B ae two index sets, b ab : R n R n, c ab : R n R +, f ab : R n R ae unifomly continuous functions and I ab is a Lévy opeato. If the Lévy measues ae symmetic and absolutely continuous with espect to the Lebesgue measue, then they can be epesented as I ab [x, u] := [u(x + z) u(x)]k ab (x, z)dz, (1.3) R n whee {K ab (x, ); x Ω, a A, b B} ae kenels of Lévy measues satisfying R n min{ z, 1}K ab (x, z)dz < + fo all x Ω. (1.4) 1

2 In fact, we will not assume ou Lévy measues to be symmetic in the following sections. Existence of viscosity solutions has been well established fo the Diichlet poblem fo integodiffeential equations by Peon s method when the equations satisfy the compaison pinciple. In [4], G. Bales and C. Imbet studied the compaison pinciple fo degeneate second ode intego-diffeential equations assuming the nonlocal opeatos ae of Lévy-Itô type and the equations satisfy the coecive assumption. Then G. Bales, E. Chasseigne and C. Imbet obtained existence of viscosity solutions fo such intego-diffeential equations by Peon s method in [3]. L. A. Caffaelli and L. Silveste poved, in Section 5 of [8], the compaison pinciple fo unifomly elliptic tanslation invaiant intego-diffeential equations whee the nonlocal opeatos ae of Lévy type. Then existence of viscosity solutions follows, if suitable baies can be constucted, by Peon s method. Late H. Chang Laa and G. Davila extended the compaison and existence esults of [8] to paabolic equations, see Section 3 in [11, 13]. The existence fo (1.1) when I is a non-tanslation invaiant nonlocal opeato is much moe difficult to tackle since we do not have a good compaison pinciple, see [33]. In [33], the authos poved compaison assuming that eithe a viscosity subsolution o a supesolution is moe egula. To ou knowledge, the only available esults fo existence of solutions fo non-tanslation invaiant equations ae the following. D. Kiventsov studied, in Section 5 of [31], existence of viscosity solutions of some unifomly elliptic nonlocal equations. In Section 4 of [38], J. Sea poved existence of viscosity solutions of unifomly elliptic nonlocal Bellman equations. H. Chang Laa and D. Kiventsov extended existence esults in [31] to a class of unifomly paabolic nonlocal equations, see Section 5 of [15]. In all these poofs, the authos used fixed point aguments. In [1], O. Alvaez and A. Touin obtained existence of viscosity solutions of degeneate paabolic nonlocal equations by Peon s method with a estictive assumption that the Lévy measues ae bounded. The boundedness of Lévy measues allows them to obtain the compaison pinciple. The eade can consult [16, 0, 1, 9] fo Peon s method fo viscosity solutions of fully nonlinea patial diffeential equations. The pobability liteatue on existence of viscosity solutions of nonlocal Bellman-Isaacs equations is enomous. It is well-known that Bellman-Isaacs equations aise when people study the diffeential games, whee the equations cay infomation about the value and stategies of the games. The pobabilists epesent viscosity solutions of nonlocal Bellman-Isaacs equations as value functions of cetain stochastic diffeential games with jump diffusion via the dynamic pogamming pinciple. Howeve, mostly in the pobability liteatue, the nonlocal tems of nonlocal Bellman-Isaacs equations ae of Lévy-Itô type and Ω is the whole space R n. We efe the eade to [, 5, 6, 7, 3, 8, 30, 34, 35, 43, 44, 45] fo stochastic epesentation fomulas fo viscosity solutions of nonlocal Bellman-Isaacs equations. In Section 3, we adapt to the nonlocal case the appoach fom [0, 1, 9] fo obtaining existence of a discontinuous viscosity solution u of (1.1) without using the compaison pinciple. Fo applying Peon s method, we need to assume that thee exist a continuous viscosity subsolution and a continuous supesolution of (1.1) and both satisfy the bounday condition. Since (1.1) involves the nonlocal tem, the poof of the existence is moe delicate than the PDE case. In Section 4, we obtain a Hölde estimate fo the discontinuous viscosity solution of (1.1) constucted by Peon s method assuming the equation is unifomly elliptic. In most of the liteatue, the nonlocal opeato I is assumed to be unifomly elliptic with espect to a class of linea nonlocal opeatos of fom (1.3) with kenels K satisfying λ Λ ( σ) K(x, z) ( σ), (1.5) z n+σ z n+σ

3 whee 0 < λ Λ. Vaious of egulaity esults wee obtained in ecent yea unde the above unifom ellipticity such as [8, 9, 10, 11, 1, 13, 14, 15, 17, 5, 6, 31, 38, 39, 40, 4] fo both elliptic and paabolic intego-diffeential equations. In this pape, we follow [37] to assume a much weake unifom ellipticity. Roughly speaking, we let I be unifomly elliptic with espect to a lage class of linea nonlocal opeatos whee the kenels K satisfy the ight hand side of (1.5) in an integal sense and the left hand side of that in a symmetic subset of each annulus domain with positive measue. The main tool we use is the weak Hanack inequality obtained in [37]. With the weak Hanack inequality, we ae able to pove the oscillation between the uppe and lowe semicontinuous envelope of the discontinuous viscosity solution u in the ball B is of ode α fo some α > 0 and any small > 0. This poves that u is Hölde continuous and thus it is a viscosity solution of (1.1). Recently, L. Silveste applied the egulaity fo nonlocal equations unde this weak ellipticity to obtain the egulaity fo the homogeneous Boltzmann equation without cut-off, see [41]. We also want to mention that M. Kassmann, M. Rang and R. Schwab studied Hölde egulaity fo a class of intego-diffeential opeatos with kenels which ae positive along some given ays o cone-like sets, see [7]. To complete the existence esults, we constuct continuous sub/supesolutions in both unifomly elliptic and degeneate cases in Section 5. In the unifomly elliptic case, we follow the idea of [36] to constuct appopiate baie functions. We then use them to constuct a subsolution and a supesolution which satisfy the bounday condition. The weak unifom ellipticity and the lowe ode tems of I make the poofs moe involved. With all these ingedients in hand, we can conclude one of the main esults in this manuscipt that (1.1) admits a viscosity solution if I is unifomly elliptic, see Theoem 5.6 in Section 5.1. This main esult genealizes nealy all the pevious existence esults fo unifomly elliptic intego-diffeential equations. In the degeneate case, it is natual to constuct a sub/supesolution only fo (1.) since we have little infomation about the nonlocal opeato I. Moeove, we need to assume the nonlocal Bellman-Isaacs equation in (1.) satisfies the coecive assumption, i.e., c ab γ fo some γ > 0. The coecive assumption is often made to study uniqueness, existence and egulaity of viscosity solutions of degeneate elliptic PDEs and intego-pdes, see [3, 4, 16, 0, 1,, 4, 3, 33]. In Section 5., we obtain a subsolution and a supesolution which satisfy the bounday condition in the degeneate case. The difficulty hee lies in giving a degeneate assumption on the kenels which allows us to constuct baie functions. Roughly speaking, we only need to assume that the kenels K ab (x, ) ae non-degeneate in the oute-pointing nomal diection of the bounday fo the points x which ae sufficiently close to the bounday. That means we allow ou kenels K ab to be degeneate in the whole domain. Then we can conclude the second main esult, the existence of a discontinuous viscosity solution of (1.), given in Theoem If the compaison pinciple holds fo (1.), we obtain the discontinuous viscosity solution is a viscosity solution. In the end, we want to notice that ou method could be adapted to the nonlocal paabolic equations fo obtaining the coesponding existence esults. Notation and definitions We wite B δ fo the open ball centeed at the oigin with adius δ > 0 and B δ (x) := B δ + x. We set Ω δ := {x Ω; dist(x, Ω) > δ} fo δ > 0. Fo each non-negative intege and 0 < α 1, we denote by C,α (Ω) (C,α ( Ω)) the subspace of C,0 (Ω) (C,0 ( Ω)) consisting functions whose th patial deivatives ae locally (unifomly) α-hölde continuous in Ω. Fo any u C,α ( Ω), 3

4 whee is a non-negative intege and 0 α 1, define { supx Ω, j = j u(x), if α = 0; [u],α;ω := sup j u(x) j u(y) x,y Ω,x y, j = x y, if α > 0, α and u C,α ( Ω) = { j=0 [u] j,0,ω, if α = 0; u C,0 ( Ω) + [u],α;ω, if α > 0. Fo simplicity, we use the notation C β (Ω) (C β ( Ω)), whee β > 0, to denote the space C,α (Ω) (C,α ( Ω)), whee is the lagest intege smalle than β and α = β. The set C β b (Ω) consist of functions fom C β (Ω) which ae bounded. We wite USC(R n ) (LSC(R n )) fo the space of uppe (lowe) semicontinuous function in R n. We will give a definition of viscosity solutions of (1.1). We fist state the geneal assumptions on the nonlocal opeato I in (1.1). Fo any δ > 0,, s R, x, x k Ω, ϕ, ϕ k, ψ C (B δ (x)) L (R n ), we assume: (A0) The function (x, ) I(x,, ϕ( )) is continuous in B δ (x) R. (A1) If x k x in Ω, ϕ k ϕ a.e. in R n, ϕ k ϕ in C (B δ (x)) and {ϕ k } k is unifomly bounded in R n, then I(x k,, ϕ k ( )) I(x,, ϕ( )). (A) If s, then I(x,, ϕ( )) I(x, s, ϕ( )). (A3) Fo any constant C, I(x,, ϕ( ) + C) = I(x,, ϕ( )). (A4) If ϕ touches ψ fom above at x, then I(x,, ϕ( )) I(x,, ψ( )). Remak.1. If I is unifomly elliptic and satisfies (A0), (A), then (A0)-(A4) hold fo I. See Lemma 4.. Remak.. The nonlocal opeato I in [37] has only two components, i.e., (x, ϕ) I(x, ϕ( )). Hee we let ou nonlocal opeato I have thee components and assume (A)-(A3) hold. It is because that we want to let I include the left hand side of the nonlocal Bellman-Isaacs equation in (1.) and, moeove, want to descibe the following two popeties I ab [x, ϕ + C] + b ab (x) (ϕ + C)(x) = I ab [x, ϕ] + b ab (x) ϕ(x), in abstact foms. c ab (x) c ab (x)s if s Remak.3. The left hand side of the nonlocal Bellman-Isaacs equation in (1.) satisfies (A0)- (A4) if (1.4) holds and its coefficients K ab, b ab, c ab and f ab ae unifomly continuous with espect to x in Ω, unifomly in a A, b B. See [19] fo when the nonlocal opeato I has a min-max stuctue. Thoughout the pape, we always assume the nonlocal opeato I satisfies (A0)-(A4). 4

5 Definition.4. A bounded function u USC(R n ) is a viscosity subsolution of I = 0 in Ω if wheneve u ϕ has a maximum ove R n at x Ω fo ϕ C b (Rn ), then I (x, u(x), ϕ( )) 0. A bounded function u LSC(R n ) is a viscosity supesolution of I = 0 in Ω if wheneve u ϕ has a minimum ove R n at x Ω fo ϕ C b (Rn ), then I (x, u(x), ϕ( )) 0. A bounded function u is a viscosity solution of I = 0 in Ω if it is both a viscosity subsolution and viscosity supesolution of I = 0 in Ω. Remak.5. In Definition.4, all the maximums and minimums can be eplaced by stict maximums and minimums. Definition.6. A bounded function u is a viscosity subsolution of (1.1) if u is a viscosity subsolution of I = 0 in Ω and u g in Ω c. A bounded function u is a viscosity supesolution of (1.1) if u is a viscosity supesolution of I = 0 in Ω and u g in Ω c. A bounded function u is a viscosity solution of (1.1) if u is a viscosity subsolution and supesolution of (1.1). We will use the following notations: if u is a function on Ω, then, fo any x Ω, u (x) = lim 0 sup{u(y); y Ω and y x }, u (x) = lim 0 inf{u(y); y Ω and y x }. One calls u the uppe semicontinuous envelope of u and u the lowe semicontinuous envelope of u. We then give a definition of discontinuous viscosity solutions of (1.1). Definition.7. A bounded function u is a discontinuous viscosity subsolution of (1.1) if u is a viscosity subsolution of (1.1). A bounded function u is a discontinuous viscosity supesolution of (1.1) if u is a viscosity supesolution of (1.1). A function u is a discontinuous viscosity solution of (1.1) if it is both a discontinuous viscosity subsolution and a discontinuous viscosity supesolution of (1.1). Remak.8. If u is a discontinuous viscosity solution of (1.1) and u is continuous in R n, then u is a viscosity solution of (1.1). 3 Peon s method In this section, we obtain existence of a discontinuous viscosity solution of (1.1) by Peon s method. We emind you that I satisfies (A0)-(A4). Lemma 3.1. Let F be a family of viscosity subsolutions of I = 0 in Ω. Let w(x) = sup{u(x) : u F} in R n and assume that w (x) < fo all x R n. Then w is a discontinuous viscosity subsolution of I = 0 in Ω. 5

6 Poof. Suppose that ϕ is a C b (Rn ) function such that w ϕ has a stict maximum (equal 0) at x 0 Ω ove R n. We can constuct a unifomly bounded sequence of C (R n ) functions {ϕ m } m such that ϕ m = ϕ in B 1 (x 0 ), ϕ ϕ m in R n, sup x B c (x 0 ){w (x) ϕ m (x)} 1 m and ϕ m ϕ pointwise. Thus, fo any positive intege m, w ϕ m has a stict maximum (equal 0) at x 0 ove R n. Theefoe, sup x B c 1 (x 0 ){w (x) ϕ m (x)} = ɛ m < 0. By the definition of w, we have, fo any u F, sup x B c 1 (x 0 ){u(x) ϕ m (x)} ɛ m < 0. Again, by the definition of w, we have, fo any ɛ m < ɛ < 0, thee exist u ɛ F and x ɛ B 1 (x 0 ) such that u ɛ ( x ɛ ) ϕ( x ɛ ) > ɛ. Since u ɛ USC(R n ) and ϕ m C b (Rn ), thee exists x ɛ B 1 (x 0 ) such that u ɛ (x ɛ ) ϕ m (x ɛ ) = sup x R n{u ɛ (x) ϕ(x)} u ɛ ( x ɛ ) ϕ m ( x ɛ ) > ɛ. Since w ϕ m attains a stict maximum (equal 0) at x 0 ove R n and u w fo any u F, then u ɛ (x ɛ ) w (x 0 ) and x ɛ x 0 as ɛ 0. Since u ɛ is a viscosity subsolution of I = 0 in Ω, we have I(x ɛ, u ɛ (x ɛ ), ϕ m ( )) 0. (3.1) Since x ɛ x 0, u ɛ (x ɛ ) w (x 0 ) as ɛ 0, ϕ m = ϕ in B 1 (x 0 ), ϕ m ϕ pointwise, {ϕ m } m is unifomly bounded, ϕ C b (Rn ), (A0) and (A1) hold, we have, letting ɛ 0 and m + in (3.1), I(x 0, w (x 0 ), ϕ( )) 0. Theefoe, w is a discontinuous viscosity subsolution of I = 0. Theoem 3.. Let u, ū be bounded continuous functions and be espectively a viscosity subsolution and a viscosity supesolution of I = 0 in Ω. Assume moeove that ū = u = g in Ω c fo some bounded continuous function g and u ū in R n. Then w(x) = sup u(x), u F whee F = {u C 0 (R n ); u u ū in R n and u is a viscosity subsolution of I = 0 in Ω}, is a discontinuous viscosity solution of (1.1). Poof. Since u F, then F. Thus, w is well defined, u w ū in R n and w = ū = u in Ω c. By Lemma 3.1, w is a discontinuous viscosity subsolution of G = 0 in Ω. We claim that w is a discontinuous viscosity supesolution of G = 0 in Ω. If not, thee exist a point x 0 Ω and a function ϕ C b (Rn ) such that w ϕ has a stict minimum (equal 0) at the point x 0 ove R n and I(x 0, w (x 0 ), ϕ( )) < ɛ 0, whee ɛ 0 is a positive constant. Thus, we can find sufficiently small constants ɛ 1 > 0 and δ 0 > 0 such that B δ0 (x 0 ) Ω and thee exists a C b (Rn ) function ϕ ɛ1 satisfying that ϕ ɛ1 = ϕ in B δ0 (x 0 ), ϕ ɛ1 ϕ in R n, inf x B c δ0 (x 0 ){w (x) ϕ ɛ1 (x)} ɛ 1 > 0 and I(x 0, ϕ ɛ1 (x 0 ), ϕ ɛ1 ( )) < ɛ 0. (3.) Thus, by (A0), thee exists δ 1 < δ 0 such that, fo any x B δ1 (x 0 ), I(x, ϕ ɛ1 (x), ϕ ɛ1 ( )) < ɛ 0 4. (3.3) By the definition of w, we have ϕ ɛ1 w ū in R n. If ϕ ɛ1 (x 0 ) = w (x 0 ) = ū(x 0 ), then ū ϕ ɛ1 has a stict minimum at point x 0 ove R n. Since ū is a viscosity supesolution of I = 0 in Ω, we have I(x 0, ϕ ɛ1 (x 0 ), ϕ ɛ1 ( )) 0, 6

7 which contadicts with (3.). Thus, we have ϕ ɛ1 (x 0 ) < ū(x 0 ). Since ū and ϕ ɛ1 ae continuous functions in R n, we have ϕ ɛ1 (x) < ū(x) ɛ in B δ (x 0 ) fo some 0 < δ < δ 1 and ɛ > 0. We define = sup x B c(x 0) {ϕ ɛ1 (x) w (x)}. Since inf x B c δ0 (x 0 ){w (x) ϕ ɛ1 (x)} ɛ 1 > 0, w ϕ ɛ1 has a stict minimum (equal 0) at the point x 0 and w USC(R n ), we have < 0 fo each > 0. Fo any y Ω \ B (x 0 ), thee exists a function v y F such that v y (y) ϕ ɛ1 (y) 3 4. Since v y and ϕ ɛ1 ae continuous in R n, thee exists a positive constant δ y such that inf x Bδy (y){v y (x) ϕ ɛ1 (x)}. Since Ω \ B (x 0 ) is a compact set in R n, thee exists a finite set {y i } n i=1 Ω \ B (x 0 ) such that Ω \ B (x 0 ) n i=1 B δ yi (y i ). Thus, we define v (x) = sup {v yi (x)}, x R n. 1 i n By Lemma 3.1 and the definition of v, we have v F and inf x Ω\B(x 0 ) {v (x) ϕ ɛ1 (x)}. Let α be a constant such that 0 < α < 1 and α < ɛ. Thus, we define { max{ϕɛ1 (x) α U(x) =, v (x)}, x B (x 0 ), v (x), x B(x c 0 ), whee 0 < < δ and 0 < α < α. By the definition of U, we obtain U C 0 (R n ), u U ū in R n, and thee exists a squence {x n } n B (x 0 ) such that x n x 0 as n + and U(x n ) > w(x n ). We claim that U is a viscosity subsolution of I = 0 in Ω. Fo any y Ω, suppose that thee is a function ψ C b (Rn ) such that U ψ has a maximum (equal 0) at y ove R n. We then divide the poof into two cases. Case 1: U(y) = v (y). Since v U ψ in R n, then v ψ has a maximum (equal 0) at y ove R n. We ecall that v is a viscosity subsolution of I = 0 in Ω. Theefoe, we have I(y, U(y), ψ( )) 0. Case : U(y) = ϕ ɛ1 (y) α. We fist notice that y B (x 0 ). Since ϕ ɛ1 α U ψ in B (x 0 ), then ϕ ɛ1 α ψ 0 in B (x 0 ). By the definition of U, we have ψ U = v in B(x c 0 ). Thus, ϕ ɛ1 α ψ ϕ ɛ1 α v α 0 in B(x c 0 ). Theefoe, we have ϕ ɛ1 α ψ has a maximum (equal 0) at y B (x 0 ) B δ1 (x 0 ) ove R n. Since (3.3), (A0), (A3)-(A4) hold, we can choose sufficiently small α independent of ψ such that I(y, ψ(y), ψ( )) I(y, ϕ ɛ1 (y) α, ϕ ɛ1 ( )) 0. Based on the two cases, we have that U is a viscosity subsolution of I = 0 in Ω. Theefoe, U F, which contadicts with the definition of w. Thus, w is a discontinuous viscosity supesolution of I = 0 in Ω. Theefoe, w is a discontinuous viscosity solution of I = 0 in Ω. Since w = g in Ω c, then w is a discontinuous viscosity solution of (1.1). Remak 3.3. Unde the assumptions of Theoem 3., if the compaison pinciple holds fo (1.1), the discontinuous viscosity solution w is the unique viscosity solution of (1.1). Fo example, if I is a tanslation invaiant nonlocal opeato, (1.1) admits a unique viscosity solution. 7

8 Befoe applying Theoem 3. to (1.), we now give the pecise assumptions on its equation. Fo any 0 < λ Λ and 0 < σ <, we conside the family of kenels K : R n R satisfying the following assumptions. (H0) K(z) 0 fo any z R n. (H1) Fo any δ > 0, B δ \B δ K(z)dz ( σ)λδ σ. (H) Fo any δ > 0, zk(z)dz B δ \B δ Λ 1 σ δ1 σ. We define ou nonlocal opeato I ab [x, u] := δ z u(x)k ab (x, z)dz, R n (3.4) whee u(x + z) u(x), if σ < 1, δ z u(x) := u(x + z) u(x) 1 B1 (z) u(x) z, if σ = 1, u(x + z) u(x) u(x) z, if σ > 1. We conside the following nonlocal Bellman-Isaacs equation sup inf { I ab[x, u] + b ab (x) u(x) + c ab (x)u(x) + f ab (x)} = 0, in Ω. (3.5) b B a A Coollay 3.4. Assume that 0 < σ <, b ab 0 in Ω if σ < 1 and c ab 0 in Ω. Let u, ū be bounded continuous functions and be espectively a viscosity subsolution and a viscosity supesolution of (3.5) whee {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b and {f ab } a,b ae sets of unifomly continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ) : x Ω, a A, b B} ae kenels satisfying (H0)-(H). Assume moeove that ū = u = g in Ω c fo some bounded continuous function g and u ū in R n. Then w(x) = sup u(x), u F whee F = {u C 0 (R n ); u u ū in R n and u is a viscosity subsolution of (3.5)}, is a discontinuous viscosity solution of (1.). Poof. We define I(x,, u( )) := sup inf { I ab[x, u] + b ab (x) u(x) + c ab (x) + f ab (x)}. b B a A It follows fom (H1) and (H) that I ab satisfies (1.4), see Lemma.3 in [37]. Then, by (1.4) and unifom continuity of the coefficients, (A0) and (A1) hold. Since c ab 0 in Ω, (A) holds. By (H0) and the stuctue of I ab, (A3) and (A4) hold. 8

9 4 Hölde estimates In this section we give Hölde estimates of the discontinuous viscosity solution constucted by Peon s method in the above section. To obtain Hölde estimates, we will assume that the nonlocal opeato I is unifomly elliptic. We define L := L(σ, λ, Λ) is the class of all the nonlocal opeatos of fom Lu(x) := δ z u(x)k(z)dz, R n whee K is a kenel satisfying the assumptions (H0)-(H) given above and (H3) Thee exist positive constants λ and µ such that, fo any δ > 0, thee is a set A δ satisfying (i) A δ B δ \ B δ ; (ii) A δ = A δ ; (iii) A δ µ B δ \ B δ ; (iv) K(z) ( σ)λδ n σ fo any z A δ. We note that we will also wite K L if the coesponding nonlocal opeato L L. We then define the extemal opeatos M + L u(x) := sup Lu(x), L L M L u(x) := inf Lu(x). L L We denote by m : [0, + ) [0, + ) a modulus of continuity. We say that the nonlocal opeato I is unifomly elliptic if fo evey, s R, x Ω, δ > 0, ϕ, ψ C (B δ (x)) L (R n ), M L (ϕ ψ)(x) C 0 (ψ ϕ)(x) m( s ) I(x,, ψ( )) I(x, s, ϕ( )) M + L (ϕ ψ)(x) + C 0 (ψ ϕ)(x) + m( s ), whee C 0 is a non-negative constant such that C 0 = 0 if σ < 1. Remak 4.1. The definition of unifom ellipticity is diffeent fom that in [37] since the nonlocal opeato I contains the second component. Lemma 4.. If the nonlocal opeato I is unifomly elliptic and satisfies (A0), (A), then I satisfies (A0)-(A4). Poof. Suppose that δ > 0, x k x in Ω, ϕ k ϕ a.e. in R n, ϕ k ϕ in C (B δ (x)) and {ϕ k } k is unifomly bounded in R n. Since I is unifomly elliptic, we have, fo any R, M L (ϕ ϕ k)(x k ) C 0 (ϕ k ϕ)(x k ) I(x k,, ϕ k ( )) I(x k,, ϕ( )) M + L (ϕ ϕ k)(x k ) + C 0 (ϕ k ϕ)(x k ). (4.1) Since K L, then, by Lemma.3 in [37], K satisfies (1.4). Letting k + in (4.1), we have, by (A0), lim k + I(x k,, ϕ k ( )) = I(x,, ϕ( )). 9

10 Theefoe, (A1) holds. Fo any constant C, we have 0 = M L ( C) C 0 C I(x,, ϕ( ) + C) I(x,, ϕ( )) M + L ( C) + C 0 C = 0. Thus, (A3) holds. If ϕ touches a C (B δ (x)) L (R n ) function ψ fom above at x, then Theefoe, (A4) holds. I(x,, ϕ) I(x,, ψ) M + L (ψ ϕ)(x) 0. The following lemma is an elliptic vesion of Theoem 6.1 in [37]. Lemma 4.3. Assume 0 < σ 0 σ <, C 0, C 1 0, and futhe assume C 0 = 0 if σ < 1. Let u be a viscosity supesolution of M L u C 0 u = C 1 in B and u 0 in R n. Then thee exist constants C and ɛ 3 such that ( B 1 u ɛ 3 dx ) 1 ɛ 3 whee ɛ 3 and C depend on σ 0, λ, Λ, C 0, n and µ. C(inf B 1 u + C 1 ), The following Lemma is a diect Coollay of Lemma 4.. Coollay 4.4. Assume 0 < σ 0 σ <, 0 < < 1, C 0, C 1 0, and futhe assume C 0 = 0 if σ < 1. Let u be a viscosity supesolution of M L u C 0 u = C 1 in B and u 0 in R n. Then thee exist constants C and ɛ 3 such that ( {u > t} B ) C n (u(0) + C 1 σ ) ɛ 3 t ɛ 3, fo any t 0, (4.) whee ɛ 3 and C depend on σ 0, λ, Λ, C 0, n and µ. Poof. Now let v(x) = u(x). By Lemma. in [37], we have M L v C 0 σ 1 v C 1 σ, in B. (4.3) Now we apply Lemma 4.3 to (4.3). Thus, fo any t 0, we have Then t {v > t} B 1 1 ɛ 3 ( v ɛ 3 dx B 1 ) 1 ɛ 3 C(inf B 1 v + C 1 σ ) C(v(0) + C 1 σ ). n {u > t} B {v > t} B 1 C(v(0) + C 1 σ ) ɛ 3 t ɛ 3 = C(u(0) + C 1 σ ) ɛ 3 t ɛ 3. Theefoe, (4.) holds. Then we follow the idea in [8] to obtain a Hölde estimate. 10

11 Theoem 4.5. Assume 0 < σ 0 σ <, C 0 0, and futhe assume C 0 = 0 if σ < 1. Fo any ɛ > 0, let F be a class of bounded continuous functions u in R n such that, 1 u 1 in R n, u is a viscosity subsolution of M + L u + C 0 u = ɛ in B 1, w = sup u F u is a discontinuous viscosity supesolution of M L w C 0 w = ɛ in B 1. Then thee exist constants ɛ 4, α and C such that, if ɛ < ɛ 4, C x α w (x) w (0) w (x) w (0) C x α, whee ɛ 4, α and C depend on σ 0, λ, Λ, C 0, n and µ. Poof. We claim that thee exist an inceasing sequence {m k } k and a deceasing sequence {M k } k such that M k m k = 8 αk and m k inf w B8 k sup w B8 M k k. We will pove this claim by induction. Fo k = 0, we choose m 0 = 1 and M 0 = 1 since 1 u 1 fo any u F. Assume that we have the sequences up to m k and M k. In B 8 k 1, we have eithe o Case 1: (4.4) holds. We define Thus, v 0 in B 1 and {w M k + m k {w M k + m k } B 8 k 1 B 8 k 1, (4.4) } B 8 k 1 B 8 k 1. (4.5) v(x) := w (8 k x) m k. M k m k {v 1} B 1 B Since w is a discontinuous viscosity supesolution of M L w C 0 w = ɛ in B 1, then v is a viscosity supesolution of M L v C 08 k(1 σ) v = 8 k(α σ) ɛ in B 8 k. We notice that C 0 = 0 if σ < 1 and choose α < σ 0. Thus, fo any 0 < σ <, v is a viscosity supesolution of M L v C 0 v = ɛ in B 8 k. By the inductive assumption, we have, fo any k j 0, Moeove, we have v m k j m k M k m k m k j M k j + M k m k M k m k = (1 8 αj ) in B 8 j. (4.6) By (4.6) and (4.7), we have v 8 αk [ 1 (1 8 αk )] = (1 8 αk ) in B c 8 k. (4.7) v(x) ( 8x α 1), fo any x B c 1. 11

12 We define v + (x) := max{v(x), 0} and v (x) := min{v(x), 0}. Since v 0 in B 1, v (x) = 0 and v (x) = 0 fo any x B 1. By (H1), we can choose sufficiently small α independent of σ such that, fo any x B 3 and σ 0 σ <, 4 M L v+ (x) M L v(x) + M + L v (x) M L v(x) + sup δ z v (x)k(z)dz K L R n M L v(x) + sup v (x + z)k(z)dz K L B c 1 {v(x+z)<0} 4 M L v(x) + sup max{( 8(x + z) α 1), 0}K(z)dz K L B c M L v(x) + ( σ)λ ( ) l σ ( ) (l+4)α 1 4 l=0 M L v(x) + 13 ( σ 0 )Λ M L v(x) + ɛ. ( 4(α σ 0) 1 α σ 0 4σ0 1 σ 0 Theefoe, we have M L v+ C 0 v + ɛ, in B 3. 4 Given any point x B 1, we can apply Coollay 4.4 in B 1 (x) to obtain 8 4 C(v + (x) + ɛ) ɛ 3 {v + > 1} B 1 4 (x) {v + > 1} B 1 8 Thus, we can choose sufficiently small ɛ 4 such that v + ɛ 4 in B 1 8 ) B 1 8. if ɛ < ɛ 4. Theefoe, v(x) = w (8 k x) m k M k m k ɛ 4 in B 1. 8 If we set m k+1 = m k + ɛ 4 M k m k and M k+1 = M k, we must have m k+1 inf B8 k 1 w sup B8 k 1 w M k+1. Case : (4.5) holds. Fo any u F, we obtain that u C 0 (R n ) is a viscosity subsolution of M + L u + C 0 u = ɛ in B 1 and u w in R n. Thus, we have We define Thus, v u 0 in B 1 and {u M k + m k } B 8 k 1 B 8 k 1. v u (x) := M k u(8 k x). M k m k {v u 1} B 1 B

13 Since u is a viscosity subsolution of M + L u + C 0 u = ɛ in B 1, then v u is a viscosity supesolution of M L v u C 0 v u = ɛ in B 8 k. Simila to Case 1, we have, if ɛ < ɛ 4, v u (x) = M k u(8 k x) M k m k ɛ 4 in B 1, 8 which implies u(8 k M k m k x) M k ɛ 4 By the definition of w, we have w (8 k x) M k ɛ 4 M k m k in B 1. 8 in B 1. 8 If we set m k+1 = m k and M k+1 = M k ɛ 4 M k m k, we must have m k+1 inf B8 k 1 w sup B8 k 1 w M k+1. Theefoe, in both of the cases, we have M k+1 m k+1 = (1 ɛ 4 )8 αk. We then choose α and ɛ 4 sufficiently small such that (1 ɛ 4 ) = 8 α. Thus we have M k+1 m k+1 = 8 α(k+1). Theoem 4.6. Assume that 0 < σ 0 σ < and I(x, 0, 0) is bounded in Ω. Assume that I is unifomly elliptic and satisfies (A0), (A). Let w be the bounded discontinuous viscosity solution of (1.1) constucted in Theoem 3.. Then, fo any sufficiently small δ > 0, thee exists a constant C such that w C α (Ω) and w C α ( Ω δ) C(C + m(c ) + I(, 0, 0) L (Ω)), whee α is given in Theoem 4.5, C := max{ u L (R n ), ū L (R n )} and C depends on σ 0, δ, λ, Λ, C 0, n, µ. Poof. It is obvious that u L (R n ) C if u F. Since I is unifomly elliptic, we have I(x, 0, 0) I(x, u(x), u( )) M + L u(x) + C 0 u(x) + m(c ), in Ω. Since u is a viscosity subsolution of I = 0 in Ω, we have Similaly, we have m(c ) I(, 0, 0) L (Ω) M + L u + C 0 u, in Ω. M L w C 0 w m(c ) + I(, 0, 0) L (Ω), in Ω. By nomalization, the esult follows fom Theoem 4.5. By applying Theoem 4.6 to Bellman-Isaacs equation, we have the following coollay. 13

14 Coollay 4.7. Assume that 0 < σ 0 σ <, b ab 0 in Ω if σ < 1 and c ab 0 in Ω. Assume that {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b, {f ab } a,b ae sets of unifomly bounded and continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ) : x Ω, a A, b B} ae kenels satisfying (H0)-(H3). Let w be the bounded discontinuous viscosity solution of (1.) constucted in Coollay 3.4. Then, fo any sufficiently small δ > 0, thee exists a constant C such that w C α (Ω) and w C α ( Ω δ) C(C + sup f ab L (Ω)), a A,b B whee α, C ae given in Theoem 4.6 and C depends on σ 0, δ, λ, Λ, sup a A,b B b ab L (Ω), sup a A,b B c ab L (Ω), n, µ. Remak 4.8. In this section we assume ou nonlocal equations satisfy the weak unifom ellipticity intoduced in [37] mainly because, to ou knowledge, this is the weakest assumption to get the weak Hanack inequality. In fact, ou appoach to get Hölde continuity of the discontinuous viscosity solution constucted by Peon s method could be applied to moe geneal nonlocal equations as long as the weak Hanack inequality holds fo such equation. 5 Continuous sub/supesolutions In this section we constuct continuous sub/supesolutions in both unifomly elliptic and degeneate cases. 5.1 Unifomly elliptic case In the unifomly elliptic case, we follow the idea in [36] to establish baie functions. We define v α (x) = ((x 1 1) + ) α whee 0 < α < 1 and x = (x 1, x,..., x n ). Lemma 5.1. Assume that 0 < σ <. Then thee exists a sufficiently small α > 0 such that M + L v α((1 + )e 1 ) ɛ 5 α σ fo any > 0 whee e 1 = (1, 0,..., 0) and ɛ 5 is some positive constant. Poof. Case 1: 0 < σ < 1. By Lemma. in [37], we have, fo any > 0 and α > 0, M + L v α((1 + )e 1 ) = sup (v α ((1 + ) e 1 + z) v α ((1 + )e 1 )) K(z)dz K L R n ( (( = sup + z1 ) +) ) α α K(z)dz K L R n ( ((1 = α σ sup + z1 ) +) ) α 1 n+σ K(z)dz K L R n = α σ sup K L ( α σ sup K L R n ( ((1 + z1 ) +) ) α 1 K(z)dz ) ((1 + z 1 ) α 1) K(z)dz inf K(z)dz. K L z 1 1 z 1 > 1 By (H3), we have, fo any K L and any δ > 0, thee is a set A δ satisfying A δ B δ \ B δ, A δ = A δ, A δ µ B δ \ B δ and K(z) ( σ)λδ n σ in A δ. It is obvious that µ δ := (B δ \ B δ ) {z; z 1 < 1} B δ \ B δ 14 0 as δ +.

15 Thus, thee exists δ 3 > 0 such that µ δ < µ if δ δ 3. Then {z; z 1 1} A δ3 B δ3 \ B δ3 By the symmety of A δ3, we have A δ 3 (B δ3 \ B δ3 ) {z; z 1 < 1} B δ3 \ B δ3 {z; z 1 1} A δ3 B δ3 \ B δ3 Theefoe, we have, fo any K L, K(z)dz K(z)dz z 1 1 {z;z 1 1} A δ3 By (H1) and (H), we have, fo any K L, ((1 + z 1 ) α 1)K(z)dz = z 1 > 1 Thus, we have lim sup α 0 + K L z 1 > 1 {z;z 1 > 1} B 1 + µ 4. µ. ( σ)λµ δ3 n σ B δ3 \ B δ3 =: ɛ 5. (5.1) 4 {z;z 1 > 1} B c 1 α 1 α zk(z)dz + B 1 ((1 + z 1 ) α 1)K(z)dz {z;z 1 > 1} B c 1 α 1 α (1 σ)λ + l=0 ( ) 1 1 σ l+ + ( ) +( σ)λ ( l 1 ) σ (1 + l ) α 1 l=0 αλ 1 σ ( α σ + 8Λ 1 σ 1 Then thee exists a sufficiently small α such that σ 1 α σ 1 σ ((1 + z 1 ) α 1) K(z)dz inf K(z)dz ɛ 5. K L z 1 1 M + L v α((1 + )e 1 ) ɛ 5 α σ. Case : σ = 1. Using (H), we have, fo any > 0 and α > 0, ). (5.) M + L v α((1 + )e 1 ) = sup (v α ((1 + ) e 1 + z) v α ((1 + )e 1 ) 1 B1 (z) v α ((1 + )e 1 ) z) K(z)dz K L R n ( (( = sup + z1 ) +) ) α α 1 B1 (z)α α 1 z 1 K(z)dz K L R n ( ((1 = α 1 sup + z1 ) +) ) α 1 1B 1 (z)αz 1 n+1 K(z)dz K L R n ( ((1 = α 1 sup + z1 ) +) ) α 1 1B 1 (z)αz 1 K(z)dz K L α 1 ( sup K L R n z 1 > 1 ((1 + z 1 ) α 1 1 B 1 (z)αz 1 ) K(z)dz inf K L 15 z 1 1 ) K(z)dz.

16 By (H1), we have, fo any K L, ((1 + z 1 ) α 1 1 B 1 (z)αz 1 )K(z)dz z 1 > 1 = ((1 + z 1 ) α 1 αz 1 )K(z)dz + ((1 + z 1 ) α 1)K(z)dz {z;z 1 > 1} B 1 {z;z 1 > 1} B c 1 α(1 α) B α z K(z)dz + ((1 + z 1 ) α 1)K(z)dz 1 {z;z 1 > 1} B c 1 α(1 α) α Λ + l=0 ( α 1 8αΛ + 4Λ ( ) 1 1 ( 1 l+ 1 1 α l+1 ). Then the est of poof is simila to Case 1. Case 3: 1 < σ <. Fo any > 0 and α > 0, we have ) + ( ) + Λ ( l 1 ) 1 (1 + l ) α 1 M + L v α((1 + )e 1 ) = sup (v α ((1 + ) e 1 + z) v α ((1 + )e 1 ) v α ((1 + )e 1 ) z) K(z)dz K L R n ( (( = sup + z1 ) +) ) α α α α 1 z 1 K(z)dz K L R n ( ((1 = α σ sup + z1 ) +) ) α 1 αz1 K(z)dz K L α σ ( sup K L R n z 1 > 1 l=0 ( ((1 + z1 ) +) ) ) α 1 αz1 K(z)dz inf (1 + αz 1 ) K(z)dz. K L z 1 1 Using (5.1) and (H), we have αλ(σ 1) inf (1 + αz 1 ) K(z)dz inf K(z)dz α sup zk(z)dz ɛ 5 K L z 1 1 K L z 1 1 K L B1 c 1 1 σ. By (H1) and (H), we have, fo any K L, ((1 + z 1 ) α 1 αz 1 )K(z)dz = z 1 > 1 {z;z 1 > 1} B 1 + {z;z 1 > 1} B c 1 α(1 α) B α z K(z)dz + α zk(z)dz 1 {z;z 1 > 1} B c 1 + ((1 + z 1 ) α 1)K(z)dz {z;z 1 > 1} B c 1 16α( σ)λ αλ(σ 1) 1 σ σ + 16( σ)λ ( ) α σ σ 1 α σ 1 σ. 16

17 Then we have lim sup α 0 + K L z 1 > 1 16α( σ)λ lim α σ + = ɛ 5. ( ((1 + z1 ) +) α 1 αz1 ) K(z)dz inf K L z 1 1 ( αλ(σ 1) α σ σ 1 1 σ + 16( σ)λ 1 α σ 1 σ Simila to Case 1, thee exists a sufficiently small α such that M + L v α((1 + )e 1 ) ɛ 5 α σ. (1 + αz 1 ) K(z)dz ) αλ(σ 1) ɛ σ Lemma 5.. Assume that 0 < σ <, C 0 0 and futhe assume C 0 = 0 if σ < 1. Then thee ae α > 0 and 0 < 0 < 1 sufficiently small so that the function u α (x) := (( x 1) + ) α satisfies M + L u α + C 0 u α 1 in B 1+0 \ B 1. Poof. We notice that u α and ae otation invaiant. By Lemma. in [37], M + L is also otation invaiant. Then we only need to pove that M + L u α((1+)e 1 )+C 0 u α ((1+)e 1 ) 1 fo any (0, 0 ] whee 0 and α ae sufficiently small positive constants. Note that, > 0, u α ((1 + )e 1 ) = v α ((1 + )e 1 ), u α ((1 + )e 1 ) = v α ((1 + )e 1 ) and that ( (1 + )e 1 + z 1) + ( + z 1 ) + C z, fo any z B 1, whee z = (z 1, z ). Theefoe, we have C α 1 z, z B, 0 (u α v α )((1 + )e 1 + z) C z α, z B 1 \ B, C z α, z R n \ B 1. Using (H1), we have, fo any 0 < σ < and L L, 0 L(u α v α )((1 + )e 1 ) = (u α v α )((1 + )e 1 + z)k(z)dz R ( n ) C α 1 z K(z)dz + z α K(z)dz + z α K(z)dz B B 1 \B R n \B 1 C α 1 z K(z)dz + z α K(z)dz B B c CΛ( α σ+1 + α σ ). Thus, we have M + L (u α v α )((1 + )e 1 ) CΛ( α σ+1 + α σ ). Theefoe, by Lemma 5.1, thee exists a sufficiently small α > 0 such that M + L u α((1 + )e 1 ) + C 0 u α ((1 + )e 1 ) M + L (u α v α )((1 + )e 1 ) + M + L v α((1 + )e 1 ) + C 0 u α ((1 + )e 1 ) CΛ( α σ+1 + α σ ) ɛ 5 α σ + αc 0 α 1. We notice that α σ + 1 > α σ, α σ > α σ and 17

18 (i) if 0 < σ < 1, then C 0 = 0; (ii) if σ = 1, then αc 0 0 as α 0; (iii) if 1 < σ <, then α 1 > α σ. Thus, thee exist sufficiently small 0 < 0 < 1 such that we have, fo any (0, 0 ], M + L u α((1 + )e 1 ) + C 0 u α ((1 + )e 1 ) 1. (5.3) In the est of this section, we assume that Ω satisfies the unifom exteio ball condition, i.e., thee is a constant Ω > 0 such that, fo any x Ω and 0 < Ω, thee exists y x Ω c satisfying B (y x) Ω = {x}. Without loss of geneality, we can assume that Ω < 1. Since Ω is a bounded domain, thee exists a sufficiently lage constant R 0 > 0 such that Ω {y; y 1 < R 0 }. Remak 5.3. At this stage, we ae not sue about whethe the exteio ball condition is necessay fo the constuction of sub/supesolution. In futue wok, we plan to constuct sub/supesolutions unde a weake assumption on Ω, such as the cone condition. Lemma 5.4. Assume that 0 < σ <, C 0 0 and futhe assume C 0 = 0 if σ < 1. Thee exists an ɛ 7 > 0 such that, fo any x Ω and 0 < < Ω, thee is a continuous function ϕ x, satisfying ϕ x, 0, in B (y x), ϕ x, > 0, in B (y c x), ϕ x, 1, in B c (y x), M + L ϕ x, + C 0 ϕ x, ɛ 7, in Ω. Poof. We define a unifomly continuous function ϕ in R n such that 1 ϕ and { ϕ(y) = 1, in y1 > R 0 + 1, We pick some sufficiently lage C 3 > α 0 ϕ(y) =, in y 1 R 0. and we define ϕ x, (y) = min{ϕ(y), C 3 u α ( y y x )} whee α and 0 ae defined in Lemma 5.. It is easy to veify that ϕ x, 0 in B (y x), ϕ x, > 0 in B c (y x), and ϕ x, 1 in B c (y x). By Lemma 5., we have M + L u α + C 0 u α 1 in B 1+0 \ B 1. It is obvious that, fo any y B (1+0 )(y x) \ B (y x), we have (M + L u α( y x ))(y) + C 0 1 σ ( u α ( y x ))(y) σ, fo any 0 < < Ω. Since C 0 = 0 if 0 < σ < 1, and 0 < < 1, then (M + L u α( y x ))(y) + C 0 ( u α ( y x ))(y) 1, fo any 0 < < Ω. Fo any y B (1+( ) α 1 ) (y x) \ B (yx), we have ϕ x, (y) = C 3 u α ( y y x ). Suppose that thee C 3 exists a test function ψ Cb (Rn ψ ) touches ϕ x, fom below at y. Thus, C 3 touches u α ( y x ) fom below at y. Thus, M + L ψ(y) + C 0 ψ(y) C 3. Fo any y Ω B c (1+( ) α 1 ) (y x), we have C 3 ϕ x, (y) = ϕ(y) = max R n ϕ x, =. Theefoe, fo any 0 < σ <, we have 18

19 (M + L ϕ x,)(y) + C 0 ϕ x, (y) = sup (ϕ x, (y + z) ϕ x, (y)) K(z)dz K L R n = sup (ϕ x, (y + z) )K(z)dz K L R n inf K(z)dz K L inf K L {z z 1 > y 1 +R 0 +1} {z z 1 >R 0 +1} K(z)dz. By a simila estimate to (5.1), thee exists a positive constant ɛ 6 such that, fo any K L, we have K(z)dz ɛ 6. {z z 1 >R 0 +1} Then, fo any y Ω B c (1+( C 3 ) 1 α ) (y x), we have M + L ϕ x,(y) + C 0 ϕ x, (y) ɛ 6. (5.4) Based on the above estimates, if we set ɛ 7 = min{c 3, ɛ 6 }, we have M + L ϕ x, + C 0 ϕ x, ɛ 7, in Ω. Theoem 5.5. Assume that 0 < σ <, I(x, 0, 0) is bounded in Ω and g is a bounded continuous function in R n. Assume that I is unifomly elliptic and satisfies (A0), (A). Then (1.1) admits a continuous viscosity supesolution ū and a continuous viscosity subsolution u and ū = u = g in Ω c. Poof. We only pove (1.1) admits a viscosity supesolution ū and ū = g in Ω c. Fo a viscosity subsolution, the constuction is simila. Since I is unifomly elliptic, we have, fo any x Ω, m( g L (R n )) I(x, g L (R n ), 0) I(x, 0, 0) m( g L (R n )). Thus, we have I(, g L (R n ), 0) L (Ω) < +. Since g is a continuous function, let ρ R be a modulus of continuity of g in B R. Let R 1 be a sufficiently lage constant such that Ω B R1 1. Fo any x Ω, we let u x, = ρ R1 (3) + g(x) + max{ g L (R n ), I(, g L (R n ),0) L (Ω) ɛ 7 }ϕ x, whee ϕ x, and ɛ 7 ae given in Lemma 5.4. It is obvious that u x, (x) = ρ R1 (3) + g(x), u x, g in R n and M + L u x, + C 0 u x, I(, g L (R n ), 0) L (Ω) in Ω. Now we define ũ = inf x Ω,0<<Ω {u x, }. Theefoe, ũ = g in Ω and ũ g in R n. Fo any x Ω and y R n, we have g(y) g(x) ũ(y) ũ(x) = ũ(y) g(x) ρ R1 (3) + max{ g L (R n ), I(, g L (R n ),0) L (Ω) ɛ 7 }ϕ x, (y) fo any 0 < < Ω. Theefoe, ũ is continuous on Ω. Fo any y Ω, we define d y = dist(y, Ω) > 0. If < dy, then we have, fo any z B dy (y), u x, (z) = ρ R1 (3) + g(x) + max{ g L (R n ), I(, g L (R n ), 0) L (Ω) ɛ 7 }, fo any x Ω. 19

20 Thus, we have, fo any z B dy (y), inf {u x, (z) u x, (y), 0} ũ(z) ũ(y) sup {u x, (z) u x, (y), 0}. x Ω, dy << Ω x Ω, dy << Ω has a unifom modulus of continuity, ũ is continuous in Ω. Thee- Since {u x, } x Ω, dy << Ω foe, ũ is a bounded continuous function in Ω. I(, g L (R n ), 0) L (Ω) in Ω. Now we define ū := { ũ, in Ω, g, in Ω c. By Lemma 3.1, we have M + L ũ + C 0 ũ By the popeties of ũ, we have ū is a bounded continuous function in R n, ū = g in Ω c and M + L ū + C 0 ū I(, g L (R n ), 0) L (Ω) in Ω. Using (A) and unifom ellipticity, we have, fo any x Ω, I(x, g L (R n ), 0) I(x, ū(x), ū( )) I(x, ū(x), 0) I(x, ū(x), ū( )) Thus, I(x, ū(x), ū( )) 0 in Ω. Now we have enough ingedients to conclude M + L ū(x) + C 0 ū(x) I(, g L (R n ), 0) L (Ω). Theoem 5.6. Let Ω be a bounded domain satisfying the unifom exteio ball condition. Assume that 0 < σ <, I(x, 0, 0) is bounded in Ω and g is a bounded continuous function. Assume that I is unifomly elliptic and satisfies (A0), (A). Then (1.1) admits a viscosity solution u. Poof. The esult follows fom Theoem 3., Theoem 4.6 and Theoem 5.5. Coollay 5.7. Let Ω be a bounded domain satisfying the unifom exteio ball condition. Assume that 0 < σ <, b ab 0 in Ω if σ < 1 and c ab 0 in Ω. Assume that g is a bounded continuous function in R n, {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b, {f ab } a,b ae sets of unifomly bounded and continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ) : x Ω, a A, b B} ae kenels satisfying (H0)-(H3). Then (1.) admits a viscosity solution u. 5. Degeneate case In the degeneate case, it is natual to constuct a sub/supesolution only fo (1.) when c ab γ fo some γ > 0. We emind you that Ω is a bounded domain satisfying the unifom exteio ball condition with a unifom adius Ω and, fo any x Ω and 0 < Ω, yx is a point satisfying B (yx) Ω = {x}. Fom now on, we will hide the dependence on x fo all vaiables and functions to make the notation simple. Fo example, we will let y := yx. Fo any x Ω, y Ω and 0 < Ω, we let (See Figue 1). n := x y x y, n y := y y y y, and v α(y) := ( ((y y ) n ) ) + α 1 0

21 Figue 1: The exteio ball centeed at y. Instead of letting {K ab (x, ); x Ω, a A, b B} satisfy (H3), we let the set of kenels satisfy the following weake assumption: (H3) Thee exist C 4 > 0, 0 < 1 < Ω, λ > 0 and µ > 0 such that, fo any x Ω, 0 < < 1 and y Ω B (y ), thee is a set A y satisfying (i) A y {z; z n y < s y} (B C4 s y \ B s y ) whee z n y := z n y and s y := y y 1; (ii) A y µ B s y ; (iii) K(y, z) ( σ)λ(s y) n σ fo any z A y. Lemma 5.8. Suppose that {K ab (x, ); a A, b B, x {y Ω; dist(y, Ω) < 1 }} satisfies (H3) fo some 1 (0, Ω ). Then (H3) holds fo the set of kenels. Poof. Fo any x Ω, 0 < < 1 and y Ω B (y ), we define (B C4 s \ B y C 4 s y ) {z; z n y s y} µ C4 := B C4 s \ B y C 4 s y. (5.5) We notice that the ight hand side of (5.5) depends only on C 4. It is obvious that By (H3), thee exists a set A satisfying and, fo any z A, A B C4 s y \ B C 4 s y lim µ C 4 = 0. C 4 +, A = A, A µ B C4 s \ B y C 4 s y, K(y, z) ( σ)λ( C 4s y ) n σ = ( σ)λ( C 4 ) n σ (s y) n σ := ( σ) λ(s y) n σ. 1

22 Thee exists a sufficiently lage constant C 4 ( ) such that µ C4 < µ. Then {z; z n y > s y} A B C4 s \ B y C 4 s y A (B C4 s \ B y C 4 s y ) {z; z n y s y} B C4 s y \ B C 4 s y Let A y := A {z; z n y < s y}. By the symmety of A, we have A y µ 4 B C 4 s \ B y C 4 s y µ 4 B s := µ B y s. y Theefoe, (H3) holds fo the set of kenels with C 4, 1, λ and µ. µ. Lemma 5.9. Assume that 0 < σ < and {K ab (x, ); x Ω, a A, b B} ae kenels satisfying (H0)-(H), (H3). Then thee exists a sufficiently small α > 0 such that, fo any x Ω, 0 < < 1 and s {l (0, 1); y + (1 + l)n Ω}, we have I ab [y + (1 + s)n, vα] ɛ 8 σ s α σ whee ɛ 8 is some positive constant. Poof. We only pove the esult fo the case 0 < σ < 1. Fo the est of cases, the poofs ae simila to those in Lemma 5.1. Fo any x Ω, 0 < < 1 and s {l (0, 1); y +(1+l)n Ω}, we have I ab [y + (1 + s)n, vα] = (vα(y + (1 + s)n + z) vα(y + (1 + s)n)) K ab (y + (1 + s)n, z)dz R n [( ( = s + z ) ) + α ] n s α K ab (y + (1 + s)n, z)dz R n [ ((1 = σ s α σ + zn ) +) ] α 1 (s) n+σ K ab (y + (1 + s)n, sz)dz R n = σ s α σ{ z n> 1 z n 1 [ ] (1 + z n ) α 1 (s) n+σ K ab (y + (1 + s)n, sz)dz } (s) n+σ K ab (y + (1 + s)n, sz)dz, whee z n := z n. Using (H3), we have (s) n+σ K ab (y + (1 + s)n, sz)dz z n 1 = (s) σ K ab (y + (1 + s)n, z)dz z n s (s) σ K ab (y + (1 + s)n, z)dz A y +(1+s)n ( σ)λµ(s) n B s := ɛ 8. We notice that the kenel (s) n+σ K ab (y + (1 + s)n, s ) still satisfies (H1) and (H). By a simila calculation to (5.), we have [ ] (1 + z n ) α 1 (s) n+σ K ab (y + (1 + s)n, sz)dz ɛ(α), z n> 1

23 whee ɛ(α) is a positive constant satisfying that ɛ(α) 0 as α 0. sufficiently small α such that Then thee exists a I ab [y + (1 + s)n, v α] ɛ 8 σ s α σ. Lemma Assume that 0 < σ <, and b ab 0 in Ω if σ < 1. Assume that {b ab } a,b ae sets of unifomly bounded functions in Ω and {K ab (x, ); x Ω, a A, b B} ae kenels satisfying (H0)-(H), (H3). Then thee ae α > 0 and 0 < s 0 < 1 sufficiently small so that, fo any x Ω and 0 < < 1, the function satisfies, fo any a A and b B, u α(y) := ( ( y y ) ) + α 1 I ab [y, u α] + b ab (y) u α(y) 1 in Ω ( B (1+s0 )(y ) \ B (y )). Poof. Note that, s > 0, u α(y + (1 + s)n) = vα(y + (1 + s)n), u α(y + (1 + s)n) = vα(y + (1 + s)n) and that ( (1 + s)n + z + ( 1) s + z ) + n C z z n, fo any z B. Thus, we have 0 (u α vα)(y + (1 + s)n + z) Cs α 1 z z n, z B s, C z zn α α, z B \ B s, C z α α, z R n \ B. Using (H1), we have, fo any 0 < σ <, a A, b B and s {l (0, 1); y + (1 + l)n Ω}, 0 I ab [y + (1 + s)n, u α vα] (u α vα)(y + (1 + s)n + z)k ab (y + (1 + s)n, z)dz R ( n C s α 1 z z n B s K ab (y + (1 + s)n, z)dz z z n α + K ab (y + (1 + s)n, z)dz + ( C B s B \B s α z Rn\B α α 1 z s CΛ σ (s α σ+1 + s α σ ). ) α K ab(y + (1 + s)n, z)dz K ab(y + (1 + s)n, z)dz + R n \B s z α ) α K ab(y + (1 + s)n, z)dz 3

24 By Lemma 5.9, we have I ab [y + (1 + s)n, u α] I ab [y + (1 + s)n, v α] I ab [y + (1 + s)n, u α v α] σ [ɛ 8 s α σ CΛ(s α σ+1 + s α σ )]. (5.6) Fo any y Ω (B (y ) \ B (y )), we have I ab [y, u α] = δ z u α(y)k ab (y, z)dz R n = δ z u α(y + (1 + s y)n y)k ab (y, z)dz R n ( ( ) ) z = δ z u α(y + (1 + s y)n)k ab y, R n z + n y n z dz. Using (H3) and a simila estimate to (5.6), we have I ab [y, u α] σ [ɛ 8 (s y) α σ CΛ((s y) α σ+1 + (s y) α σ )]. By a simila estimate to (5.3), thee exists a sufficiently small constant 0 < s 0 < 1 such that we have, fo any y Ω ( B (1+s0 )(y ) \ B (y )), I ab [y, u α] + b ab (y) u α(y) 1. Lemma Assume that 0 < σ <, b ab 0 in Ω if σ < 1 and c ab γ in Ω fo some γ > 0. Assume that {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b, {f ab } a,b ae sets of unifomly bounded and continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ); x Ω, a A, b B} ae kenels satisfying (H0)-(H), (H3). Then, fo any x Ω and 0 < < 1, thee is a continuous viscosity supesolution ψ of (3.5) such that ψ 0 in B (y ), ψ > 0 in B c (y ) and ψ sup a A,b B f ab L (Ω) + 1 γ in B c (1+s 0 ) (y ), (5.7) whee s 0 is given by Lemma Poof. Without loss of geneality, we assume that 0 < γ < 1. We pick a sufficiently lage C 5 > 0 such that C 5 > sup a A,b B f ab L (Ω) + 1 s α 0 γ. (5.8) We then define, fo any x Ω and 0 < < 1, { } supa A,b B f ab L ψ (y) = min (Ω) + 1, C 5 u γ α(y). It is easy to veify that ψ 0 in B (y ), ψ > 0 in B c (y ) and ψ is a continuous function in R n. Using (5.8), we know that C 5 u α C 5 s α 0 sup a A,b B f ab L (Ω) + 1, in B(1+s c γ 0 ) (y ). 4

25 Theefoe, (5.7) holds. Since c ab γ > 0 in Ω, sup a A,b B f ab L (Ω) +1 γ is a viscosity supesolution of (3.5) in Ω. By Lemma 5.10 and (5.8), we have, fo any y Ω ( B (1+s0 )(y ) \ B (y )), sup inf { I ab[y, C 5 u α] + C 5 b ab (x) u α(y) + C 5 c ab (x)u α(y) + f ab (y)} b B a A sup a A,b B f ab L (Ω) f ab (y) 0. (5.9) Theefoe, ψ is a continuous viscosity supesolution of (3.5) in Ω. Theoem 5.1. Assume that 0 < σ <, b ab 0 in Ω if σ < 1 and c ab γ in Ω fo some γ > 0. Assume that g is a bounded continuous function in R n, {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b, {f ab } a,b ae sets of unifomly bounded and continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ); x Ω, a A, b B} ae kenels satisfying (H0)-(H), (H3). Then (1.) admits a continuous viscosity supesolution ū and a continuous viscosity subsolution u and ū = u = g in Ω c. Poof. We only pove (1.) admits a viscosity supesolution ū such that ū = g in Ω c. Since g is a continuous function, let ρ R be a modulus of continuity of g in B R. Let R 1 be a sufficiently lage constant such that Ω B R1 1. Fo any x Ω, we let ( ) γ u = ρ R1 (3) + g(x) + g L (R n ) sup a A,b B f ab L (Ω) ψ, whee ψ is given in Lemma Using Lemma 5.11, u (x) = ρ R1 (3) + g(x), u g in R n and u is a continuous viscosity supesolution of (3.5) in Ω. Then the est of the poof is simila to Theoem 5.5. Theoem Let Ω be a bounded domain satisfying the unifom exteio ball condition. Assume that 0 < σ <, b ab 0 in Ω if σ < 1 and c ab γ in Ω fo some γ > 0. Assume that g is a bounded continuous function in R n, {K ab (, z)} a,b,z, {b ab } a,b, {c ab } a,b, {f ab } a,b ae sets of unifomly bounded and continuous functions in Ω, unifomly in a A, b B, and {K ab (x, ); x Ω, a A, b B} ae kenels satisfying (H0)-(H), (H3). Then (1.) admits a discontinuous viscosity solution u. Poof. The esult follows fom Coollay 3.4 and Theoem 5.1. Acknowledgement. impoved the pape. We would like to thank the efeee fo valuable comments which Refeences [1] O. Alvaez and A. Touin, Viscosity solutions of nonlinea intego-diffeential equations, Ann. Inst. H. Poincaé Anal. Non Linéaie, 13 (1996), no. 3, [] G. Bales, R. Buckdahn and E. Padoux, Backwad stochastic diffeential equations and integal-patial diffeential equations, Stochastics Stochastics Rep., 60 (1997), no. 1-, [3] G. Bales, E. Chasseigne and C. Imbet, On the Diichlet poblem fo second-ode elliptic intego-diffeential equations, Indiana Univ. Math. J. 57 (008), no. 1,

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