Remarks on L p -viscosity solutions of fully nonlinear parabolic equations with unbounded ingredients

Transcription

1 Remarks on L p -viscosity solutions of fully nonlinear parabolic equations with unbounded ingredients Shigeaki Koike Andrzej Świe ch Mathematical Institute School of Mathematics Tohoku University Georgia Institute of Technology Aoba, Sendai Atlanta, GA JAPAN USA and Shota Tateyama Mathematical Institute Tohoku University Aoba, Sendai JAPAN Abstract The weak Harnack inequality for L p -viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of L p -viscosity solutions is derived from the weak Harnack inequality for L p -viscosity supersolutions. The local maximum principle for L p -viscosity subsolutions and the Harnack inequality for L p -viscosity soultions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives. Keywords: fully nonlinear parabolic equations, viscosity solutions, Harnack inequality, maximum principle. 200 MSC: 49L25, 35D40, 35B65, 35K55, 35K20, 35K0. Supported in part by Grant-in-Aid for Scientific Research No. 6H06339, 6H03948, 6H03946 of JSPS, koike@m.tohoku.ac.jp swiech@math.gatech.edu Mathematical Institute, Tohoku University, Aoba, Sendai, Miyagi, , Japan. Supported by Grant-in-Aid for JSPS Research Fellow 6J02399, shota.tateyama.p3@dc.tohoku.ac.jp

2 Introduction The seminal paper [3] of L. A. Caffarelli was the most influential in the development of modern regularity theory for viscosity solutions of fully nonlinear uniformly elliptic partial differential equations PDE for short. Various results were proved there, including Harnack inequality, C α, C,α, C 2,α and W 2,p estimates, and the reader can find a more detailed and complete account of them in [4]. Around the same time similar results like Harnack inequality, C α and C,α estimates for viscosity solutions were also proved by different methods in [20, 2, 22]. In order to treat PDE with measurable terms, the notion of L p -viscosity solution of fully nonlinear uniformly elliptic PDE was introduced in [5] and similar idea was also considered in [23]. L. Wang in [23, 24] extended regularity results of [3] to viscosity solutions of fully nonlinear uniformly parabolic PDE. Later, L p -viscosity solutions of parabolic PDE were studied in [6, 7]. The main ingredient in the theory is the Aleksandrov-Bakelman-Pucci ABP for short maximum principle, which gives the L -estimates in terms of the L p -norms of the inhomogeneous terms. The ABP maximum principle for viscosity solutions of fully nonlinear uniformly parabolic PDE was proved in [23]. In [5], the ABP maximum principle was proved for L p -viscosity solutions of uniformly elliptic PDE which are uniformly Lipschitz continuous in the first derivatives. It was later extended for elliptic and parabolic PDE to equations which are not uniformly Lipschitz continuous in the first derivative terms in [2], where the Lipschitz coefficient functions as functions of x and t belong to some L q spaces. The second ingredient of the regularity theory of [3] is the Harnack inequality for viscosity solutions as well as the weak Harnack inequality and the local maximum principle. Such results for non-divergence form equations started with the work of Krylov and Safonov [6] and the results for strong solutions can be found in classical books [9, 8]. Results for viscosity solutions first appeared in [3, 20] see [4]. General form of the weak Harnack inequality for L p -viscosity supersolutions of fully nonlinear elliptic PDE which implies the Hölder continuity of L p -viscosity solutions was proved in [3], using the ABP estimates of [2], while a general local maximum principle for L p -viscosity solutions can be found in [5]. The corresponding results for viscosity solutions of uniformly parabolic PDE were proved in [23], however only for equations which are uniformly Lipschitz continuous in the first derivatives. In this paper we want to extend them to L p -viscosity solutions of more general equations. The relevant equations are the parabolic extremal equations u t + P ± D 2 u ± µ Du + f = 0 in Q, where f L p Q and µ L q Q. In this manuscript, combining the argument from [0] with the ABP maximum principle of [2], we first show the weak Harnack inequality when the L q -norm of the coefficient function µ is small. We then avoid this smallness assumption by the introduction of a new heat kernel like barrier function in our proof of the weak Harnack inequality. We 2

3 will use global estimates on strong solutions of fully nonlinear parabolic equations from a recent paper by Dong, Krylov and Xu [8]. We remark that the weak Harnack inequality yields the local Hölder estimate. In order to establish the Harnack inequality, following the argument of [4] see also [5], we also obtain the corresponding local maximum principle. We refer to [23] and [0] for the other approach. We also present some results when the PDE contains first space derivative terms which may grow superlinearly. This paper is organized as follows. In section 2, we recall the definition of L p -viscosity solution for parabolic PDE, its properties and known results. Section 3 is devoted to a proof of the weak Harnack inequality for L p -viscosity supersolutions. In section 4 we first establish the local Hölder continuity estimate using the weak Harnack inequality. For the completeness of the theory, we show the local maximum principle for L p -viscosity subsolutions by a parabolic version of the argument of [4] and then obtain the Harnack inequality. In section 5, we present some results for PDE which may contain superlinearly growing gradient terms. 2 Preliminaries We fix n N, a bounded domain Ω R n, and T > 0. We denote by S n the set of all n n symmetric matrices with the standard order. Given F : Ω 0, T ] R n S n R, we are concerned with the following fully nonlinear parabolic PDE: u t + F x, t, Du, D 2 u = 0 in Ω 0, T ], 2. where Du and D 2 u, respectively, denote the first and second derivatives with respect to x R n, u t is the time derivative, and F is at least measurable with respect to all the variables. We will write u xk, u xk x l for u x k, 2 u x l x k, respectively. In what follows, we assume that F is uniformly parabolic, i.e. that there exist 0 < λ Λ < such that P λ,λ X Y F x, t, ξ, X F x, t, ξ, Y P+ λ,λ X Y 2.2 for all x, t, ξ, X, Y Ω 0, T ] R n S n S n, where P ± λ,λ : Sn R are defined by P + λ,λ X := max{ TrAX A Sn, λi A ΛI}, P λ,λ X := min{ TrAX A Sn, λi A ΛI} for X S n, where I denotes the n n identity matrix. Since we fix 0 < λ Λ in this paper, we simply write P ± for P ± λ.λ. For properties of P±, we refer for instance to [5]. Setting Q := Ω 0, T ], we denote the parabolic boundary of Q by p Q := Ω {0} Ω [0, T. 3

4 The parabolic distance is defined by dx, t, y, s := x y 2 + t s. For U, V R n+, we define the distance between U and V distu, V := inf {dx, t, y, s x, t U, y, s V }. We will write diamq for the diameter of Q measured with respect to the parabolic distance and diamω for the diameter of Ω. We will use the anisotropic Sobolev spaces. For p, and W 2, p Q := {f L p Q f xk, f xk x l, f t L p Q k, l n}, W 2, p,loc Q := { f W 2, p Q Q Q }. Here and later, Q Q means distq, p Q > 0. We define the norm for f Wp 2, Q by f W 2, p Q := f L p Q + f t L p Q + We will also use the anisotropic Sobolev spaces n f xk L p Q + k= n f xk x l L p Q. k,l= Wp,0 Q := {v L p Q f xk L p Q k, l n} for p, equipped with the norm f W,0 p Q := f L p Q + n f xk L p Q. We denote by C 2, Q be the space of functions u CQ such that u t, u xk, u xk x l CQ for k, l n. For 0 < α we denote by C α Q the space of functions which are α-hölder continuous in Q with respect to the parabolic distance. We denote by Wp k Ω, k =, 2,..., the standard Sobolev spaces. We recall the notion of L p -viscosity solutions of parabolic PDE 2.. To this end, we denote by B r x the open ball in R n with the radius r > 0 and the center x, and define the parabolic cylinders k= Q r x, t := x, t + r, r n r 2, 0]. Definition 2.. Let Q be a relatively open subset of Q. A function u CQ is said to be an L p -viscosity subsolution resp., supersolution of 2. if for φ W 2, p,loc Q, we have lim inf { φt y, s + F y, s, Dφy, s, D 2 φy, s } 0 ess ε 0 y,s Q εx,t 4

5 resp., lim ess sup ε 0 y,s Q εx,t { φt y, s + F y, s, Dφy, s, D 2 φy, s } 0 provided that u φ attains a maximum resp., minimum at x, t Q over some parabolic cylinder Q r x, t Q. A function u CQ is said to be an L p -viscosity solution of 2. if u is an L p -viscosity subsolution and supersolution of 2.. Remark 2.2. We note that W 2, n+2 p,loc Q CQ for p > and if Ω is regular enough 2 e.g. if Ω is C, then Wp 2, Q C α Q for α = 2 n + 2/p is a bounded imbedding for n+2 < p < n + 2. If u W 2, 2 p,loc Q for p > n + 2 then u x i C α for α = n + 2/p see e.g. [7]. Also, it is known that if p > n+2 and u W 2, 2 p Q, then u t, u xi and u xi x j i, j n exist a.e. in Q see [6]. In this section, we recall the ABP maximum principle for L p -viscosity subsolutions of the following extremal uniformly parabolic equations; where u t + P D 2 u µ Du f = 0 in Q, 2.3 f L p Q, and µ L q Q. We will suppose that the powers p and q satisfy the condition q > n + 2, p < p q, 2.4 where p = p n, Λ [ n+2, n + is the constant, which gives a range where the ABP λ 2 maximum principle holds, see e.g. [2]. Proposition 2.3. cf. Theorem 2.8 in [7], Proposition 3.3 in [2] For p > p, there exists a constant C = Cn, Λ, λ, p > 0 such that for f L p Q, there exists u CQ W 2, p,loc Q such that { u t + P + D 2 u fx, t = 0 a.e. in Q, u = 0 on p Q, and C f L p Q u C f + L p Q in Q. Moreover, for Q Q, there is C = C n, Λ, λ, p, T, diamω, distq, p Q > 0 such that u W 2, p Q C f L p Q. We emphasize that the dependence of constants on various parameters sometimes may mean that a constant may blow up as a parameter converges to 0, for instance the constant 5

6 C in Proposition 2.3 may blow up as λ 0. The precise dependence of constants on T, diamω, diamq can often be found by scaling. We state below in Proposition 2.5 a scaled version of the ABP maximum principle for L p -strong and L p -viscosity solutions of 2.3 based on the results of [2]. For u CQ, we introduce the set { } Q + [u] := x, t Q ux, t > sup u +. pq We denote by L p +Q for the set of all nonnegative functions in L p Q. Remark 2.4. The non-scaled statement of the classical ABP maximum principle for strong solutions in Proposition 3.2 of [2] was slightly incorrect and might be confusing. The exact ABP inequality from [9] is u L Q ψ L pq + C d n n+ Ω exp C 2 d Ω µ n+ L n+ Q f L n+ Q, 2.5 where d Ω = diamω, which behaves differently from the one in [2] when diamω is small. This however does not affect the results of [2] since the proofs there only used 2.5 in parabolic cylinders of fixed size which contained Q and did not depend on diamω. Proposition 2.5. see Proposition 3.6, Theorem 3.0 of [2] Let 2.4 hold and let f L p +Q, µ L q +Q. There exists a constant C = C n, Λ, λ, p, q, d n+2/q Q µ L q Q > 0 such that if u CQ is either an L p -strong or an L p -viscosity subsolution of 2.3, then sup Q sup Q u sup pq v sup pq n+2 2 p u + C dq f L p Q, 2.6 where d Q = diamq. Moreover, if v CQ is an L p -viscosity subsolution of 2.3 in Q + [v] then v + n+2 2 p + C dq f L p Q + [u]. 2.7 We remark that when p n+2 then 2.6 can be made more precise based on 2.5 or on a scaled version of 2.5 in a unit cylinder. We also remark that it can be proved that if 2.4 holds then an L p -strong subsolution of u t + P ± D 2 u ± µ Du = f is an L p -viscosity subsolution of those. We refer to [3], Section 3, for such a proof in the elliptic case. Similar statement holds for L p -viscosity supersolutions. Proof of Proposition 2.5. To see why 2.6 is true we notice that the function wx, t = ud Q x, d 2 Q t is an Lp -strong or an L p -viscosity subsolution of w t + P D 2 w µ Dw f = 0 6

7 in a unit cylinder Q, where µx, t = d Q µd Q x, d 2 Q t and fx, t = d 2 Q fd Qx, d 2 Qt. Then, by the estimates of [2] sup w sup w + C n, Λ, λ, p, q, µ L q Q f L p Q. Q pq n+2 2 p Q It remains to notice that µ L q Q = d n+2/q Q µ L q Q and f L p Q = d f L p Q. Estimate 2.7 is proved similarly by rescaling and adding to w a subsolution of an extremal equation in a bigger cylinder to eliminate f, which can be found using Proposition 3.5 of [2] or using Proposition 2.6 below if q p > n + 2. The reader can find a similar argument in the proof of Proposition 2.8 of [2]. A result similar to Proposition 2.6 can be found in [2] see Proposition 3.5 there. Using global Wp 2, estimates by Dong, Krylov and Xu in [8], we present a slightly different existence result. Proposition 2.6. Assume that Ω is C, and q p > n + 2. Let µ L q +Q, ψ Q CQ and f L p Q. The equation W 2, p { ut + P + D 2 u + µ Du f = 0 a.e. in Q, u = ψ on p Q 2.8 has an L p -strong solution u CQ Wp 2, Q. The solution u satisfies n+2 2 u L Q ψ L p pq + C dq f L p Q 2.9 where C is the constant from 2.6 and u W 2, p Q C 2 ψ W 2, p Q + f L p Q, 2.0 for some constant C 2 = C 2 n, Λ, λ, p, q, µ L q Q, T, diamω, Ω > 0. Remark 2.7. The function u in Proposition 2.6 is also an L p -viscosity solution of 2.8. We also note that p > n + is assumed in [8] while we assume p > n + 2. Proof. Let f j, µ j CQ be such that f j f L p Q + µ j µ L q Q 0, and f j, µ j f, µ almost everywhere in Q as j. Let u j CQ C 2, Q be the classical solution of { u j t + P + D 2 u j + µ j Du j f j = 0 in Q, u j 2. = ψ on p Q. 7

8 Here and later, C > 0 stands for various constants depending only on known quantities. We know from [8] that µ u j W 2, p Q C j Du j f j + ψ L p Q W 2, Q. 2.2 It is also known e.g. Lemma 3.3 in [7] that for sufficiently small small ε > 0, Du j L Q ε α D 2 u j L p Q + u j t L p Q + ε α 2 C u j L p Q, 2.3 where α = n+2 > 0 and α p 2 = + n+2 > 0. Combining 2.3 with 2.2, in view of p the global estimates in [8], we have u j W 2, p Q C f j L p Q + µ j Du j L p Q + ψ Wp 2, Q { } µ j L p Q ε α D 2 u j L p Q + u j t L Q p + Cε α 2 u j L p Q 2.4 +C f j L p Q + ψ W 2, p Q Hence, for an appropriate ε > 0 depending on µ j L p Q, using the ABP maximum principle, we obtain u j W 2, p Q C f j L p Q + ψ W 2, Q. 2.5 Since by anisotropic Sobolev imbeddings the functions u j are equicontinuous in CQ and u j x i, i =,..., n, are locally equicontinuous, by taking a subsequence, we can assume that there exists u Wp 2, Q CQ satisfying 2.9 and 2.5 such that u j u in Wp 2, Q, u j u in CQ and u j x i u xi locally uniformly. Thus µ j Du j + f j µ Du + f in L p loc Q. It is then standard by the techniques of [7] to obtain that u is an L p -viscosity solution and hence an L p -strong solution of. u t + P + D 2 u = g, where g = µ Du + f, which concludes the proof. p p 3 The weak Harnack inequality In what follows, we set Ω := 0, 0 n, T = 0 and Q := 0, 0 n 0, 0]. Although we need to suppose Ω C, to use Propostion 2.6, for the sake of simplicity of the presentation, we will assume that the boundary of cubes are C,. Otherwise we 8

9 would have to use a smooth domain similar to 0, 0 n. We refer to [3] for such an argument. In this section, we show the weak Harnack inequality for nonnegative L p -viscosity supersolutions of u t + P + D 2 u + µ Du + f = 0 in Q, 3. where f L p +Q and µ L q +Q. 3. A restricted case In order to show the weak Harnack inequality for nonnegative L p -viscosity supersolutions of 3. with f L p +Q and µ L q +Q, we follow the standard argument as in [0] except for a new barrier function, which will be constructed in Lemma 3.7. However, for this purpose, we first have to show the weak Harnack inequality under a restricted setting. Theorem 3.. Assume that 2.4 holds, f L p +Q and µ L q +Q. Then, there exist constants ε 0 = ε 0 n, Λ, λ, p, q > 0, δ 0 = δ 0 n, Λ, λ, p, q > 0 and C 0 = C 0 n, Λ, λ, p, q > 0 such that if µ L p Q δ 0, 3.2 then any nonnegative L p -viscosity supersolution u of 3. satisfies u ε ε 0 0 dxdt C0 inf J J 2 u + f L p Q, 3.3 where J :=, n 0, 2 ], and J 2 :=, n 9, 0]. We remark that the statement of Theorem 3. also holds for nonnegative L p -strong supersolutions of 3.. In order to prove Theorem 3., we first construct a strong subsolution of an extremal equation. To this end, we use the following cubes: K :=, n 0, ], and K 2 := 3, 3 n, 0]. We recall a barrier function from Lemma of [0] see also [23]. We can also construct one by the same manner as in Lemma 3.7 here. 9

10 t 0 3 J 2 9 K 2 Q 2 J }K K /4 0 0 x Figure. Lemma 3.2. There exists a nonnegative function φ C 2, Q and a function g CQ such that φ t + P + D 2 φ gx, t in Q, φ 2 in K 2, φ = 0 on p Q, supp g K. Letting K as above, we denote by C the set of all 2 n+2 cubes + i, i + i n, i n j, j+ ] for i 4 4 k = 0, k =, 2,..., n, and j = 0,, 2, 3. For each cube L C, we divide it into 2 n+2 cubes. We denote by C 2 the set of such cubes constructed by the same procedure from each cube L C. Inductively, we construct C k whose elements have length 2 k+ in each space direction and 4 k in time. We call L k= C k a dyadic cube of K. When L C k is constructed from an element of C k by the above procedure, we denote by L C k the predecessor of L. For L C k and its predecessor L := J τ, τ + ] C 4 k k for a cube J = a, a + 2 k 2 2 k 2 an, an+ with some integers a 2 k 2 2 k 2,..., a n, and τ [0,, we set L m = J τ + 4 k+, τ + 4 k+ m + ], which is the union of m cubes of the translated predecessor in the future direction. We define C := k= C k. Moreover, for m N, we define Cm := {L C L m Q}. Notice that when m 36, we have Cm = C. 0

11 We recall a parabolic version of the Calderón-Zygmund decomposition, which is a modification of Lemma of [0]. Since fine cubes are needed in the proof of Lemma of [0], we can follow the argument there to prove the next lemma. Lemma 3.3. Let m be an integer, and K R n+ be as above. Let measurable sets A B K and σ 0, satisfy { i A σ K, Then, it follows that ii if L Cm is such that A L > σ L, then L m B. A σ m + m B. Proof of Theorem 3.. For ε > 0, which will be fixed later, we set ũx, t = N 0 ux, t, where N 0 = inf J2 u + ε f L p Q + η for η > 0, which will be sent to 0 at the end of the proof. By considering ũ instead of u, it is enough to show that there are ε 0, C 0 > 0 such that u ε ε 0 0 dxdt C0 3.4 J under the assumptions inf J 2 u, and f L p Q ε. 3.5 Let φ be the function in Lemma 3.2. By letting w := φ u, it is immediate to see that w is an L p -viscosity subsolution of w t + P D 2 w µ Dw h = 0 in Q, where h := µ Dφ + g + f. In view of Proposition 2.5, we have sup w C h L p Q + [w]. Q Hence, by recalling supp g K in Lemma 3.2, it is easy to verify that this inequality implies sup J 2 w C g L p Q + [w] + ε + δ 0. Thus, for fixed ε, δ 0 > 0, there is θ 0, such that {x, t K wx, t > 0} θ K. Hence, setting M := sup φ, M 2, we have K {x, t K ux, t M} θ K. 3.6

12 We next fix δ θ, and select large m N such that θ < θ m + m Letting J k :=, n 0, + m+ ] for k, we note that 2 9 mk 3 9 m We choose k 0 N such that J k+ J k k N, and lim k J k = J. m + 9 mk m < 2 δ <. 3.7 i.e. J k K for k k 0. Finally, putting Ĉ0 := J k 0 m θ m + k 0, by our choice of δ i.e. 3.7, we observe J k 0 Ĉ0δ k We will show that {x, t J k ux, t M km } Ĉ0δ k k k Notice that 3.8 yields 3.9 for k = k 0. For any fixed k k 0 +, we suppose that 3.9 holds for k. Set A = {x, t J k ux, t M km } and B = {x, t J k ux, t M k m }. It is immediate to see that A B K, and A θ K from 3.6 because A {x, t K ux, t M}. If the hypotheses in Lemma 3.3 are satisfied for A, B and σ = θ, then using B Ĉ0δ k, we have A θ m + m Ĉ0δ k. Hence, 3.9 holds for any k k 0 by our choice of δ and m in 3.7. Therefore, the standard argument implies {x, t J ux, t s} A 0 s β 0 s > 0, 3.0 where A 0 = Ĉ0δ and β 0 := log δ m log M > 0. We thus obtain 3.4 when ε 0 0, β 0. In order to check ii in Lemma 3.3, we take a dyadic cube L Cm such that We can find j N and x 0, t 0 K such that A L > θ L. 3. L = x 0, t j, 2 j n 0, 2 2j ]. 2

13 We claim that if 3. holds then inf u > M km l for l {, 2,, km}. 3.2 N l {R n 0,0]} Here, we set N := x 0, t K 4 j 2 j,, N l := x 0, t 0 + 9l for σ > 0, σk := σ, σ n 0, σ 2 ]. We will prove this claim later l K 4 j 2 j,, where t N 3 N 2 L N x 0, t 0 x Figure 2. One direct consequence of this claim for l =, 2,..., m is the following assertion: under 3., it follows that m u > M k m in N l {R n 0, 0]}. 3.3 It is obvious from the definition that l= L k Γ k for k N, 3.4 where Γ k := k l= N l for k N. We also write Γ = l N N l. We easily verify the following inclusions: x 0, t 0 + S Γ 4 j, x 0, t 0 + S + 4j,, 3.5 3

14 where for 0 α < β, paraboloid type domains S ± α,β are given by S α,β := { x, t R n α, β] t > x 2 4 j }, S + α,β := { x, t R n α, β] t > 2 3 x 2 4 j }. Here, x := max{ x,..., x n } for x = x,..., x n R n. For extreme cases when x 0, t 0 = x, 0 or x 0, t 0 = x,, where x =, 0,..., 0, we observe Hence, we obtain J 2 x, + S 0,9 and x, 0 + S + 0,0 Q. J 2 Γ {R n 0, 0]} Q. 3.6 Now, assuming 3., we will prove L m B. To this end, by 3.3 and 3.4, it is enough to show that L m J k. On the other hand, since 3. yields J k L > θ L > 0, we have, n 0, + m+ ] L. By the definition of L m, we have 2 9 mk 3 9 m L m, n 0, 2 + m + 9 mk 3 9 m + m + ] j Setting l = min{k N L k+ R n 0, 0] = }, we have Since inf J 2 J 2 Γ l {R n 0, 0]}. u, by 3.2 again for l =, 2,..., km, we thus have which implies km < l. Hence, noting inf u > inf u Γ km {R n 0,0]} Γ l {R n 0,0]} t j 3 9 l 0, we have 2 2j 80 9 km which, together with 3.7, yields L m, n 0, ] 2 + m + 320m mk 3 9 m 9 mk 4

15 Therefore, noting 9 mk 3 9 m mk 9 mk 3 9 m, we can apply Lemma 3.3 to conclude the proof. It remains to show that 3.2 holds under 3.. By setting vx, t = M km ux j x, t j t, 3. implies {x, t K vx, t M} > θ K. 3.8 However, we note that v is a nonnegative L p -viscosity supersolution of v t + P + D 2 v + µ Dv f = 0 in Q, where µx, t = 2 j µx j x, t j t, fx, t = M km 4 j fx j x, t j t. We notice that f L p Q ε, and µ L q Q µ L q Q because q > n + 2 and p > n+2. Thus, if inf v holds, then the same argument to 2 K 2 obtain 3.6 yields {x, t K vx, t M} θ K, 3.9 which contradicts 3.8. Hence, we have v > in K 2, namely, 3.2 holds for l = by the definition. Next, for l 2, we suppose that 3.2 holds for l. We may suppose that N l R n 0, 0] since otherwise N l {R n 0, 0]} =, which concludes 3.2 for l. Thus, since inf u = inf u > M km l+, N l N l {R n 0,0]} we have a trivial inequality { x, t N l ux, t M km l+} = Nl > θ N l Set wx, t := ux M km l 0 + 3l x, t 2 j 0 + 9l that x 0, t 0 + 9l 0 3l +, 8 4 j 2 j 8 4 j + 9l 0 3l 2 j t. In view of 3.6, we easily see 4 j n ] 0 9l 0, Q. 4 j Hence, it follows that w > in K 2 because, if inf K2 w, then the above argument again implies 3.9 for w in place of v, which contradicts 3.20 for w. 5

16 3.2 A general case In order to show the weak Harnack ineuality without assuming 3.2, we use a new barrier function, which will be constructed in Lemma 3.7 below. Theorem 3.4. Let 2.4 hold, f L p +Q and µ L q +Q. There exist ε 0 = ε 0 n, Λ, λ, p, q, µ L q Q > 0 and C 0 = C 0 n, Λ, λ, p, q, µ L q Q > 0 such that any nonnegative L p -viscosity supersolution u of 3. satisfies u ε ε 0 0 dxdt C0 inf J J 2 u + f L p Q. 3.2 Remark 3.5. The constants ε 0, C 0 above depend on µ L q Q in a sense that even if we consider a different µ L q Q such that µ L q Q µ L q Q in place of µ in Theorem 3.4, the same conclusion holds true with the same constants as in Theorem 3.4. Remark 3.6. When µ L Q, φ in the next lemma can be given by a modified heat kernel from [23]. However, since we have unbounded µ, it is not possible to construct such a precise function for φ below. Lemma 3.7. Let q > n + 2 and µ L q +Q. There exist a nonnegative function φ CQ Wq 2, Q and a function g L q Q such that φ t + P + D 2 φ + µx, t Dφ gx, t a.e. in Q, φ 2 in K 2, φ = 0 on p Q, supp g K. Proof. Choose a nonnegative function ξ C Q such that ξ = 0 in Q \ K /4, where K /4 :=, 2 2 n 0, ] see Fig, and ξx, 0 > 0 for x, n. In view of Proposition 2.6, we can find a nonnegative function ψ CQ Wq 2, Q satisfying { ψt + P + D 2 ψ + µ Dψ = 0 a.e. in Q, ψ = ξ on p Q. We claim that there exists σ > 0 such that ψ σ in K 2. In fact, assuming ψx 0, t 0 = 0 for x 0, t 0 K 2, we will obtain a conradiction. For r 0, 0 ], we set v 0 x, t = ψx 0 + rx, t 0 + r 2 t 0 = 0 for x, t Q, v 0 0, 0 = 0 and the function v 0 is a solution of v 0 t + P + D 2 v 0 + µ Dv 0 = 0 in Q, 6

17 where Since it follows that if we choose r := δ 0 µ L p Q µx, t = rµx 0 + rx, t 0 + r 2 t 0. n+2 q µ L q Q, µ L p Q r q q n+2, where δ 0 is from Theorem 3. for p = q, then Theorem 3. yields v 0 = 0 in J. To continue the proof we will assume without loss of generality that t 0 =. If x 0 [ 4, 4 ]n, then Theorem 3. implies ψx 0, 0 = 0, which contradicts our choice of ψ. Thus, without loss of generality, it is enough to consider x 0 3, 3 n. Therefore, we can choose x 3, 3 n such that x 0 x + r, r n. x 0, 0rk 2{ x, x 2, 0rk 2 r k... x 3, 20rk 2 t x k, 0k rk K 2 K /4 0, 0 3 x 2 Figure 3. Setting r k = 5 for k + 5 i.e. r 2k 2r k r, if we fix k max{ 253, + 5 }, then 3 2r 0r 2 kk 3 4. Thus, using Theorem 3. finitely many times, we can find x k, 0k rk 2 [, 2 2 ]n [, ] such that ux 4 k, 0k rk 2 = 0. See Fig 3 for this procedure. Hence, by Theorem 3. again, we arrive at a contradiction. 7

18 Thererfore, for a large number M > 0, we verify that Mψ 2 in K 2. η C Q be a nonnegative function such that η = in Q \ K, and η = 0 in K /4. Now, let It is easy to observe that φ := Mηψ satisfies the desired properties. In fact, we may choose g = M[ψη t + P + ψd 2 η + 2Dη Dψ + µψ Dη ]. Remark 3.8. We notice that the global Wp 2, Q estimate of Proposition 2.6 is necessary to verify that g L p Q in the final step of the above proof. Proof of Theorem 3.4. For ε > 0, which will be fixed later, we set ũx, t = N 0 ux, t, where N 0 = inf J2 u + ε f L p Q + η for η > 0, which will be sent to 0 at the end of the proof. As in the proof of Theorem 3., it is enough to show that there are ε 0, C 0 > 0 such that 3.4 holds under assumptions 3.5. Let φ be the function from Lemma 3.7. By letting w := φ u, it is immediate to see that w is an L p -viscosity subsolution of w t + P D 2 w µ Dw h = 0 in Q, where h := g f. In view of Proposition 2.5, we have sup w C h L p Q + [w]. Q Hence, it is easy to verify that this inequality implies sup w C h L p Q + [w]. J 2 Recalling that supp g K in Lemma 3.7, we can find Ĉ = Ĉn, Λ, λ, p, q, µ L q Q > 0 such that Ĉ g L p Q + [w] K + ε. Thus, for some fixed ε > 0, there is θ 0, such that {x, t K wx, t > 0} θ K. Hence, as before, we obtain 3.6. We can follow the same arguments as those in the proof of Theorem 3. to conclude the proof. Remark 3.9. In the above proof, we have shown that there exist A 0, β 0, ε > 0 such that if u CQ is an L p -viscosity supersolution of 3. satisfying and if f L p Q ε, then 3.0 holds true. inf J 2 u, 8

19 4 Applications In this section, we consider L p -viscosity solutions of u t + Gx, t, Du, D 2 u fx, t = 0 in Q, 4. where Q = 0, 0 n 0, 0], and G : Q R n S n R and f : Q R are given. We assume the following hypotheses for G and f: { there exists µ L q Q for q > n + 2 such that Gx, t, ξ, O µx, t ξ for x, t Q and ξ R n 4.2, Remark 4.. We note that 4.2 yields f L p Q for p p, q]. 4.3 Gx, t, 0, O = 0 for x, t Q. Under 4.2 and 4.3, if we suppose that G satisfies 2.2, then it is easy to observe that if u CQ is an L p -viscosity subsolution resp., supersolution of 4., then it is an L p -viscosity subsolution resp., supersolution of u t + P D 2 u µ Du f = 0 resp., ut + P + D 2 u + µ Du f = 0 in Q Hölder continuity We show that the weak Harnack inequality for L p -viscosity supersolutions of 3. yields the Hölder continuity of solutions of 4. under the above hypotheses. This was remarked in [3] for elliptic PDE. For r 0,, we set Q r := 0r, 0r n 0 0r 2, 0]. Notice that Q 0r 0, 0 defined in section 2 is slightly different from this Q r. Theorem 4.2. Let G satisfy 2.2, 4.2 and 4.3. There exist C > 0 and α 0, such that if u CQ is an L p -viscosity solution of 4., then ux, t u x, t C x x 2 + t t α 2 u L Q + f L p Q for x, t, x, t Q. Proof. Working with extremal equations 4.4 and considering u := u u L Q + f L p Q 9 2

20 we can assume that u L Q and f L p Q. Fix r 0,. Setting M r := sup Qr u and m r := inf Qr u, we define ωr := M r m r for r 0,. It is easy to observe that for x, t Q, vx, t := M r urx, 0 + r 2 t 0 and wx, t := urx, 0+r 2 t 0 m r are nonnegative, L p -viscosity supersolutions of 3.. Hence, in view of Theorem 3.4, we find constants ε 0, C 0 > 0 such that U ε ε 0 0 dxdt C0 inf J J 2 U + r 2 n+2 p f L p Q for U = v and U = w, where ε 0, C 0 > 0 are the constants from Theorem 3.4. Setting a 0 = 2 n+2 and C p = 2 max{0, ε } 0 C 0 J ε 0, we have ωr = ωr ε ε 0 0 dx C inf v + inf w + r α 0 J J J 2 J 2 C ωr sup u + inf u + r α 0 J2 r J2 r where J2 r := r, r n 0 r 2, 0]. Since we may suppose C >, noting Q r J r 0 2, we have ω 0 r sup u inf u γωr + r α 0, J2 r J2 r where γ = C C. Therefore, in view of the standard argument e.g. Lemma 8.23 in [9], setting α = min{ log γ log 0, α 0} 0,, we conclude the proof. 4.2 Harnack inequality In order to prove the Harnack inequality we need the local maximum principle for L p - viscosity subsolutions of u t + P D 2 u µ Du f = 0 in Q. 4.5 Following the arguments of [4], we show that the weak Harnack inequality implies the local maximum principle. We note that to show Proposition 4.5 below, we can apply the arguments of [0], which is based on the standard one e.g. [9]. In this paper, we present a parabolic version of the method of [4] see also [5]. We first show a blow-up lemma., 20

21 Lemma 4.3. Let 2.4 hold and let f L p +Q and µ L q +Q. Assume that f L p Ω ε, where ε > 0 is the constant in the proof of Theorem 3.4. Suppose that v CQ is an L p -viscosity subsolution of 4.5 satisfying {x, t J vx, t s} A 0 s β 0 s 4.6 where β 0 > 0 and A 0 >. Then, there exist ν = νn, Λ, λ, p, q, β 0, A 0 >, n 0 = n 0 n, Λ, λ, p, q, β 0, A 0 N and l j = l j n, Λ, λ, p, q, β 0, A 0 0, for j N such that j= l j <, and if v satisfies vx 0, t 0 ν j for some j n 0 and x 0, t 0 J 3, then it follows that sup Q j v ν j, where J 3 = 2, 2 n 4, 2 ] and Q j = x 0, t 0 + l j, l j n l2 j 0, 0]. Remark 4.4. The constants A 0 and β 0 in Lemma 4.3 will be those in Remark 3.9. t x 0, t 0 Q j 2 J 3 } K J 4 0, 0 x Figure 4. Proof. We first fix ν := α > i.e. α = α ν, where ν α := 22A 0 β 0 >. Assume sup Qj v ν j. We will arrive at a contradiction provided l j := 22 β 0+ A 0 ν jβ 0 n Choose j 0 N such that l j 2 0 for j j 0. For j j 0, setting w j x, t = ν ν { ν j v x 0 + l j x, t 0 + l 2 jt 0 } 0 in Q, 2

22 we note that inf w j w j 0, 0 = ν { ν j vx 0, t 0 } J 2 ν and that w j is an L p -viscosity supersolution of w j t + P + D 2 w j + µ j Dw j f j = 0 in Q, where µ j x, t = l j µx 0 + l j x, t 0 + l 2 jt 0 and f j x, t = 0. Since it follows that n+2 µ j L q q Q = lj µ L q Q j, and f j L p Q = ν j l 2 j ν ν j fx 0 + l j x, t 0 + l 2 jt n+2 p ν l2 j f L p Q j, there exists an integer n 0 = n 0 n, λ, Λ, p, q, β 0, A 0 j 0 such that µ j L q Q µ L q Q and f j L p Q ε for j n 0. In view of Remark 3.9, we thus have { x, t J w j x, t } β0 2 2 α A 0 = α 2. Hence, we have { y, s Ĉj vy, s } 2 νj 2 ln+2 j, where Ĉ j = x 0, t 0 + l j, l j n 0l 2 j, 9 2 l2 j. On the other hand, since Ĉj J, by 4.6, we have { y, s Ĉj vy, s } 2 νj A 0 2ν j β 0. Thus, noting we have which contradicts 4.7. Ĉj = 2 n l n+2 j 2 ln+2 j + A 0 2ν j β 0, l j 2 β 0+ A 0 ν jβ 0 n+2, We can now show the local maximum principle for L p -viscosity subsolutions. 22

23 Proposition 4.5. Let 2.4 hold and let f L p +Q and µ L q +Q. Then, for any ε 0 0, β 0, there exists a constant C 3 = C 3 n, Λ, λ, p, q, µ L q Q, ε 0 > 0 such that any L p -viscosity subsolution u CQ of 4.5 satisfies sup u C 3 u + L ε 0 J + f L p Q J 3 where β 0 > 0 is the constant in Remark 3.9. Proof. Choose y 0, s 0 J 3 such that Setting N 0 = A 0 sup J 3 u = uy 0, s 0. J u + ε 0 dxdt ε 0 + 2ε f L p Q,, 4.8 where ε > 0 is from the proof of Theorem 3.4, we observe that v := N 0 u is an L p viscosity subsolution of We note that for s, we have v t + P D 2 v µ Dv N 0 f = 0 in Q. {x, t J vx, t s} s ε 0 J v ε 0 dxdt A 0 s ε 0. Let ν >, n 0 N and l j > 0 be the constants in Lemma 4.3 when β 0 = ε 0. There exists n n 0 such that j=n l j 4. Now, suppose that there is y 0, s 0 J 3 such that vy 0, s 0 ν n. In view of Lemma 4.3, for j N, we can find y j, s j y j, s j +[ l j+n, l j+n ] n [ l2 j+n 0, 0] such that vy j, s j ν n +j. Because y j, s j [ 3, ]n [, ], this contradicts that v CQ. Therefore, we conclude 8 2 the proof. Using the weak Harnack inequality, together with Proposition 4.5, we can obtain the Harnack inequality which we state without proof. 23

24 Corollary 4.6. Let 2.4 hold and let f L p Q and µ L q Q. There is a constant C 4 = C 4 n, Λ, λ, p, q, µ L q Q > 0 such that any nonnegative L p -viscosity solution u CQ of 4. satisfies sup u C 4 inf u + f L J 3 J p Q, where J 3 = 2, 2 n 4, 2 ] and J 2 =, n 9, 0]. 5 Remarks on the superlinear growth case In this section, we exhibit several properties of L p -viscosity solutions of 4., where G satisfies 2.2, 4.3 and, in place of 4.2, { there are m > and µ L q +Q for q > n + 2 such that Gx, t, ξ, O µx, t ξ m for x, t Q and ξ R n 5.. More precisely, we present a remark on the ABP maximum principle in [2], and an existence result corresponding to that in [4], with which we show the weak Harnack inequality for L p -viscosity supersolutions 4. under 5.. If 2.2, 4.3 and 5. are satisfied then if u CQ is an L p -viscosity subsolution resp., supersolution of 4., then it is an L p -viscosity subsolution resp., supersolution of u t + P D 2 u µ Du m f = 0 resp., ut + P + D 2 u + µ Du m f = 0 in Q. 5. A remark on the ABP maximum principle In this section, to comply with the setup of [2], Q = Ω 0, T ], where 0 < T and the domain Ω satisfies Ω {x R n x < }. 5.2 We recall the ABP maximum principle from [2]. The estimates there seem a little complicated. However, if we carefully examine them, we can give simple statements as below. Proposition 5.. Theorems 3. and 3.2 of [2] Let 2.4 hold with q < +. Let 2.2, 4.3 and 5. be satisfied and let p > m qn mq n 2 There exist δ = δn, Λ, λ, m, p, q > 0 and C = Cn, Λ, λ, m, p, q > 0 such that if u CQ is an L p -viscosity subsolution of 4., and f m L p Q µ L q Q δ,

25 then sup Q u sup pq u + C f L p Q. Remark 5.2. We note that 5.3 is satisfied when n + 2 p q, q > n + 2, and 5.3 is equivalent to mqn + 2 p < n + 2q p, which is iv of 5.6. We also remark that when q = +, the ABP maximum principle does not require any smallness condition and can be found in Theorems 3.7 and 3.8 of [2]. Contidion 5.3 then reduces to p > m n + 2/m, which is the inequality in i of 5.6. We show here that the smallness condition 5.4 can be removed, however the estimate becomes more complicated. Theorem 5.3. Let 2.4 hold with q < +. Let 2.2, 4.3, 5. and 5.3 be satisfied. There exists C = Cn, Λ, λ, m, p, q > 0 such that if u CQ is an L p -viscosity subsolution of 4., then sup Q u sup pq u + C + f m q L p Q µ q L q Q p p f L p Q. Proof. By considering u := u sup pq u, we may assume that sup pq u 0. When 5.4 does not holds, it is easy to see that we can find a partition 0 = t 0 < t < < t k = T such that, setting Q i := Ω [t i, t i ], i =,, k, and δ := f m L p Q δ, we have µ L q Q i δ for i =,, k, where k + δ q f m q L p Q µ q L q Q. By Proposition 5., we then have sup Q i u sup pq i u + C f L p Q i for i =,, k. Let x, t Q i satisfy sup Q u = u x, t for some i {,..., k}. Then But sup Q u sup pq i u + C f L p Q i max0, sup Q i u + C f L p Q i. sup u sup u + C f L p Q i max0, sup u + C f L p Q i. Q i pq i Q i 2 Therefore, continuing this procedure, we obtain sup Q u C k f L p Q i. i= 25

26 Now k i= f L p Q i k p p f L p Q + δ q f m q L p Q µ q L q Q p p f L p Q. 5.2 Existence of strong solutions In this subsection, for the sake of simplicity, Ω is as in 5.2 and we assume that Ω is C,. We discuss the existence of L p viscosity solutions of parabolic extremal PDE, u t + P ± D 2 u ± µ Du m = f in Q := Ω 0, ], 5.5 where m >, f L p Q and µ L q Q. Since we do not know a precise proof of Wp 2, -estimates near p Q of [23], possibly for p n +, though it was mentioned in [23] without a proof, we will use global estimates for p > n + from [8] to show a different type of estimates. Thus we will assume that p > n +. We first recall a global estimate for L p -strong solutions of extremal PDE with no first derivative terms. Proposition 5.4. e.g. Theorem. of [8] Let Ω be C, and p > n +. Then, there exists a constant C = Cn, p, Λ, λ, diamω, Ω > 0 such that for every f L p Q and ψ Wp 2, Q, there exists a unique u CQ Wp 2, Q such that { ut + P + D 2 u = f a.e. in Q, u = ψ on p Q, and u W 2, p Q C f L p Q + ψ W 2, p Q. For the elliptic case, in [4], the existence of L p -strong solutions of extremal PDE with superlinear growth in the first derivatives was obtained assuming that µ L q Q is small enough. Following the idea of [4], we establish the corresponding existence result for L p -strong solutions of 5.5. Theorem 5.5. Let Ω be C,, n + < p q, q > n + 2, f L p Q, µ L q Q and ψ Wp 2, Q. Assume that one of the following conditions holds: i n + < p < n + 2, mn + 2 p < n + 2, q =, ii p n + 2, q =, iii n + 2 < p = q <, iv n + < p < q, q > n + 2, mqn + 2 p < n + 2q p

27 Assume also that r = pm r = r = mpq q p for i, ii for iii, for iv. 5.7 Then, there exists δ = δ n, Λ, λ, p, q, m > 0 such that if m µ L q Q f L p Q + ψ W 2, p Q δ, 5.8 then there exist L p -strong solutions u Wp 2, Q of { ut + P ± D 2 u ± µ Du m = f a.e. in Q, u = ψ on p Q. 5.9 Moreover, there exists Ĉ = Ĉn, Λ, λ, p, q, m, diamω, Ω > 0 such that u W 2, p Q Ĉ f L p Q + ψ W 2, p Q. 5.0 Remark 5.6. We note that in iv of 5.6, if p n + 2, then the third inequality automatically holds. Proof. We will do the proof only for the case of P +. For r in 5.7, we define a mapping K : Wr,0 Q Wp 2, Q in the following way. For v Wr,0 Q, in view of Proposition 5.4, we find a unique solution u := Kv Wp 2, Q of { ut + P + D 2 u + µ Dv m = f a.e. in Q, u = ψ on p Q. Since Kv W 2, p Q C f L p Q + µ Dv m L p Q + ψ W 2, p Q holds for some C > 0, noting µ Dv m L p Q C µ L q Q Dv m L Q, r we can argue like in the proof of Theorem 3. of [4] to find a sufficiently large α and small δ > 0 such that if R = α f L p Q + ψ W 2, p Q, then K : B R Wp 2, Q B R is a continuous map when 5.8 holds, where { } B R = v Wr,0 Q v W,0 r Q R. Since Wp 2, Q is compactly imbedded in Wr,0 Q see the next proposition, we conclude the proof by the Schauder fixed point theorem as in [4]. For the reader s convenience, we provide a proof of compact imbeddings of parabolic Sobolev spaces. More general results for compact imbeddings of anisotropic Sobolev spaces can be found in [] and [2] see in particular Theorem 0.2 of [] and Theorem of [2]. 27

28 Proposition 5.7. Let Ω be Lipschitz. Assume that p r satisfy one of the following conditions: i p < n + 2, p r < pn+2, n+2 p ii p = n + 2 r <, 5. iii n + 2 < p <, r =. Then, Wp 2, Q is compactly imbedded in Wr,0 Q. Proof. Under assumption 5., by Lemma 3.3 of [7], it follows that there exist ε > 0 and C > 0 such that for any ε 0, ε, we have for u Wp 2, Q u L r Q + Du L r Q Cε α D 2 u L p Q + u t L p Q + Cε 2 α u L p Q, 5.2 where α = n+2 + n+2 > 0 for r <, or α = n+2 > 0 for r =. Here, C is p r p independent of u and ε 0, ε. A better inequality is true for u L r Q but we do not need it here. In view of 5.2, it is thus enough to show that a bounded subset of Wp 2, Q is compact in L p Q. However this is clear since Wp 2, Q Wp Q when we consider Q as a subset of R n+ and the mapping I : Wp Q L p Q is compact by the standard compact Sobolev imbedding theorem see e.g. Theorem 7.26 of [9]. Remark 5.8. We remark that for case 5.-iii a stronger result is true, namely that Wp 2, Q is compactly imbedded in the parabolic space C +α Q for α = n+2. p 5.3 Weak Harnack inequality Using Theorem 5.5, we establish the weak Harnack inequality for L p -viscosity supersolutions of uniformly parabolic PDE with superlinear growth in Du. We refer to [4] for an analogous elliptic result. In this subsection, we again set Q := 0, 0 n 0, 0]. In what follows, we will utilize the same notation as that in Figure. We will construct a barrier function for 5.5 when m >. This will require a slightly more careful analysis than that in the elliptic case. Lemma 5.9. Assume that 2.4 holds. Then, there exists δ 2 = δ 2 n, Λ, λ, q, m > 0 such that if µ L q Q satisfies µ L q Q δ 2, then there exist φ Wq 2, Q CQ and g L q Q such that φ t + P + D 2 φ + µ Dφ m g a.e. in Q, φ 2 in K 2, φ = 0 in p Q, supp g K

29 Proof. We first introduce a smooth, nonnegative η : Q [0, ] satisfying i ηx, t = for x, t Q if x or t, ii iii ηx, 0 = 0 for x, 2 η W 2, Q. 5.4 We next choose a nonnegative function ξ 0 C R n [0, such that { i ξ0 = 0 in R n [0, \ {x, t R n [0, 4 x < 2 }, ii ξ 0 x, 0 > 0 for x < As in the proof of Lemma 4. in [4], we claim that there exist δ2 0 > 0 and σ > 0 such that if µ L q Q satisfies µ L q Q δ2, 0 then the strong solution ψ Wq 2, Q of { ψt + P + D 2 ψ + µ Dψ m = 0 a.e. in Q, ψ = ξ 0 on p Q 5.6 satisfies ψ σ in K 2. Indeed, otherwise, there are nonnegative ψ k Wq 2, Q CQ and µ k L q Q such that µ k L q Q and ψ k k is a strong solution of 5.6 with µ replaced by µ k, such that inf K2 ψ k, then by 5.0 a subsequence {ψ k k j } j= converges uniformly in Q to some ψ Wq 2, Q, and inf K2 ψ = 0. Since ψ is a strong solution of { ψt + P + D 2 ψ = 0 a.e. in Q, ψ = ξ 0 on p Q, we can find x, t K 2 such that ψ x, t = 0, which gives a contradiction as in the proof of Lemma 3.7. We now choose δ 2 > 0 small enough so that 4σ m δ 2 δ 0 2 and 4σ m δ 2 ξ 0 m W 2, p Q δ, where δ is from Theorem 5.5. In view of Theorem 5.5 and the above choice of δ 2, if µ L q Q satisfies µ L q Q δ 2, then there exists ψ 0 CQ Wq 2, Q such that { ψ 0 t + P + D 2 ψ 0 + 4σ η m µ Dψ 0 m = 0 a.e. in Q, ψ 0 = ξ 0 on p Q, and ψ 0 σ in K 2. Setting ψ = 2/σψ 0 and ξ = 2/σξ 0, we observe that { ψt + P + D 2 ψ + 2η m µ Dψ m = 0 a.e. in Q, ψ = ξ on p Q. 29

30 Furthermore, it is easy to check that φ := ηψ satisfies φ t + P + D 2 φ + µ Dφ m g a..e. in Q, where g = ψη t + 2 m µ ψdη m + P + Dη Dψ + Dψ Dη + ψd 2 η, and φ and g satisfy all the conditions required in 5.3. We will now show that the weak Harnack inequality holds under a smallness condition. Since we separate the weak Harnack inequality from the L -estimate, similarly to Theorem 4.2 in [4], we assume boundedness of supersolutions. Theorem 5.0. Suppose that 5.6 holds and assume < m < 2 n q Let M 0, f L p +Q and µ L q Q. Then, there exist δ 3 = δ 3 n, λ, Λ, p, q, m, M > 0, C = Cn, λ, Λ, p, q, m > 0 and ε 0 = ε 0 n, λ, Λ, p, q, m > 0 such that if µ L q Q + f m L p Q < δ 3, 5.8 and u CQ is an L p -viscosity supersolution of u t + P + D 2 u + µ Du m + f = 0 in Q satisfying 0 u M in Q, then u ε ε 0 0 dxdt C inf J J 2 u + f L p Q. Proof. The proof follows the arguments of the proof of Theorem 4.2 of [4] so we just sketch it. We first reduce to the case of f = 0. Let δ be from Theorem 5.5 and let µ L q Q2 f L p Q m δ. We notice that if 5.7 holds then 5.6 is satisfied. Let w Wp 2, Q be from Theorem 5.5 such that { wt + P D 2 w 2 m µ Dw m f = 0 a.e. in Q, w = 0 on p Q. By Theorem 5.3, we have 0 w C f L p Q,

31 and it is easy to see that v := u + w is an L p -viscosity supersolution of v t + P + D 2 v + 2 m µ Dv m = 0 in Q Thus, if we can prove that J v ε 0 dxdt ε 0 C inf J 2 v, the claim will follow using 5.9. Thus we can assume that f = 0. We now set m 0 := inf J2 u. We may suppose m 0 > 0 by adding a positive constant, which will be sent to 0 in the end. Considering v := m 0 u, we verify that inf J v, and it is an L p -viscosity supersolution of In view of Lemma 5.9, if v t + P + D 2 v + m m 0 µ Dv m = 0 in Q. 2M m µ L q Q δ 2, where δ 2 > 0 is from Lemma 5.9, we can find a strong solution φ Wq 2, Q of φ t + P + D 2 φ + 2m 0 m µ Dφ m g a.e. in Q, φ = 0 on p Q, φ 2 in K 2, supp g K. Then w := φ v is an L p -viscosity subsolution of Hence, Theorem 5.3 yields w t + P D 2 w 2m 0 m µ Dw m g = 0 in Q. sup J 2 w sup Q w C g L p {x,t K φ vx,t 0}, where C is a constant which depends on various absolute constants, δ 2, and g L p Q, which is also bounded by various absolute constants. The above inequality now implies {x, t K vx, t > M} θ K for some M > and θ 0,. The rest of the proof follows the arguments in the proof of Theorem 4.2 of [4]. References [] O.V. Besov, V.P. Il in and S.M. Nikol skii, Integral representations of functions and imbedding theorems. Vol. I. Translated from the Russian. Scripta Series in Mathematics. Edited by Mitchell H. Taibleson. V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London,

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