Remarks on Regularity for $ P $-Laplacian Type Equations in Non

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Birindelli and Demengel [11] and Imbert and Silvestre [34] proved C1,α regularity results for related equations. In [11, Proposition 3.6] the authors proved Hölder regularity of the gradient for solutions of (1.1) for γ 0 and p 2 by using approximations and a fixed point argument. In≤ [34] the authors≥ used improvement of flatness to show local C1,α regularity in the degenerate case γ > 0 for viscosity solutions of the equation Du γ F (x, D2u) = f, where F is a uniformly elliptic operator, and the |result| was extended to the full range γ > 1 in [13]. Notice that this result does not cover equation (1.1) because of discontinuous− gradient dependence. In [2] with Parviainen, we showed local C1,α regularity of solutions of (1.1) when γ = 0 and p > 1. For the special case of radial solutions, Birindelli and Demengel [12] showed C1,α regularity of solutions of (1.1) for γ > 1 and p> 1. − In this paper we extend these regularity results to viscosity solutions of (1.1) when γ > 1 and p > 1, and provide some results on the existence and the integrability− of the second derivatives. Our first result concerns the interior regularity of the gradient. Theorem 1.1. Let γ > 1, p > 1, and f L∞(Ω) C(Ω). Then there exists α = α(p, n, γ) > 0−such that any viscosity∈ solution∩ u of (1.1) is in C1,α(Ω), and for any Ω′ Ω, loc ⊂⊂ ′ [u] 1,α ′ C = C p,n,d,γ,d , u ∞ , f ∞ , C (Ω ) ≤ || ||L (Ω) || ||L (Ω) where d = diam(Ω) and d′ = dist(Ω ′, ∂Ω). An iterative argument leads us to analyze compactness and regularity of deviations of solutions u from planes, w(x) = u(x) q x for some q Rn, and the key estimate for these deviations is called improvem− · ent of flatness,∈ Lemma 3.6. A key step in the proof is to obtain Arzelà-Ascoli type compactness for deviations u(x) q x. The main difficulty is that for γ = 0, the ellipticity constants of the− equation· satisfied by w depend on q .6 To overcome this problem, we use both Ishii-Lions’ method and Alexandrov-| | Bakelman-Pucci (ABP for short) estimate in the proofs of Lemmas 3.2, 3.1, and 3.3. Our second result concerns integrability of the second derivatives when γ is negative and the range of p is restricted. 2 ∞ Theorem 1.2. Let γ ( 1, 0], p (1, 3+ n−2 ), and f L (Ω) C(Ω). ∈ − ∈ 2,2∈ ∩ Then any viscosity solution u of (1.1) belongs to Wloc (Ω), and for any Ω′′ Ω′ Ω, we have ⊂⊂ ⊂⊂ ′ ′′ u 2,2 ′′ C = C(p,n,γ,d , d , u ∞ , f ∞ ), (1.2) || ||W (Ω ) ≤ || ||L (Ω) || ||L (Ω) where d′ = diam(Ω′) and d′′ = dist(Ω′′, ∂Ω′). The proof starts from the observation that when γ 0, viscosity solutions of (1.1) are viscosity solutions of the inhomogeneous≤ normalized p-Laplace p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 3

N −γ equation with a drift, ∆p u = f Du , see Lemma 2.5. This equation is singular but uniformly− elliptic. The| main| idea is to regularize the equation (see equation (4.6)) and use the Cordes condition, see Theorem 4.1. This condition guarantees a uniform estimate for solutions of regularized problems, which together with equi-H¨older continuity and compactness argument gives a local W 2,2 estimate for solutions of equation (1.1). Obtaining integrability for second derivatives in the case γ > 0 is harder because of the degeneracy of equation (1.1). However, for a small γ and p close to two, we obtain local W 2,2 estimate by again using an approximation with uniformly elliptic regularized problems, and considering equation (1.1) as a perturbation of the (γ + 2)-Laplacian.

Theorem 1.3. Assume that f W 1,1(Ω) L∞(Ω) C(Ω) and γ (0, β], where β (0, 1) and ∈ ∩ ∩ ∈ ∈ 1 β √n p 2 γ β(p 2 γ)+ > 0. (1.3) − − | − − |− − − 2,2 Then any viscosity solution u of (1.1) belongs to Wloc (Ω), and for any Ω′′ Ω′ Ω, we have ⊂⊂ ⊂⊂ ′ ′′ u 2,2 ′′ C(p,n,γ,β,d , d , u ∞ , f ∞ , f 1,1 ), (1.4) || ||W (Ω ) ≤ || ||L (Ω) || ||L (Ω) || ||W (Ω) where d′ = diam(Ω′) and d′′ = dist(Ω′′, ∂Ω′).

This paper is organized as follows. In Section 2 we deﬁne viscosity solutions for the problem (1.1). In Section 3 we prove Theorem 1.1, and in Section 4 Theorems 1.2 and 1.3.

Acknowledgment. A.A. is supported by the Academy of Finland, project number 307870.

2. Preliminaries and definitions of solutions In this section we define viscosity solutions of equation (1.1) and fix the γ N notation. For γ > 0, the operator Du ∆u + (p 2)∆∞u is continuous, and we can use the standard definition| | of viscosity− solutions, see [36]. For γ 0, the operator (1.1) is undefined when Du = 0, where it has a bounded discontinuity≤ when γ = 0 and is very singular when γ < 0. This can be remediated adapting the notion of viscosity solution using the upper and lower semicontinuous envelopes (relaxations) of the operator, see [23]. The definition for viscosity solutions for the normalized p-Laplacian (γ = 0) is the following.

Deﬁnition 2.1. Let Ω be a bounded domain and 1

N ∆p φ(x0) f(x0), if Dφ(x0) = 0, − ≤ 2 6  ∆φ(x0) (p 2)λmax(D φ(x0)) f(x0), if Dφ(x0) = 0 and p 2, − − − 2 ≤ ≥  ∆φ(x0) (p 2)λmin(D φ(x0)) f(x0), if Dφ(x0) = 0 and 1

N ∆p φ(x0) f(x0), if Dφ(x0) = 0, − ≥ 2 6  ∆φ(x0) (p 2)λmin(D φ(x0)) f(x0), if Dφ(x0) = 0 and p 2, − − − 2 ≥ ≥  ∆φ(x0) (p 2)λmax(D φ(x0)) f(x0), if Dφ(x0) = 0 and 1 1. A lower semicontinuous − function u is a viscosity super-solution of (1.1) in Ω, if for every x0 Ω one of the following conditions hold. ∈ p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 5

2 i) Either for all φ C (Ω) such that u φ has a local minimum at x0 and Dφ(x ) = 0, we∈ have − 0 6 Dφ(x ) γ ∆N φ(x ) f(x ). −| 0 | p 0 ≥ 0 ii) Or there is an open ball B(x , δ) Ω, δ > 0 such that u is constant in 0 ⊂ B(x0, δ) and 0 f(x) for all x B(x , δ). ≥ ∈ 0 An upper semicontinuous function u is a viscosity sub-solution of (1.1) in Ω if for all x Ω one of the following conditions hold. 0 ∈ i) Either for all φ C2(Ω) such that u φ reaches a local maximum at x and Dφ(x ) =∈ 0, we have − 0 0 6 Dφ(x) γ ∆N φ(x)) f(x ). (2.3) − | | p ≤ 0 ii) Either there exists an open ball B(x , δ) Ω, δ > 0 such that u is 0 ⊂ constant in B(x0, δ) and 0 f(x) for all x B(x , δ). (2.4) ≤ ∈ 0 Proposition 2.4. Definitions 2.2 and 2.3 are equivalent. Proof. We show first that if u is a supersolution of (1.1) in the sense of Definition 2.2, then it is a supersolution in the sense of Definition 2.3. The case i) is immediate, so we only need to consider the case where u is constant in a ball B(x0, δ). We see that for all y B(x0, δ), the function φ(x) = γ + 2 ∈ u(y) x y q with q > > 2 is a smooth test function with an isolated −| − | γ + 1 critical point, and it holds γ N lim sup Dφ(z) ∆p φ(z) = 0, r→0 z∈Br(y), − | | z6=y so that f 0 in B(x0, δ), and the statement follows. Now we≤ assume that u is a supersolution in the sense of Definition 2.3. Let x Ω and φ an admissible test function such that u φ has a local 0 ∈ − minimum at x0. We assume without a loss of generality, that x0 is a strict local minimum in B(x , η). If Dφ(x ) = 0, then the conclusion is immediate. 0 0 6 If Dφ(x0)=0 and φ has only one critical point in a neighborhood B(x0, r0) of x0, we have to consider two cases: either u is constant or not. If u is constant in a small ball around x0, then u is smooth and it follows that D2φ(x ) 0. Using that 0 ≤ ∆N φ(x) ∆φ(x) (p 2)λ (D2φ(x)) if p 2, − p ≥− − − max ≥ ∆N φ(x) ∆φ(x) (p 2)λ (D2φ(x)) if p< 2, − p ≥− − − min we get that for p 2 ≥ N 2 lim sup ∆p φ(x) lim sup ∆φ(x) (p 2)λmax(D φ(x)) 0, r→0 r→0 x∈Br(x0), − ≥ x∈Br(x0), − − − ≥ x=6 x0 x6=x0 6 ATTOUCHIANDRUOSTEENOJA and for 1

Next we consider the case where u is not constant in any ball around x0. We use the argument of [27, Lemma 2.1]. For y B(0, r) where r > 0 is ∈ small enough, we consider the function φy(x) = φ(x + y). Then we have that u φy reaches a local minimum at some point xy in B(x0, η). We can show that− there exists a sequence y 0 such that Dφ (x ) = 0 for all k. k → yk yk 6 Testing the equation at xyk we get that for k large enough γ N γ N sup Dφ(z) ∆p φ(z) Dφyk (xyk ) ∆p φyk (xyk ) f(xyk ). z∈Br(x0), − | | ≥ −| | ≥ z6=x0 Letting r 0 and k , we get that → →∞ γ N lim sup Dφ(x) ∆p φ(x) f(x0). r→0 x∈Br(x0), − | | ≥ x6=x0

If such a sequence (yk) does not exist, then it follows by the property that x0 is the only critical point of φ in a small neighborhood of x0, that for all y B(0, r), we must have xy + y = x0 and also that for all y B(0, r) and x ∈ B(x , η) we have ∈ ∈ 0 u(x y) φ(x )= u(x ) φ (x ) u(x) φ (x)= u(x) φ(x + y). (2.5) 0 − − 0 y − y y ≤ − y − That is, u(x y) u(x) φ(x ) φ(x + y). (2.6) 0 − − ≤ 0 − For a vector z B(0, r) and a unit vector e , taking x = x + z and ∈ i 0 y = z tei (resp. x = x0 + z + tei and y = z) in (2.6), dividing by t> 0 and− taking− lim sup (resp. lim inf), we see that− the directional derivatives of u near x0 vanish everywhere in the neighborhood. This implies that u is constant around x0, which is a contradiction. p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 7

We can see from the proof that the three definitions are also equivalent when γ = 0. When the operator is continuous (γ > 0), then Definitions 2.2 and 2.3 are equivalent with the usual definition of viscosity solution (see [27, Lemma 2.1]). Next we observe that the singular case can be reduced to the study of the normalized p-Laplacian problem with a lower order term. Lemma 2.5. Let γ ( 1, 0). Assume that u is a viscosity solution to (1.1). Then u is a viscosity∈ solution− to ∆N u = f Du −γ. (2.7) − p | | Proof. It is sufficient to prove the supersolution property since the sub- solution property is similar. We give the proof for p > 2 the other case being analogous. Let φ be a test function touching u strictly from below at x . The case Dφ(x ) = 0 is immediately clear, so we focus on the case 0 0 6 Dφ(x0) = 0. 2 We distinguish two cases: either D φ(x0) is invertible or not. If it is, then x0 is the only critical point of φ in a small ball around x0. In this case using the definition of x0, we have that for all ε> 0 there exist r0 such for r r0 there exist x x such that Dφ(x ) = 0 and ≤ r → 0 r 6 Dφ(x ) γ ∆N φ(x ) f(x ) ε, −| r | p r ≥ 0 − hence ∆N φ(x ) (f(x ) ε) Dφ(x ) −γ . − p r ≥ 0 − | r | It follows that ∆φ(x ) (p 2)λ (D2φ(x )) (f(x ) ε) Dφ(x ) −γ . − r − − min r ≥ 0 − | r | Letting ε, r 0, we get → ∆φ(x ) (p 2)λ (D2φ(x )) 0= f(x ) Dφ(x ) −γ . − 0 − − min 0 ≥ 0 | 0 | 2 Now we suppose that D φ(x0) is not invertible. Then we take a symmetric 2 matrix B, semi-positive definite such that D φ(x0) ηB is invertible, for all η> 0. We consider the test function − φ (x)= φ(x) η(x x )B(x x ). η − − 0 − 0 Applying the previous argument we have ∆φ (x ) (p 2)λ (D2(φ (x )) 0= f(x ) Dφ (x ) −γ . − η 0 − − min η 0 ≥ 0 | η 0 | Passing to the limit η 0 we get the desired result. → The following lemma will be useful in the next section concerning the proof of the improvement of flatness. Lemma 2.6. Let γ > 1 and p> 1. Assume that w is a viscosity solution of − D2w(Dw + q), (Dw + q) Dw + q γ ∆w + (p 2)h i = 0. −| | − Dw + q 2 | | 8 ATTOUCHIANDRUOSTEENOJA

Then w is a viscosity solution of D2w(Dw + q), (Dw + q) ∆w (p 2)h i = 0. − − − Dw + q 2 | | Proof. We can reduce the problem to the case q = 0 since v = w q x γ N − · solves Dv ∆p v = 0. It is suﬃcient to prove the super-solution property since the−| sub-solution| property is similar. Let φ be a test function touching u strictly from below at x0. We use the fact that for the homogeneous case and for 1

3. Local Holder¨ regularity of the gradient In this section we give a proof for Theorem 1.1. We assume that γ ( 1, ), p > 1 and f L∞(Ω) C(Ω), and we want to show that there∈ exists− ∞α = α(p, n, γ) >∈0 such that∩ any viscosity solution u of (1.1) is in C1,α(Ω), and for any Ω′ Ω, loc ⊂⊂ ′ [u] 1,α ′ C = C p,n,γ,d,d , u ∞ , f ∞ , C (Ω ) ≤ || ||L (Ω) || ||L (Ω) where d = diam(Ω) and d′ = dist(Ω′, ∂Ω).

A standard method to investigate the regularity of solutions is through their approximations by linear functions. The goal is to get good estimates of the error of these approximations using a compactness method and the scaling properties of the equation. We will use the following characterization of C1,α functions: there exists a positive constant C such that for any x Ω and r> 0, there exists a vector l for which ∈ osc (u(y) u(x) l (x y)) Cr1+α. (3.1) y∈Br(x) − − · − ≤ By rescaling we may assume osc u 1, and by proceeding by iteration it is enough to ﬁnd ρ (0, 1) such that inequality≤ (3.1) holds true for r = r = ρk, ∈ k l = lk and C = 1. The balls Br(x) for x Ω and r < dist (x, ∂Ω) covering the domain Ω, we may work on balls. Moreover,∈ by using a translation argument and the following scaling,

γ+2 − +1 ur(y)= r γ u(x + ry), we may work on the unit ball B1(0) and prove the regularity only at the origin. Considering u u(0) if necessary, we may suppose that u(0) = 0. We also reduce the− problem by rescaling. Let −1 1/(1+γ) −1 κ = (2 u ∞ + (ε f ∞ ) ) . || ||L (B1) 0 || ||L (B1) Settingu ˜ = κu, thenu ˜ satisfies Du˜ γ ∆N (˜u)= f˜ −| | p p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 9 with 1 u˜ ∞ and f˜ ∞ ε . || ||L (B1) ≤ 2 || ||L (B1) ≤ 0 ∞ Hence, without loss of generality we may assume that u L (B1) 1/2 ∞ || || ≤ and f L (B1) ε0, where ε0 = ε0(p, n, γ) is chosen later. To prove the inductive|| || step,≤ the strategy is to study the deviations of u from planes, w(x)= u(x) q x, which satisfy in B − · 1 Dw + q Dw + q Dw + q γ ∆w + (p 2) D2w , = f (3.2) − | | − Dw + q Dw + q | | | | in the viscosity sense. The existence of the vector qk+1 can be reduced to prove an ”improvement of flatness” for solutions of (3.2). The proof of the improvement of flatness consists of two steps. First we show equicontinuity for uniformly bounded solutions of (3.2) in Lemma 3.1 and Lemma 3.3. Next, by the Arzelà-Ascoli theorem we get compactness, which, together with the regularity of the limiting solutions (Lemma 3.5), allow to show improvement of flatness for solutions of (3.2) in Lemma 3.6 via a contradiction argument. Finally, we prove C1,α regularity for solutions of (1.1) in Lemma 3.7 by using Lemma 3.6 and iteration.

3.1. Equicontinuity for deviations from planes. First, we need to prove some compactness result for the deviations from planes. We will provide H¨older regularity results independently of q using diﬀerent arguments for large and small slopes. In the next lemmas we use the following notation for Pucci operators: +(X):= sup tr(AX) M A∈Aλ,Λ − and −(X) := inf tr(AX), M A∈Aλ,Λ − n where λ,Λ S is a set of symmetric n n matrices whose eigenvalues belongA to [λ,⊂Λ]. ×

−γ Lemma 3.1. Let γ ( 1, 0] and p (1, ). Assume that q f ∞ ∈ − ∈ ∞ | | || ||L (B1) ≤ a0(p, n, γ). Let w be a viscosity solution of (3.2) in B1. For all r 3 ∈ (0, 4 ) there exist a constant β = β(p, n, γ) (0, 1), and a constant C = C(p, n, γ) > 0 such that for all x,y B , ∈ ∈ r 1 1+γ β w(x) w(y) C 1+ w ∞ + f ∞ x y . (3.3) | − |≤ || ||L (B1) || ||L (B1) | − | Proof. For 1 < γ 0, we observe that − ≤ Dw + q −γf C f Dw −γ + C q −γ f | | ≤ | || | | | | | 1 Dw + C q −γ f + C f 1+γ . ≤ | | | | | | | | 10 ATTOUCHIANDRUOSTEENOJA

It follows that w satisfies +(D2w)+ Dw + g 0 M−(D2w) |Dw| |g|≥ 0 M − | | − | |≤ 1 where g(x) := C( f(x) q −γ + f(x) γ+1 ) is a bounded function (uniformly | | | || | | | 1 γ+1 with respect to q ) satisfying g ∞ C(a0 + f ∞ ). Then the | | || ||L (B1) ≤ || ||L (B1) Hölder estimate for w follows from the classical theory of uniformly elliptic equations. Lemma 3.2. Let γ (0, ) and p (1, ). Assume that q Γ = ∈ ∞ ∈ ∞ | | ≥ 0 Γ0(p, n, γ) 1. Let w be a viscosity solution of (3.2) in B1 with oscB1 (w) ≥ 3 ≤ 1 and f ∞ 1. For all r (0, ) there exist a constant β = || ||L (B1) ≤ ∈ 4 β(p, n, γ) (0, 1) and a constant C = C(p, n, γ) > 0 such that for all x,y B , ∈ ∈ r w(x) w(y) C x y β . (3.4) | − |≤ | − | For the proof, see Appendix A. The following lemma provides a uniform Hölder estimate for small slopes. Lemma 3.3. Let γ (0, ) and p (1, ). For all r (0, 3/4), there exist a constant β = β(p,n∈) ∞(0, 1) and∈ a positive∞ constant∈C = C(p, n, γ) such ∈ that any viscosity solution w of (3.2) with osc (w) 1, f ∞ 1 B1 ≤ || ||L (B1) ≤ and q Γ =Γ (p, n, γ), satisfies | |≤ 0 0 [w] 0,β C. (3.5) C (Br) ≤ −γ Proof. If Dw 2Γ0 then Dw+q Dw q Γ0 1 and Dw+q 1. Hence| we have|≥ | | ≥ || |−| || ≥ ≥ | | ≤ +(D2w)+ f 0 M−(D2w) |f|≥ 0. M − | |≤ Using the result of [35] (see also [28]), there exists β = β(p,n) (0, 1) such that ∈

0,β [w]C (Br) C = C p, n, r, osc(w), f Ln(B1) . ≤ B1 || || !

As a consequence of Lemma 3.2 and Lemma 3.3, we have the following uniform H¨older estimate. Lemma 3.4. Let γ (0, ), p (1, ) and f L∞(B ) C(B ). Let w be ∈ ∞ ∈ ∞ ∈ 1 ∩ 1 a viscosity solution to (3.2) with osc (w) 1, f ∞ 1. Then, for B1 ≤ || ||L (B1) ≤ all r (0, 3/4), there exist a constant β = β(p, n, γ) (0, 1) and a positive constant∈ C = C(p, n, γ) such that w satisﬁes ∈

[w] 0,β C. (3.6) C (Br) ≤ p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 11

3.2. Proof of the improvement of flatness. The next lemma gives uniform Hölder estimates for the limit equation independent of q , and is needed in the contradiction argument in the proof of the key Lemma| | 3.6, where we show improvement of flatness. We refer the reader to [2, Lemma 3.2]. Lemma 3.5. Let v be a viscosity solution of Dv + q Dv + q ∆v (p 2) D2v , = 0 in B (3.7) − − − Dv + q Dv + q 1 | | | | 1 with osc v 1. For all r (0, 2 ], there exist constants C0 = C0(p,n) > 0 B1 ≤ ∈ and β1 = β1(p,n) > 0 such that

[v] 1,β C . (3.8) C 1 (Br) ≤ 0 Lemma 3.6. Let γ ( 1, ) and p (1, ). When γ < 0 suppose that −γ ∈ − ∞ ∈ ∞ f ∞ q a (p, n, γ). Then there exist ε = eps (p, n, γ) (0, 1) || ||L (B1) | | ≤ 0 0 0 ∈ and ρ = ρ(p, n, γ) (0, 1) such that, for any q Rn and any viscosity solu- ∈ ∈ −γ tion w of (3.2) with osc (w) 1, f ∞ ε and q f ∞ B1 ≤ || ||L (B1) ≤ 0 | | || ||L (B1) ≤ c (p, n, γ)ε when γ < 0, there exists q′ Rn with q′ C¯(p, n, γ) such that 0 0 ∈ | |≤ 1 osc (w(x) q′ x) ρ. x∈Bρ − · ≤ 2 Proof. We use a contradiction argument. Assume that there exist a sequence ∞ of functions (fj) with fj L (B1) 0, a sequence of vectors (qj) such that −γ || || → q f ∞ 0 when γ < 0 and a sequence of viscosity solutions (w ) | j| || j||L (B1) → j with osc (w ) 1 of B1 j ≤ Dw + q Dw + q Dw +q γ ∆w + (p 2) D2w j j , j j = f , (3.9) −| j j| j − j Dw + q Dw + q j | j j| | j j| such that, for all q′ Rn and any ρ (0, 1) ∈ ∈ ′ ρ osc (wj(x) q x) > . (3.10) x∈Bρ − · 2 Relying on the compactness result of Lemma 3.1 and Lemma 3.4, there exists a continuous function w∞ such that wj w∞ uniformly in Bρ for any ρ (0, 3/4). Passing to the limit in (3.10),→ we have that for any vector q′, ∈ ′ ρ osc (w∞(x) q x) > . (3.11) x∈Bρ − · 2

We treat separately the cases where the sequence (qj) is bounded or unbounded. Suppose ﬁrst that the sequence (qj) is bounded. Using a compactness argument and relaxed limits, we extract a subsequence (wj) converging to a limit w∞, which satisﬁes in B1

γ Dw∞ + q∞ Dw∞ + q∞ 2 Dw∞ +q∞ tr I + (p 2) D w∞ = 0 −| | − Dw∞ + q∞ ⊗ Dw∞ + q∞ | | | | (3.12) 12 ATTOUCHIANDRUOSTEENOJA in a viscosity sense. (Here qj q∞ up to the same subsequence.) Using the result of Lemma 2.6, we know→ that viscosity solutions to (3.12) are viscosity solutions to (3.7) and using the regularity result of Lemma 3.5, there exist β1 = β1(p,n) > 0 and C0 = C0(p,n) > 0 such that

w∞ 1,β1 C0. || ||C (B1/2) ≤ If the sequence (qj) is unbounded, we can extract a converging subse- qj −γ quence from ej = such that ej e∞. Multiplying (3.9) by qj and |qj | → | | passing to the limit, we obtain ∆w (p 2) D2w e , e =0 in B , (3.13) − ∞ − − ∞ ∞ ∞ 1 with e = 1. Noticing that equation (3.13) can be written as | ∞| tr ((I + (p 2)e e )D2w ) = 0, − − ∞ ⊗ ∞ ∞ we see that equation (3.13) is uniformly elliptic with constant coeﬃcients and linear. Using the regularity result of [15, Corollary 5.7], there is β2 = β (p,n) > 0 so that w C1,β2 and there exists C = C (p,n) > 0 such 2 ∞ ∈ loc 0 0 that w∞ 1,β2 C0. || ||C (B1/2) ≤ 1,β We have thus shown that in both cases w∞ Cloc for β = min(β1, β2) > 0. Choose ρ (0, 1/2) such that ∈ ∈ 1 C ρβ . (3.14) 0 ≤ 4 1,β By Cloc regularity, there exists a vector kρ with kρ C(p, n, γ) such that | | ≤ 1+β 1 osc (w∞(x) kρ x) C0ρ ρ. (3.15) x∈Bρ − · ≤ ≤ 4 This contradicts (3.11) so the proof is complete. Once we have proved the improvement of ﬂatness for deviations from planes, the proof of Theorem 1.1 proceeds by standard iteration. Lemma 3.7. Let ρ and ε (0, 1) be as in Lemma 3.6 and let u be a viscos- 0 ∈ ity solution of (1.1) with γ > 1, p> 1, osc (u) 1 and f ∞ ε . − B1 ≤ || ||L (B1) ≤ 0 Then, there exists α 0, 1 such that for all k N, there exists q Rn ∈ 1+γ ∈ k ∈ such that i 1+α osc (u(y) qk y) rk , (3.16) y∈Brk − · ≤ where r := ρk and q q Crα with C = C(p, n, γ). k | k+1 − k|≤ k

Proof. For k = 0, the estimate (3.16) follows from the assumption oscB1 (u) 1 α ≤ 1. Next we take α (0, min(1, γ+1 )) such that ρ > 1/2. We assume for ∈ n k 0 that we already constructed qk R such that (3.16) holds true. To prove≥ the inductive step k k + 1, we∈ rescale the solution considering for x B → ∈ 1 w (x)= r−1−α u(r x) q (r x) . k k k − k · k p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 13

By induction assumption, we have osc (wk) 1, and wk satisﬁes B1 ≤ Dw + ξ Dw + ξ Dw + ξ γ ∆w + (p 2) D2w k k , k k = f , − | k k| k − k Dw + ξ Dw + ξ k | k k| | k k| 1−(γ+1)α α ∞ where ξk = (qk/rk ) and fk(x) = rk f(rkx) with fk L (B1) ε0 1 || || ≤ −γ ∞ since α< γ+1 . Moreover, for 1 <γ< 0, we have that ξk fk L (B1) 1−α −γ − | | || || ≤ f ∞ r q ε c (p, n, γ). Here we used that, by assumption it || ||L (B1) k | k| ≤ 0 0 holds k k−1 −1 q q q C ραi c γ (p, n, γ). | k|≤ | i − i−1|≤ ≤ 0 Xi=0 Xi=1 Using the result of Lemma 3.6, there exists l Rn such that k+1 ∈ 1 osc (wk(x) lk+1 x) ρ. x∈Bρ − · ≤ 2 α Setting qk+1 = qk + lk+1rk , we get ρ 1+α 1+α osc (u(x) qk+1 x) rk rk+1 . x∈Brk+1 − · ≤ 2 ≤ Since the estimate (3.16) holds for every k, the proof of Theorem 1.1 is complete.

Remark 3.8. 1) For γ 0, when the boundary data g C1,β and the domain Ω has a C1,β boundary,≤ the boundary H¨older regularity∈ of the gradient is a direct consequence of the uniform ellipticity of the operator and the result of [50]. 2) For γ > 0, the regularity of the gradient up to the boundary could be obtained by adapting the arguments of [13]. 3.3. An alternative proof for the regularity of the gradient when γ p 2. In this range we can rely on known results of weak theory. When γ ≤ p − 2, we have that viscosity solutions of (1.1) are viscosity solution of ≤ − ∆ u = f Du p−2−γ. (3.17) − p | | Since 0 p 2 γ

Remark 3.9. In the case 0 < γ p 2, it is possible to relax the dependence on f in the local C1,αestimate from≤ −L∞-norm to Lq-norm for some q < . Here we just outline the idea and refer to [2, Section 4], where a similar∞ technique was used by relying on the paper of Duzaar and Mingione [29]. Fix a viscosity solution u of (1.1) and a small λ > 0, then consider the approximation process which consists in studying the equation p−2 p−2−γ div (( Dv 2 + ε2) 2 Dv ) = (f + λu λv )( Dv 2 + ε2) 2 in Ω − | ε| ε ε − ε | ε| (vε = u in ∂Ω. q One can provide uniform Lipschitz estimates for vε depending on the the L p norm of f for some q > max n, , . Indeed using the fact that u is a γ + 2 weak solution we can control the Lp norm of the gradient by a Caccioppoli inequality, and the fact that the L∞ norm of u is controlled by the Ln norm of f, see the ABP estimate [27, Theorem 1.1] for viscosity solutions of (1.1). Combining these estimates with the computations of [29], we get the uniform estimates which depends on a lower norm of f. Then one can prove that ′′ [u] 1,α ′′ C = C p,q,n,d,,γ,d , u ∞ , f q , C (Ω ) ≤ || ||L (Ω) || ||L (Ω) where d = diam (Ω) and d′′ = dist(Ω′′, ∂Ω′). Here Ω′′ Ω′ Ω. ⊂⊂ ⊂⊂ 4. W 2,2 regularity In this section we study the integrability properties of the second derivatives of the viscosity solutions to (1.1). The idea is to divide the study into different cases. When γ 0, we reduce the problem to the case of the normalized p-Laplacian with≤ a continuous right hand term. We rely on the uniform ellipticity of the operator and on the Cordes condition (see Theorem 4.1). When γ > 0, the operator is degenerate and we have to use another trick. Then we study a regularized problem. When 1 >γ > 0 but still γ p 2 δ for some δ small enough and p close to 2, we prove uniform estimates| − − |≤ on the second derivatives of the approximate solutions. This will provide the desired result by passing to the limit problem. Let us recall some known results and open problems about the existence and integrability of the second derivatives of solutions for p-Laplacian type problems. The W 2,2-estimates for elliptic equations with measurable coefficients in smooth domains were obtained by Bers and Nirenberg [5] in the two dimensional case in 1954, and by Talenti [61] in any dimensions under the Cordes condition. In [16] Campanato established the W 2,q-estimate for elliptic equations with measurable coefficients in two dimensions for q close to 2. There are also some available results for p-harmonic functions [14, 48, 63] based on difference quotient, which assert that the nonlinear expression of the gradient p−2 1,2 Du 2 Du W (Ω). | | ∈ loc p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 15

This property and the local boundedness of the gradient guarantee that for p (1, 2) the second derivative exists and u W 2,2(Ω). ∈ ∈ loc ∂ p−2 ∂u For p > 2, the derivative Du 2 exists but the passage to ∂x | | ∂x i j D2u is difficult at the critical points. The existence of second derivatives of W 1,p-solutions is not clear due to the degeneracy of the problem. In 1,α 3,q dimension 2 and for p (1, ), p-harmonic function are in Cloc (Ω) Wloc (Ω) ∈ ∞ 2 ∩ where q is any number 1 q < , see [37, Theorem 1]. ≤ 2 α The nonhomogeneous case − ∆ u = g (4.1) − p n was also treated with partial results. For g Lr(Ω) with r > max(2, ) ∈ p and p (1, ), Lou proved in [49] that weak solutions satisfy Du p−1 ∈ ∞ | | ∈ W 1,2(Ω). Tolksdorf [62] proved W 2,2-regularity for p (1, 2] and g L∞(Ω), loc loc ∈ ∈ see also [1, 47, 48, 56, 59]. The W 2,2 regularity for 2 1 shows. These functions are local solutions to ∆pu = g |for| some g L∞ (Rn) provided that p is large enough, but they fail− to be in ∈ loc W 2,2(Rn) if β 3 . loc ≤ 2 The Cordes condition for operators in nondivergence form. Here we recall some available results on the summability of the second derivative for operators in non-divergence form with measurable coefficients. The first are due to [5, 21, 61] and require that the second order linear operator is close to the Laplacian. The case with lower order term was also treated in [20]. In [16], Campanato extended the W 2,2 estimate to W 2,q for q sufficiently 16 ATTOUCHIANDRUOSTEENOJA close to 2. We refer the reader to [53, Theorem 1.2.1, Theorem 1.2.3] for a review on the Cordes condition. 2,2 Theorem 4.1. Consider a linear operator L defined on the space Wloc (Ω) as n Lv(x) := ai,j(x)DiDjv(x) i,jX=1 where aij(x) is a symmetric matrix with measurable coefficients satisfying the ellipticity condition

n Λ ξ 2 a (x)ξ ξ Λ ξ 2 (4.2) 1| | ≤ i,j i j ≤ 2| | i,jX=1 for some 0 < Λ1 < Λ2 and satisfying the Cordes condition: there exists δ (0, 1] such that for a.e x Ω ∈ ∈ n n 2 2 1 (aij(x)) aii(x) . (4.3)   ≤ n 1+ δ ! i,jX=1 − Xi=1 Then any strong solution v W2,2(Ω) L2(Ω) of the equation Lv = f in Ω ∈ loc ∩ with f L2(Ω) satisﬁes, for any Ω′ Ω ∈ ⊂⊂ 2 2 2 2 D v dx C f L2(Ω) + v L2(Ω) (4.4) ′ | | ≤ || || || || ZΩ ′ where C = C(n, Λ1, Λ2, δ, Ω , Ω). Moreover, under the same hypothesis on the operator L, there exist two real numbers 1 0 such that any C solution of Lv = f with f Ln(Ω) satisﬁes ∈

v 2,r ′ C v ∞ + f n , || ||W (Ω ) ≤ || ||L (Ω) || ||L (Ω) where C = C(n, Ω, Λ1, Λ2) > 0. p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 17

The second results can be found in [15] and relies on the smallness of the oscillation of the operator measured in the Ln norm.

Theorem 4.2. Let v be a bounded viscosity solution in B1 of F (D2v,x)= f(x). Assume that F is uniformly elliptic with ellipticity constants λ and Λ, F,f 2 1,1 are continuous in x, F (0, ) 0 and F (D w,x0) has C interior estimates · ≡ q (with constant ce) for any x0 in B1. Suppose that f L (Ω) for some n

w 2,q C w ∞ + f q . || ||W (B1/2) ≤ || ||L (B1) || ||L (B1) 4.1. The case γ 0 and p close to 2. Using the result of Lemma 2.5, we can reduce the≤ study to the case of the normalized p-Laplacian with a bounded right hand term. In this case the operator is singular but uniformly elliptic. We will thus use the result of Theorem 4.1 for regularized problems and provide uniform local W 2,2 estimates for p in the range where the assumptions of Theorem 4.1 are satisﬁed, 2 1 0, the function u is a viscosity solution of ∆N u(x)+ λu(x)= h(x) := f(x) Du(x) −γ + λu(x), x Ω. − p | | ∈ Let Ω′ Ω with Ω′ smooth enough. In the sequel we ﬁx small enough ⊂⊂ λ > 0 and a viscosity solution u of (1.1). We take smooth functions fε C1(Ω) L∞(Ω) converging uniformly to f in Ω′. ∈ We consider∩ the following regularized problem, 2 D vεDvε·Dvε 2 2 −γ/2 ′ ∆v (p 2) 2 2 + λv = f ( Dv + ε ) + λu in Ω ε |Dvε| +ε ε ε ε − − − | | ′ ( vε = u on ∂Ω . (4.6) 18 ATTOUCHIANDRUOSTEENOJA

It follows, using the fact that the problem is uniformly elliptic without sin- 2 gularities, that solutions vε are classical C solutions and solve the equation in the classical sense, see [31, Theorem 15.18] and [41, Theorem 3.3]. From the comparison principle we have an estimate

f ∞ ′ || ||L (Ω ) vε L∞(Ω′) 2 u L∞(Ω) + , || || ≤ || || λ ! In Appendix B we also show that

Dvε L∞(Ω′) (4.7) || || 1 ′ 1+γ C(p,n,γ, Ω )( f ∞ ′ + v ∞ ′ + h¯ ∞ ′ , u 1,∞ ′ ) ≤ || ||L (Ω ) || ε||L (Ω ) || ε||L (Ω ) || ||W (Ω ) 1 ′ 1+γ C(p,n,γ, Ω )( f ∞ ++ u ∞ + f ∞ ), (4.8) ≤ || ||L (Ω) || ||L (Ω) || ||L (Ω) where we denoted h¯ε := λ(u vε). In the sequel we take λ =− 1. Consider the operator D2vDv Dv L (v) := ∆v (p 2) ε · ε . vε − − − Dv 2 + ε2 | ε| The operator Lvε is uniformly elliptic with Λ1 = min(1,p 1) and Λ2 = max(1,p 1) and satisﬁes the Cordes condition with δ = δ(p,n− ) for 1

′ with v∞ = u on ∂Ω . By the uniqueness of viscosity solutions of (4.9) (see [27, Theorem 6.1] and [12, Proposition 2.2]), we conclude that v∞ = u and hence the estimate holds for u.

Notice that for f 0, using the result of Lemma 2.6 and the previous ≡ 2 result, we have that for γ > 1 and 1

Du γ F (D2u, Du, u, x)= f, (4.10) − | | where F is uniformly elliptic. This equation includes equation (1.1). How- ever, both their result and method are different from ours. They prove that there exists some δ > 0 depending on the ellipticity constants, dimension and γ, such that viscosity solutions of (4.10) are globally in W 2,δ(Ω), and the estimate depends on the Ln-norm of f. Their method is based on an ABP estimate, barrier function method, touching by paraboloids, a local- ization argument, and a certain covering lemma, which plays a similar role than the standard Calderón-Zygmund cube decomposition lemma. The classical results of fully non linear elliptic equations require a small oscillation condition on the coefficients. Applying the result of Theorem 4.2, we would also get uniform W 2,q estimates for p close to 2 (using that β(x,x0) 2 p 2 ). Nevertheless, the results using the Cordes condition gives a more≤ | precise− | range on the values of p where we are granted that W 2,2 estimates hold. In our case, since we have a uniform Lipschitz bound for u, Theorem 7.4 of [15] gives the existence of δ = δ(p, n, γ) such that for every solution u of (1.1) with 1 < γ 0 and p> 1, we have u W 2,δ(Ω), and − ≤ ∈ loc

1 ′ 1+γ u 2,δ ′ C(p,n,γ, Ω )( f ∞ + u ∞ ). || ||W (Ω ) ≤ || ||L (Ω) || ||L (Ω) This provides an alternative proof for the result of Li and Li [43] in the special case of equation (1.1)

4.2. The case γ > 0 but close to 0 and p close to 2. In case that γ > 0, we still have some results for γ < 1 and p 2 γ close to 0. Consider | − − | 20 ATTOUCHIANDRUOSTEENOJA smooth solutions to the following problem,

γ 2 2 2 D vεDvε·Dvε ′ ( Dv + ε ) 2 ∆v + (p 2) 2 2 + λv = h in Ω ε ε |Dvε| +ε ε ε − | | − ′ ( vε = u h i on ∂Ω , (4.11) where hε := fε + λu, with fε smooth and converging locally uniformly to f. The problem being uniformly elliptic without singularities and the right 1 2,α hand side being C -continuous, the function vε are in C . The existence of smooth solutions vε is ensured by the classical theory (see [31, Theorem 15.18]).

Proof of Theorem 1.3. Step 1: Uniform Lipschitz estimates. Applying the comparison principle for elliptic quasilinear equation in general form (see [31, Theorem 10.1]), we have that

h ∞ ′ || ε||L (Ω ) vε L∞(Ω′) C u L∞(Ω) + || || ≤ || || λ !

f ∞ ′ || ||L (Ω ) C u L∞(Ω) + . (4.12) ≤ || || λ ! Next, if γ p 2 then v are solutions to ≤ − ε p−2 p−2−γ div ( Dv 2 + ε2) 2 Dv = (h λv )( Dv 2 + ε2) 2 . − | ε| ε ε − ε | ε| Using that the boundary data u is in C1,α(Ω′), it follows from [45, Theorem 1,α ′ 1] that vε are uniformly bounded in C (Ω ). We also have a uniform Lipschitz bound on vε using the Ishii-Lions method in the case where we lack the divergence structure (γ>p 2) − ′ Dv ∞ ′ C(p,n,γ, Ω )( f ∞ + u ∞ ). (4.13) || ε||L (Ω ) ≤ || ||L (Ω) || ||L (Ω) The proof of this estimate is provided in the appendix, see Proposition B.2. Step 2: Uniform estimates for the Hessian. Equation (4.11) can be regarded as a perturbation of the regularized γ + 2- Laplacian. Indeed, we can rewrite

2 γ D vεDvε Dvε ( Dv 2 + ε2) 2 ∆v + (p 2) · − | ε| ε − Dv 2 + ε2 ε | | 2 γ γ D vεDvε Dvε 2 2 2 2 2 2 = div ( Dvε + ε ) Dvε (p 2 γ)( Dvε + ε ) 2 · 2 − | | − − − | | Dvε + ε | | p 2 γ p 2 γ γ = − div ( Dv 2 + ε2) 2 Dv + − − ( Dv 2 + ε2) 2 ∆v . − γ | ε| ε γ | ε| ε p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 21

Hence we have 2 γ γ D vεDvε Dvε 2 2 2 2 2 2 div ( Dvε + ε ) Dvε = (p 2 γ)( Dvε + ε ) 2 · 2 + gε, − | | − − | | Dvε + ε | | (4.14) where gε := fε + λu λvε. − 2 ′ Multiplying (4.14) by a smooth test function φ C0 (Ω ) and integrating by parts, we have ∈

γ 2 2 2 Dvε + ε Dvε Dφdx ′ | | · ZΩ 2 2 2 γ D vεDvε Dvε = (p 2 γ)( Dvε + ε ) 2 · + gε φ dx. ′ − − | | Dv 2 + ε2 ZΩ | ε| Choosing Dsφ instead of φ for s 1, ..., n as a test function and integrating by parts, we get ∈ { }

−β φ := η2( Dv 2 + ε2) 2 D v , | ε| s ε where η C∞ is a non negative cut-oﬀ function, we have ∈ c 2 −β −β Diφ = η H(Dvε) DiDsvε + 2ηDiηH(Dvε) Dsvε βη2 D D v D v H(Dv )−β−2D v . − i l ε l ε ε s ε Xl 22 ATTOUCHIANDRUOSTEENOJA

Summing up over s 1, ..., n , it follows that ∈ { } 2 γ−β ˜ε I1 + I2 + I3 := η H(Dvε) AijDjDsvεDiDsvε dx Ω′ Z Xi,j,s 2 γ−β−2 ˜ε β η H(Dvε) AijDjDsvεDsvε DiDlvεDlvε dx − Ω′ ! Z Xi,j,s Xl γ−β ˜ε + 2ηH(Dvε) AijDjDsvεDsvε Diη dx Ω′ Z Xi,j,s 2 2 γ−β D vεDvε Dvε = (p 2 γ) η DsDsvεH(Dvε) · dx − − − ′ Dv 2 + ε2 ZΩ s | ε| X 2 2 γ−β−2 D vεDvε Dvε + β(p 2 γ) η DsvεDsDlvεDlvεH(Dvε) 2 · 2 dx − − Ω′ Dvε + ε Z Xs,l | | 2 γ−β D vεDvε Dvε 2(p 2 γ) η DsηDsvεH(Dvε) · dx − − − ′ Dv 2 + ε2 Ω s ε Z X | | 2 −β + DsgεDsvεη H(Dvε) dx ′ Ω s Z X = : II1 + II2 + II3 + II4. That is, I + I = II + II + II + II I . 1 2 1 2 3 4 − 3 The estimates of I1,I2,I3 are given as follows. We have

2 γ−β 2 2 β 2 2 −2 I1 + I2 = η H(Dvε) D vε D( Dvε ) H(Dvε) dx ′ | | − 4 | | | | ZΩ 2 γ 2 γ−β−2 2 2 2 −1 + η H(Dvε) D( Dvε ) β Dvε D( Dvε ) H dx, 4 ′ | | | | − · | | | ZΩ " #

≥0 so that for| γ > 0 {z } 2 γ−β 2 2 I1 + I2 (1 β) η H(Dvε) D vε dx ≥ − ′ | | ZΩ (1 β)γ 2 γ−β−2 2 2 + − η H(Dvε) D( Dvε ) dx. 4 ′ | | | | ZΩ Next, using Young’s inequality, it follows that

2H(Dv )γ−βη A˜ε D D v D v D η δ D( Dv 2) 2η2H(Dv )γ−β−2 ε ij j s ε s ε i ≤ 1| | ε| | ε i,j,s X 2 γ−β+2 + C(p, δ1) Dη H(Dvε) , | | p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 23 and hence 2 2 2 γ−β−2 I3 δ1 D( Dvε ) η H(Dvε) dx | |≤ ′ | | | | ZΩ 2 γ−β+2 + C(p, δ1) Dη H(Dvε) dx. ′ | | ZΩ 2 To estimate the terms II1 and II2, we will use that D vεDvε, Dvε D2v Dv 2 and ∆v √n D2v . We obtain | | ≤ | ε|| ε| | ε|≤ | ε| 2 2 2 γ−β II1 p 2 γ √n D vε η H(Dvε) dx | |≤ ′ | − − | | | ZΩ + 2 2 2 γ−β II2 β(p 2 γ) D vε η H(Dvε) dx. | |≤ − − ′ | | ZΩ Using Young’s inequality, we can estimate II3 by

2 2 2 γ−β 2 γ−β+2 II3 δ2 D Dvε η H(Dvε) dx+C(p, δ2, γ) Dη H(Dvε) dx. | |≤ ′ | | | | ′ | | ZΩ ZΩ Finally we estimate II4 by

1−β 2 II4 H(Dvε) ∞ ′ η Dgε dx. | |≤ ′ || ||L (Ω ) | | ZΩ Choosing δ1 and δ2 small enough, such that γ (1 β) δ δ = 0, (4.15) − 4 − 1 − 2 we obtain

2 γ−β 2 2 + 2 2 2 γ−β (1 β) η H(Dvε) D vε β(p 2 γ) D vε η H(Dvε) dx − ′ | | ≤ − − ′ | | ZΩ ZΩ 2 2 2 γ−β + p 2 γ √n D vε η H(Dvε) dx ′ | − − | | | ZΩ 1−β 2 + H(Dvε) ∞ ′ η Dgε dx ′ || ||L (Ω ) | | ZΩ 2 γ−β+2 + C(p, n, γ) Dη H(Dvε) dx. ′ | | ZΩ

Now, if 1 β √n p 2 γ β(p 2 γ)+ = κ> 0, (4.16) − − | − − |− − − we get

2 γ−β 2 2 1−β 2 κ η H(Dvε) D vε H(Dvε) ∞ ′ η Dgε dx ′ | | ≤ ′ || ||L (Ω ) | | ZΩ ZΩ 2 γ−β+2 + C(p, n, γ) Dη H(Dvε) dx. ′ | | ZΩ 24 ATTOUCHIANDRUOSTEENOJA

We ﬁx λ = 1. By using (4.12) and (4.13), we have ′ H(Dv ) C(p,n,γ, Ω )( f ∞ + u ∞ ) ε ≤ || ||L (Ω) || ||L (Ω) and

Dgε dx = Dfε + λDu λDvε dx ′ | | ′ | − | ZΩ ZΩ ′ Df dx + λ Ω ( Du L∞(Ω′) + Dvε L∞(Ω′)) ≤ ′ | | | | || || || || ZΩ C(p,n,γ, Ω )( f 1,1 + u ∞ + f ∞ ). ≤ | | || ||W (Ω) || ||L (Ω) || ||L (Ω) ′′ ′ We take η such that η 1 on Ω Ω , 0 η 1 and Dη ∞ ′ C. ≡ ⊂⊂ ≤ ≤ || ||L (Ω ) ≤ For any β (0, 1) and 0 < γ β, assuming that (4.16) holds and f W 1,1(Ω) C∈(Ω), we get that D2≤v is uniformly bounded in L2(Ω′′), ∈ ∩ ε 2 2 ′ D vε dx C n,p,γ,β,d,d , f W 1,1(Ω) , f L∞(Ω) , u L∞(Ω) . Ω′′ | | ≤ || || || || || || Z (4.17) If γ 0 and p > 1, then using Proposition B.1 below together with [32, ≥ α ′ Theorem 1], we see that vε are uniformly bounded in C (Ω ). It follows from the Arzelà-Ascoli theorem, that up to a subsequence vε converges to some function v. Using the stability result, we have that v is a viscosity solution of Dv γ ∆N v + λv = f + λu (4.18) − | | p in Ω′ with v = u on ∂Ω′. By uniqueness of solutions of (4.18) for λ > 0 (see [8, Theorem 1.1]), we conclude that u = v in Ω′. Passing to the limit in (4.17), we get the desired local W 2,2-estimate for u, so the proof of Theorem 1.3 is complete. 1,1 Remark 4.3. For γ = p 2, 2 0 and let w be a solution to (3.2) with osc (w) 1 B1 ≤ ∞ and f L (B1) 1. We are going to show that w is Hölder in B3/4, and this || || ≤ 3 will imply that w is Hölder in any smaller ball Bρ for ρ 0, 4 with the same Hölder constant. First observe thatw ˜(x) := w(x)+∈q x is a solution · p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 25 to (1.1). Proceeding as in the Appendix B, we can first show thatw ˜ is Lipschitz continuous and that for x,y Q , we have ∈ 7/8 w˜(x) w˜(y) C(p, n, γ)(1 + w˜ ∞ + f ∞ ) x y . | − |≤ || ||L (B1) || ||L (B1) | − | This implies that w is Lipschitz continuous in B7/8 and that

w(x) w(y) C(p, n, γ)(1 + w ∞ + f ∞ + q ) x y . | − |≤ || ||L (B1) || ||L (B1) | | | − | Hence if q 2+ w ∞ + f ∞ := Γ (p, n, γ), then for x,y B | |≥ || ||L (B1) || ||L (B1) 0 ∈ 7/8 it holds w(x) w(y) c¯ (p, n, γ) q x y . (A.1) | − |≤ 1 | || − | Now we will provide uniform H¨older estimates using the estimate (A.1). We ﬁx x0,y0 B 3 , and consider the auxiliary function ∈ 4 M M Φ(x,y) := w(x) w(y) Lφ( x y ) x x 2 y y 2 , − − | − | − 2 | − 0| − 2 | − 0| where φ(t) = tβ for some 0 <β< 1. Our goal is to show that Φ(x,y) 0 3 ≤ for (x,y) Br Br, where r = 4 . We will argue by contradiction, write two viscosity inequalities∈ × and combine them in order to get a contradiction using the second order terms and the strict ellipticity in the gradient direction. We argue by contradiction and assume that for all L > 0, M > 0 and β (0, 1), Φ has a positive maximum. We assume that the maximum ∈ is reached at some point (x1,y1) B¯r B¯r. Since w is continuous and bounded, we get ∈ × 2 M x x 2 w ∞ , | 1 − 0| ≤ || ||L (B1) 2 (A.2) M y y 2 w ∞ . | 1 − 0| ≤ || ||L (B1) Notice that x1 = y1, otherwise the maximum of Φ would be non positive. 6 2 32 Taking M w ∞ , we have x x < r/8 and y y < r/8 ≥ || ||L (B1) r | 1 − 0| | 1 − 0| so that x1 and y1 are in Br. M 2 Next we apply Jensen-Ishii’s lemma to w(x) 2 x x0 and w(y)+ M 2 − | − | 2 y y0 . By Jensen-Ishii’s lemma (also known as theorem of sums, see [23|, Theorem− | 3.2]), there exist

2,+ M (ζ , X) w(x ) x x 2 , x ∈ J 1 − 2 | 1 − 0| 2,− M (ζ ,Y ) w(y )+ y y 2 , y ∈ J 1 2 | 1 − 0| that is 2,+ (a, X + MI) w(x ), ∈ J 1 2,− (b, Y MI) w(y ), − ∈ J 1 26 ATTOUCHIANDRUOSTEENOJA where (ζx = ζy) x y a = Lφ′( x y ) 1 − 1 + M(x x )= ζ + M(x x ), | 1 − 1| x y 1 − 0 x 1 − 0 | 1 − 1| x y b = Lφ′( x y ) 1 − 1 M(y y )= ζ M(y y ). | 1 − 1| x y − 1 − 0 y − 1 − 0 | 1 − 1| If L is large enough (depending on M, actually since x x 2, y y | 1 − 0|≤ | 1 − 0|≤ 2, x y 2, it is enough that L> M 24−β), we have | 1 − 1|≤ β L 2Lβ x y β−1 a Lφ′( x y ) M x x β x y β−1 . | 1 − 1| ≥ | |≥ | 1 − 1| − | 1 − 0|≥ 2 | 1 − 1| L 2Lβ x y β−1 b Lφ′( x y ) M y y β x y β−1 . | 1 − 1| ≥ | |≥ | 1 − 1| − | 1 − 0|≥ 2 | 1 − 1| Moreover, by Jensen-Ishii’s lemma [22], for any τ > 0, we can take X,Y n such that for all τ > 0 such that τB

Denoting η1 = a + q, η2 = b + q, we get q 2 q η q a | | 2Lβ x y β−1 , | | ≥ | 1| ≥ | | − | |≥ 2 ≥ | 1 − 1| q 2 q η q b | | 2Lβ x y β−1 . (A.7) | | ≥ | 2| ≥ | | − | |≥ 2 ≥ | 1 − 1| Notice also that η q /2 1. The viscosity inequalities read as | i| ≥ | | ≥ (X + MI)(a + q), (a + q) f(x ) η γ tr(X + MI) + (p 2)h i , − 1 ≤ | 1| − a + q 2 | | (Y MI)(b + q), (b + q) f(y ) η γ tr(Y MI) + (p 2)h − i . − 1 ≥ | 2| − − b + q 2 | | In other words,

f(y ) η −γ f(x ) η −γ tr(A(η )(X + MI)) tr(A(η )(Y MI)). 1 | 2| − 1 | 1| ≤ 1 − 2 − It results (using that η −γ 1) that | i| ≤ 2 f ∞ tr(A(η )(X Y )) − || ||L (B1) ≤ 1 − (I) + tr((A(η ) A(η ))Y ) | {z1 − } 2 (II)

+ M| tr(A(η{z1)) + tr(A(}η2)) . (A.8) (III) Estimate of (I). Remark that all| the eigenvalues{z of X Y} are non positive. Applying the previous matrix inequality (A.3) to the− vector (ξ, ξ) where 1 1 − ξ := x −y and using (A.4), we obtain |x1−y1|

β 1 (X Y )ξ,ξ 4 Bτ ξ,ξ 8Lβ x y β−2 − < 0. (A.9) h − i≤ h i≤ | 1 − 1| 3 β − This means that at least one of the eigenvalue of X Y that we denote by − λ is negative and smaller than 8Lβ x y β−2 β−1 . The eigenvalues of i0 | 1 − 1| 3−β 28 ATTOUCHIANDRUOSTEENOJA

A(η ) belong to [min(1,p 1), max(1,p 1)]. Using (A.9), it results that 1 − − tr(A(η )(X Y )) λ (A(η ))λ (X Y ) 1 − ≤ i 1 i − Xi min(1,p 1)λ (X Y ) ≤ − i0 − β 1 min(1,p 1)8Lβ x y β−2 − . ≤ − | 1 − 1| 3 β − Estimate of (II). We have A(η ) A(η ) = (η η η η )(p 2) 1 − 2 1 ⊗ 1 − 2 ⊗ 2 − = [(η η ) η η (η η )](p 2) 1 − 2 ⊗ 1 − 2 ⊗ 2 − 1 − and it follows that tr((A(η ) A(η ))Y ) n Y A(η ) A(η ) 1 − 2 ≤ || || || 1 − 2 || n p 2 Y η η ( η + η ) ≤ | − | || || | 1 − 2| | 1| | 2| 2n p 2 Y η η . ≤ | − | || || | 1 − 2| By using (A.7) and η η 4M, we have | 1 − 2|≤ η η η η η η η η = 1 2 max | 2 − 1|, | 2 − 1| | 1 − 2| η − η ≤ η η 1 2 2 1 | | | | | | | | 16M β−1 . ≤ βL x y | 1 − 1| Combining the previous estimate with (A.5), we obtain tr((A(η ) A(η ))Y ) 64n p 2 M x y −1 . 1 − 2 ≤ | − | | 1 − 1| Estimate of (III). We have M(tr(A(η )) + tr(A(η ))) 2Mn max(1,p 1). 1 2 ≤ − Gathering the previous estimates with (A.8), we get −1 0 2 f ∞ + 64n p 2 M x y ≤ || ||L (B1) | − | | 1 − 1| β 1 + 2Mn max(1,p 1) + min(1,p 1)8Lβ x y β−2 − . − − | 1 − 1| 3 β −

Choosing L satisfying

L C(p, n, γ)(M + f ∞ +1) C(p, n, γ)( w ∞ + f ∞ +1), ≥ || ||L (B1) ≥ || ||L (B1) || ||L (B1) we get min(1,p 1)β(β 1) 0 − − L x y β−2 < 0, ≤ 200(3 β) | 1 − 1| − which is a contradiction and hence Φ(x,y) 0 for (x,y) B B . The ≤ ∈ r × r desired result follows since for x0,y0 B 3 , we have Φ(x0,y0) 0, so we get ∈ 4 ≤ w(x ) w(y ) L x y β. | 0 − 0 |≤ | 0 − 0| p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 29

We conclude that w is H¨older continuous in B 3 and 4 β w(x) w(y) C(n, p, γ) w ∞ + f ∞ + 1 x y . | − |≤ || ||L (B1) || ||L (B1) | − | Appendix B. Uniform estimates for the approximating problem

Proposition B.1. Let vε be a smooth solution of (4.11) with γ [0, ), p (1, ) and ε (0, 1). Then there exists a positive constants∈ α >∞ 0 ∈ ∞ ∈ and C = C (n,p,γ, h ∞ ′ , v ∞ ′ , u α ′ ) such that for every α α || ε||L (Ω ) || ε||L (Ω ) || ||C (Ω ) x,y Ω′, we have ∈ v (x) v (y) C x y α. (B.1) | ε − ε |≤ α| − | Proof. The uniform interior estimate follows from [33, Theorem 1.1], [28, Theorem 1.1] and [35, Corollary 2] since the operator

γ Xη η F (x,s,η,X) := ( η 2 + ε2) 2 tr(X) + (p 2) · + λs h − | | − η 2 + ε2 − ε | | satisﬁes

We are going to make use of the above H¨older estimate and the Ishii-Lions’ method [23] again to prove the following Lipschitz estimate.

Proposition B.2. Let vε be a smooth solution of (4.11) with γ [0, ), p (1, ) and ε (0, 1). Then there exists a positive constant∈ C∞= ∈ ∞ ∈ C(n,p,γ, h ∞ ′ , v ∞ ′ , u 1,∞ ′ ) such that for every x,y || ε||L (Ω ) || ε||L (Ω ) || ||W (Ω ) ∈ Ω′, we have v (x) v (y) C x y . (B.4) | ε − ε |≤ | − | Proof. First we provide an interior estimate. Using a scaling and a translation argument combined with a covering argument, it is enough to prove the Lemma in the unit ball B1 and for osc vε 1 in B1. Like in the proof of the Hölder continuity (see≤ Lemma 3.2), we fix x ,y 0 0 ∈ B3/4 and introduce the auxiliary function M M Φ(x,y) := v (x) v (y) Lφ( x y ) x x 2 y y 2 , ε − ε − | − | − 2 | − 0| − 2 | − 0| where φ is defined below. Our goal is to show that Φ(x,y) 0 for (x,y) B B , where r = 4 . We take ≤ ∈ r × r 5 t tδφ 0 t t := ( 1 )1/(δ−1) φ(t)= − 0 ≤ ≤ 1 δφ0 (φ(t1) otherwise, 30 ATTOUCHIANDRUOSTEENOJA

δ−1 where 2 >δ> 1 and φ0 > 0 is such that t1 > 2 and γφ02 1/4. Then ≤ 1 δtδ−1φ 0 t t φ′(t)= − 0 ≤ ≤ 1 (0 otherwise, δ(δ 1)tδ−2φ 0

2,+ M (ζ , X) v (x ) x x 2 , x ∈ J ε 1 − 2 | 1 − 0| 2,− M (ζ ,Y ) v (y )+ y y 2 , y ∈ J ε 1 2 | 1 − 0| that is 2,+ (a, X + MI) v (x ), ∈ J ε 1 2,− (b, Y MI) v (y ), − ∈ J ε 1 where (ζx = ζy) x y a = Lφ′( x y ) 1 − 1 + M(x x )= ζ + M(x x ), | 1 − 1| x y 1 − 0 x 1 − 0 | 1 − 1| x y b = Lφ′( x y ) 1 − 1 M(y y )= ζ M(y y ). | 1 − 1| x y − 1 − 0 y − 1 − 0 | 1 − 1| If L is large enough (depending on the H¨older constant Cα), we have L 2L a 2 + ε2, b 2 + ε2 Lφ′( x y ) C x y α/2 . (B.7) ≥ | | | | ≥ | 1 − 1| − α | 1 − 1| ≥ 2 q q p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 31

The Jensen-Ishii’s lemma ensures that for any τ > 0, we can take X,Y n such that ∈ S I 0 X 0 τ + 2 B (B.8) − || || 0 I ≤ 0 Y − and X 0 B B 2 B2 B2 − + − (B.9) 0 Y ≤ B B τ B2 B2 − − − 1 2 = D2φ(x ,y )+ D2φ(x ,y ) , 1 1 τ 1 1 where x y x y B =Lφ′′( x y ) 1 − 1 1 − 1 | 1 − 1| x y ⊗ x y | 1 − 1| | 1 − 1| Lφ′( x y ) x y x y + | 1 − 1| I 1 − 1 1 − 1 x y − x y ⊗ x y | 1 − 1| | 1 − 1| | 1 − 1|! and 2 ′ 2 2 L (φ ( x1 y1 )) x1 y1 x1 y1 B = | − 2 | I − − x1 y1 − x1 y1 ⊗ x1 y1 ! | − | | − | | − | x y x y + L2(φ′′( x y ))2 1 − 1 1 − 1 . | 1 − 1| x y ⊗ x y | 1 − 1| | 1 − 1| φ′(t) Observe that φ′′(t)+ 0, φ′′(t) 0 for t (0, 2) and hence t ≥ ≤ ∈ B Lφ′( x y ), (B.10) || || ≤ | 1 − 1| φ′( x y ) 2 B2 L2 φ′′( x y ) + | 1 − 1| . (B.11) ≤ | | 1 − 1| | x y 1 1 | − | Besides, for ξ = x1−y1 , we have |x1 − y1|

Bξ,ξ = Lφ′′( x y ) < 0, B2ξ,ξ = L2(φ′′( x y ))2. h i | 1 − 1| h i | 1 − 1| ′ ′′ φ ( x1 y1 ) x1−y1 Taking τ = 4L φ ( x1 y1 ) + | − | , we obtain for ξ = , | | − | | x y |x1−y1| | 1 − 1| 2 2 Bξ,ξ + B2ξ,ξ = L φ′′( x y )+ L(φ′′( x y ))2 h i τ h i | 1 − 1| τ | 1 − 1| L φ′′( x y ) < 0. (B.12) ≤ 2 | 1 − 1| Applying inequalities (B.8) and (B.9) to any vector (ξ,ξ) with ξ = 1, we have that X Y 0 and X , Y 2 B + τ. | | − ≤ || || || || ≤ || || 32 ATTOUCHIANDRUOSTEENOJA

We denote η = a 2 + ε2, η = b 2 + ε2, and g = f + λu λv = 1 | | 2 | | ε ε − ε hε λvε, and we writeq the viscosity inequalitiesq − (X + MI)a, a g (x ) η −γ tr(X + MI) + (p 2)h i − ε 1 | 1| ≤ − η 2 | 1| (B.13) (Y MI)b, b g (y ) η −γ tr(Y MI) + (p 2)h − i. ε 1 | 2| ≥ − − η 2 | 2| We end up with

g (x ) η −γ tr(A (a)(X + MI)) − ε 1 | 1| ≤ ε g (y ) η −γ tr(A (b)(Y MI)) ε 1 | 2| ≤− ε − where η η A (η) := I + (p 2) ⊗ . ε − η 2 + ε2 | | Adding the two inequalities and using that η −δ, η −δ cL−δ, we get | 1| | 2| ≤ −δ 2c g ∞ L tr(A (a)(X + MI)) tr(A (b)(Y MI)). (B.14) − || ε||L (B1) ≤ ε − ε − It follows that

−γ 2c g ∞ L tr(A (a)(X Y )) + tr((A (a) A (b))Y ) − || ε||L (B1) ≤ ε − ε − ε (I) (II)

|+ M tr({zAε(a)) +} tr(|Aε(b)) . {z }(B.15) (III) Estimate of (I). Observe that| all the eigenvalues{z of X} Y are non positive. Moreover, applying the previous matrix inequality (B.9)− to the vector (ξ, ξ) 1 1 − where ξ := x −y and using (B.12), we obtain |x1−y1|

2 (X Y )ξ,ξ 4 Bξ,ξ + B2ξ,ξ ) h − i≤ h i τ h i 2Lφ′′( x y ) < 0. (B.16) ≤ | 1 − 1|

We deduce that at least one of the eigenvalue of X Y denoted by λi0 is ′′ − negative and smaller than 2Lφ ( x1 y1 ). The eigenvalues of Aε(a) belong to [min(1,p 1), max(1,p 1)].| Using− (|B.16), it follows that − − tr(A (a)(X Y )) λ (A (a))λ (X Y ) ε − ≤ i ε i − Xi min(1,p 1)λ (X Y ) ≤ − i0 − 2 min(1,p 1)Lφ′′( x y ). ≤ − | 1 − 1| p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 33

Estimate of (II). We have a a b b A (a) A (b)= (p 2) ε − ε η ⊗ η − η ⊗ η − 1 1 2 2 a b a b a b = + (p 2). η − η ⊗ η η ⊗ η − η − 1 2 1 2 1 2 It follows that a b A (a) A (b) 2 p 2 . | ε − ε |≤ | − | η − η 1 2

Hence,

tr((A (a) A (b))Y ) n Y A (a) A (b) ε − ε ≤ || || || ε − ε || a b 2n p 2 Y . ≤ | − | || || η − η 1 2

We have a b a b a b 2max | − |, | − | η − η ≤ η η 1 2 | 2| | 1| 8C α x y α/2 , ≤ L | 1 − 1| where we used (B.7) and (B.6).

Using the estimates (B.8)–(B.11), we obtain 4 Y = max Y ξ, ξ 2 Bξ, ξ + B2ξ, ξ || || ξ |h i| ≤ |h i| τ |h i| φ′( x y ) 4L | 1 − 1| + φ′′( x y ) . ≤ x y | | 1 − 1| | | 1 − 1| Hence, remembering that x y 2, we end up with | 1 − 1|≤ tr((A (a) A (b))Y ) 128n p 2 C φ′( x y ) x y −1+α/2 ε − ε ≤ | − | α | 1 − 1| | 1 − 1| + 128n p 2 C φ′′( x y ) . | − | α| | 1 − 1| | Estimate of (III). Finally, we have M(tr(A (a)) + tr(A (b))) 2Mn max(1,p 1). ε ε ≤ − Putting together the previous estimates with (B.15) and recalling the deﬁnition of φ, we end up with 0 128n p 2 C φ′( x y ) x y α/2−1 + φ′′( x y ) ≤ | − | α | 1 − 1| | 1 − 1| | | 1 − 1| | ′′ −γ + 2 min(1,p 1)Lφ ( x y )++2Mn max(1,p 1) 2cL g ∞ − | 1 − 1| − − || ε||L (B1) 128n p 2 C x y α/2−1 + 2nM max(1,p 1) ≤ | − | α | 1 − 1| − δ−2 −γ + 128n p 2 C δ(δ 1)φ x y 2cL g ∞ | − | α − 0 | 1 − 1| − || ε||L (B1) 2 min(1,p 1)δ(δ 1)φ L x y δ−2 . − − − 0 | 1 − 1| 34 ATTOUCHIANDRUOSTEENOJA

Taking δ =1+ α/2 > 1 and choosing L large enough depending on p,n,Cβ, g ∞ we get || ε||L min(1,p 1)δ(δ 1)φ0 δ−2 −γ 0 − − − L x y 2cL g ∞ < 0, ≤ 200 | 1 − 1| − || ε||L (B1) which is a contradiction. Hence Φ(x,y) 0 for (x,y) B B . The ≤ ∈ r × r desired result follows since for x0,y0 B 3 , we have Φ(x0,y0) 0, we get ∈ 4 ≤ v (x ) v (y ) Lφ( x y ) L x y . | ε 0 − ε 0 |≤ | 0 − 0| ≤ | 0 − 0| The uniform Lipschitz estimate up to the boundary of Ω′ follows from a barrier argument as in the proof of [13, Lemma 2.2]. We use the following notation. d( , ∂Ω′) is the distance function of ∂Ω′, · Ω′ := x Ω′ : d(x, Ω′) δ , δ { ∈ ≤ } ¯ ′ ′ ′ and d(x)= d(x, ∂Ω ) in Ωδ is smooth and bounded in Ω . We construct a barrier function u¯(x) =v ¯(x) +w ¯(x), wherev ¯ is a solution of (D2(¯v))=0 in Ω′ M ′ (B.17) (v¯ = u on ∂Ω , and C d¯(x) w¯ (x)= with ν (0, 1). δ δ 1+ d¯(x)ν ∈ Recall that the Pucci operators are convex, uniformly elliptic and depend only on the Hessian. The boundary condition in (B.17) belonging to C1,α(Ω′), 2 ′ 1,α ′ we get thatv ¯ is in C (Ω ) C (Ω ). Moreover, v¯ ∞ ′ u ∞ ′ and ∩ || ||L (Ω ) ≤ || ||L (Ω ) ′ Dv¯ ∞ ′ C(p,n, Ω ) Du ∞ ′ . || ||L (Ω ) ≤ || ||L (Ω ) Taking ′ ν C (1 + Ω )( v ∞ ′ + u ∞ ′ ), ≥ | | || ε||L (Ω ) || ||L (Ω ) we get u¯ v on ∂Ω′ . ≥ ε δ Proceeding as in [13], we can ﬁnd ′ δ = δ (p,n, Ω , f ∞ , u 1,∞ ′ , γ) > 0 0 0 || ||L (Ω) || ||W (Ω ) such that

2 2 γ/2 2 ( Dv¯ + Dw¯ + ε ) (D w¯) 2 f ∞ λ( u ∞ ′ | δ| M ≤− || ||L (Ω) − || ||L (Ω ) on Ω′ for any δ δ . δ ≤ 0 p-LAPLACIAN TYPE PROBLEMS IN NON-DIVERGENCE FORM 35

′ It follows that in Ωδ, D2uD¯ u,¯ Du¯ ( Du¯ 2 + ε2)γ/2 ∆¯u + (p 2)h i | | − Du¯ 2 + ε2 | | ( Du¯ 2 + ε2)γ/2 (D2u¯) ≤ | | M ( Dv¯ + Dw¯ 2 + ε2)γ/2 (D2w¯) ≤ | | M 2 f ∞ λ( u ∞ ′ + v¯ ∞ ′ )+ λw¯ ≤ || ||L (Ω) − || ε||L (Ω ) || ||L (Ω ) f λ(u u¯). ≤− ε − − ′ By comparison principle, we have uε u¯ in Ωδ, and the Lipschitz estimate follows. ≤

Proposition B.3. Let vε be a smooth solution of (4.6) with γ ( 1, 0], p (1, ) and ε (0, 1). Then there exists a positive constant∈ −C = ∈ ∞ ′ ∈ C(n,p,γ, Ω , h ∞ ′ , v ∞ ′ , u 1,∞ ′ ) such that for every x,y || ε||L (Ω ) || ε||L (Ω ) || ||W (Ω ) ∈ Ω′, we have v (x) v (y) C x y . | ε − ε |≤ | − | Proof. It follows from Lemma 3.3 that solutions of (4.6) are H¨older continuous with uniform H¨older bound −1 1+γ C(p, n, γ)( v ∞ ′ + h ∞ ′ + f ∞ + u ∞ ) || ε||L (Ω ) || ε||L (Ω ) || ||L (Ω) || ||L (Ω) The proof proceeds similarly to the previous proof of Proposition B.2. The main changes occur when writing the viscosity inequalities: Now we write h (x ) f (x ) η −γ tr(D(a)(X + MI)) − ε 1 − ε 1 | 1| ≤ h (y )+ f (y ) η −γ tr(D(b)(Y MI)). ε 1 ε 1 | 2| ≤− − and adding the two inequalities we have −δ 2 h ∞ ′ 2c f ∞ ′ L tr(D(a)(X +MI)) tr(D(b)(Y MI)). − || ε||L (Ω ) − || ε||L (Ω ) ≤ − − By choosing −1 1+γ L c( v ∞ ′ + h ∞ ′ + f ∞ ′ ), ≥ || ε||L (Ω ) || ε||L (Ω ) || ||L (Ω ) the proof is completed as the proof of Proposition B.2. The uniform up to the boundary Lipschitz estimate follows from [60] due to the regularity of the boundary data and the uniform ellipticity of the operator and the subquadratic growth with respect to the gradient of the right hand term of the equation.

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