Higher integrability for obstacle problem related to the singular porous medium equation

In this paper we study the self-improving property of the obstacle problem related to the singular porous medium equation by using the method developed by Gianazza and Schwarzacher (J. Funct. Anal. 277(12):1–57, 2019). We establish a local higher integrability result for the spatial gradient of the mth power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function. In comparison to the work by Cho and Scheven (J. Math. Anal. Appl. 491(2):1–44, 2020), our approach provides some new aspects in the estimations of the nonnegative weak solution of the obstacle problem.


Introduction
Kinnunen and Lewis [12,13] proved the higher integrability of weak solutions to parabolic systems of p-Laplacian type by using an intrinsic scaling method. The method of intrinsic scaling is also applied to the study of regularity problems of porous medium type equations of the form ∂ t uu m = 0, m > 0, where u = u(x, t) and (x, t) ∈ R n+1 . There are two different approaches to the study of higher integrability of weak solutions to porous medium equations, one is the approach developed by Gianazza and Schwarzacher [9,10], the other is the approach developed by Bögelein et al. [1,2]. In fact, the second method can also be used to study the vector-valued weak solutions of porous medium system.
It is of interest to compare these two approaches and we restrict ourselves to the singular case (n-2) + n+2 < m < 1 in the discussion. In [7] the authors introduced a certain intrinsic cylinders of the form Q r,θr 2 (z 0 ) = B r (x 0 ) × t 0θ r 2 , t 0 + θ r 2 , θ ≈ u 1-m , (1.1) and used this kind of cylinders to study the boundedness, the Hölder continuity and the Harnack inequality of nonnegative weak solutions to porous medium equations. The intrinsic scaling method in [10] is very close to this kind of idea and the intrinsic cylinder takes the form Q r,θr 2 (z 0 ) = B r (x 0 ) × t 0θ r 2 , t 0 + θ r 2 , θ ≈ --Q r,θ r 2 (z 0 ) u m+1 dx dt 1-m 1+m . (1.2) In fact, if we use a dimensional analysis, then the relation in (1.2) actually implies [θ ] = [u] 1-m , which is similar to (1.1). This enables us to use some ideas from [7] in this approach. Motivated by the proof of [7,Proposition B.4.1], Gianazza and Schwarzacher obtained a mean value result for nonnegative weak solutions [10,Proposition 5.2], which is a key ingredient in the proof of the reverse Hölder inequality in the degenerate regime. Therefore, the approach in [10] relies heavily on the regularity of the solution and this is the weak point in the study of the higher integrability. In fact, the intrinsic cylinder (1.2) shows no information for the spatial gradient. Heuristically, the intrinsic cylinder should be modified so that the factor θ should be related to the spatial gradient of u m . To this end, we apply the dimensional analysis again. We first note that the dimensional relation holds. From (1.2), we obtain the dimensional relation for the time direction as follows: Heuristically, the modified intrinsic cylinder should be taken in the following form: This kind of cylinders should be the correct form for the study of higher integrability of singular porous medium equations. This idea was also used in [14] to study the higher integrability of singular parabolic systems with non-standard growth. In [2] the authors considered the intrinsic cylinders of the form Du m 2 dx dt, (1.4) which are compatible with (1.3). On the other hand, a key ingredient in the approach [1,2] is an application of the properties of the boundary term b u m , a m = m m + 1 |a| m+1 -|u| m+1u a mu m , (1.5) which was first introduced by Bögelein et al. [3]. In summary, there are some differences between these two approaches and the second approach has more advantages.
Recently, Cho and Scheven [5,6] proved the higher integrability of weak solutions to obstacle problems related to the porous medium equation and their proofs followed the approach in [1,2]. In [6] the authors used the boundary term (1.5) to establish an energy estimate and a gluing lemma for weak solutions of the obstacle problem. Moreover, the intrinsic cylinders in [6] takes the form (1.4). Admittedly, the approach in [2] is effective in treating the higher integrability of obstacle problem for the singular porous medium equation, but it is natural to try to use the old idea in [10] to study the same problem. To this end, the present work is intended as an attempt to follow the approach in [10] to establish a self-improving result for the obstacle problem. In this paper, we shall use the intrinsic cylinder of the type (1.2) and we will not make any use of boundary term (1.5) which is a basic tool in [1,2,5,6]. The result of this paper was first announced in [15].
The present paper is built up as follows. In Sect. 2, we set up notations and state the main result. Section 3 presents some preliminaries and we explain the construction of the sub-intrinsic cylinders. In Sect. 4, we establish the energy estimates, while in Sect. 5 we prove a gluing lemma which describes the difference of two spatial averages. In Sect. 6, we establish the intrinsic reverse Hölder inequalities for the gradient on intrinsic cylinders. Finally the proof of the main result is presented in Sect. 7. Throughout this paper, we also compare our arguments with [2,6,10].

Statement of the main result
In the present section, we introduce the notations and give the statement of the main result. Throughout the paper, we assume that is a bounded domain in R n with n ≥ 2. For T > 0, let T denote the space-time cylinder × (0, T). Given a point z 0 = (x 0 , t 0 ) ∈ R n+1 and two parameters r, s > 0, we set B r (x 0 ) = {x ∈ R n : |xx 0 | < r}, s (t 0 ) = (t 0s, t 0 + s) and Q r,s (z 0 ) = B r (x 0 ) × s (t 0 ). If the reference point z 0 is the origin, then we simply write B r , s and Q r,s for B r (0), s (0) and Q r,s (0). In this work we study obstacle problems related to the quasilinear parabolic equations of the form Here, the vector field A is only assumed to be measurable and satisfies where ν 0 and ν 1 are fixed positive constants. Throughout the work, we only consider the singular case m ∈ ( (n-2) + n+2 , 1). The obstacle problem for the porous medium type equation (2.1)-(2.2) can be formulated as follows. Given an obstacle function ψ : T → R + with Dψ m ∈ L 2 ( T ) and ∂ t ψ m ∈ L m+1 m ( T ), we define the function classes be a cut-off function in time and η ∈ W 1,∞ 0 ( , R + ) be a cut-off function in space. We define The definition of weak solutions to the obstacle problems related to the porous medium equation was first introduced by Bögelein et al. [3]. Cho and Scheven [4] later extended the definition to the general quasilinear structure. In this paper, we adopt the definition from [4].
In this work, we shall make two regularity assumptions on the obstacle function under consideration. More precisely, we assume that the obstacle function ψ satisfies the following regularity properties: (1) The function ψ m is locally Lipschitz continuous in T .
(2) The time derivative ∂ t ψ 1-m is locally bounded in T . The first assumption will be needed for the proof of Lemma 6.2 in Sect. 6, and the second assumption will be used to simplify estimating the weighted spatial averages from Sect. 5. We emphasize that the second assumption can be removed, but the proof is too long to give here. We refer the interested reader to Remark 5.4, which explains the idea of the proof.
According to [4], the assumption (1) implies that the weak solution u is locally bounded and Hölder continuous in T . There is no loss of generality in assuming for all (x, t) ∈ T . For simplicity of notation, we write = ψ m+1 + |∂ t ψ m | m+1 m + |Dψ m | 2 . We are now in a position to state our main theorem. Theorem 2.2 Let z 0 ∈ T be a fixed point, and let R < 1 be a fixed positive number such that Q 8R,64R 2 (z 0 ) ⊂ T . Assume that there exists a constant M 0 > 0 such that Let u be a nonnegative weak solution to the obstacle problem in the sense of Definition 2.1 that satisfies (2.4). Then there exists a constant ε = ε(n, m, ν 0 , ν 1 ) > 0 such that where the constant γ depends only upon n, m, ν 0 and ν 1 .
Remark 2.3 Contrary to [6], our assumption (2.5) is much stronger than [6, (2.9)]. However, the Lipschitz continuity of ψ m will be used to establish a mean value result for the nonnegative weak solutions and this condition seems to be optimal for the study of the obstacle problem if we follow the approach in [10].
Remark 2.4 Contrary to [10,Theorem 7.4], which established a Calderón-Zygmund type estimate for the porous medium equation, we only derive the reverse Hölder inequality for the obstacle problem. Finally, for the proof of Theorem 2.2, we will write z 0 = (0, 0) for simplicity of presentation.

Preliminary material
In this section, we provide some preliminary lemmas. All the materials in this section are stated without proof. We first note that the weak solution to the obstacle problem may not be differentiable in the time variable. In order to handle the problem with the time derivative, we will use the following time mollification. For a fixed h > 0, we set where v ∈ L 1 ( T ). Next, we recall the inequalities which were obtained from [10, Proposition 2.1].
As mentioned earlier, we will not make use of the boundary term (1.5). Instead, we will use Lemma 3.1 to establish the energy estimates and a gluing lemma for the nonnegative weak solutions to the obstacle problem. Furthermore, we recall the definitions of intrinsic and sub-intrinsic cylinders which were introduced from [10, Sect. 3].

Definition 3.2 ([10]
) Let z 0 ∈ T be a fixed point, and let r, θ > 0 such that Q r,θr 2 (z 0 ) ⊂ T . We say that Q r,θr 2 (z 0 ) is a sub-intrinsic cylinder if and only if the following inequality holds: where the constant K 1 ≥ 1. Moreover, we say that Q r,θr 2 (z 0 ) is an intrinsic cylinder if and only if At this point, we follow the idea in [10] to construct the sub-intrinsic cylinders which will be used in the covering argument in Sect. 7. Let z 0 = (x 0 , t 0 ) ∈ T be a point such that Q R,R 2 (z 0 ) ⊂ T . For any s ∈ (0, R 2 ], we denote byr(s) the quantitỹ for any s ∈ (0, R 2 ]. Subsequently, we write Q s (z 0 ) = Q r(s),s (z 0 ) and denote by θ s (z 0 ) the quantity θ s (z 0 ) = s r(s) 2 . (3.5) If z 0 = (0, 0), then we abbreviate Q s := Q s ((0, 0)) and θ s := θ s ((0, 0)). We now summarize the results obtained from [10] for this kind of cylinder as follows.
In the applications, we can use the assumption (2.4) to deduce that -- This enables us to take K = 1 when we apply Lemma 3.3(6) and (7). As indicated in [10], the properties (4) and (7) imply the following Vitali-type covering property which will be used only in Sect. 7.
Remark 3.5 In the construction of the sub-intrinsic cylinder Q s (z 0 ), we first determine the radius r in terms of a fixed s > 0 and this determines the value of θ s (z 0 ) by (3.5). As mentioned in the introduction, this kind of sub-intrinsic cylinders shows no information for the spatial gradient. In contrast to [2, Sect. 7.1], the inequality in (3.3) is coincident with which is the inequality in the definition ofθ in [2, Sect. 7.1]. In [2, Sect. 7.1] the authors first determine the value of θ z 0 ; in terms of a fixed > 0 and a quantity λ 0 . This determines a new type of sub-intrinsic cylinder Q (θ z 0 ; ) (z 0 ) and this construction also establishes a relationship between the factor θ and the spatial gradient.

Caccioppoli type inequalities
The aim of this section is to establish energy estimates for the weak solution of the obstacle problem. Here, we state and prove the energy estimates on the condition that the function is locally integrable in T . Our main result in this section states as follows.
Lemma 4.1 Let 0 < m < 1 and let u be a nonnegative weak solution to the obstacle problem in the sense of Definition 2.1.
Moreover, for any c ≥ 0 we have ess sup where the constant γ depends only upon ν 0 , ν 1 and m.
Proof We begin with the proof of (4.1), which is the most difficult part of the proof. In the variational inequality as a comparison map, where the function ψ c is defined by ψ m c = max{c m , ψ m }. It is easy to check that v ∈ K ψ . We first remark that, since u ≥ ψ, the two superlevel sets {u ≥ c} and {u ≥ ψ c } are equal. More precisely, the relation holds true for any t ∈ s 2 (t 0 ). Let η = φ 2 and α ∈ W 1,∞ 0 ([0, T], R + ) be a fixed cut-off function which will be determined later.
We now proceed to establish an energy estimate from the variational inequality (2.3). For the first term on the left-hand side of (2.3) we compute ] h >ψ m c } and the second term on the right-hand side of (4.5) is estimated above by Integrating by parts, we obtain Combining (4.6) with (4.5), we infer that we use integration by parts to get with the obvious meaning of V 1 and V 2 . We first observe that Our next aim is to obtain lower and upper bounds for V 1 . To this end, we need to determine the cut-off function in time α(t). For a fixed time level t 1 ∈ s 1 (t 0 ) ⊂ (0, T), we define where 0 < ε 1. We now turn our attention to the estimate of V 1 . From (3.1), we find that Applying Lebesgue's dominated convergence theorem, we pass to the limit h ↓ 0 on the right-hand side and conclude that From the preceding arguments, we infer from (4.7) that, for any t 1 ∈ s 1 (t 0 ), we have with the obvious meaning of VI, VII and VIII. To estimate VI, we note that uψ c ≤ uc on the set {u ≥ ψ c }. From this inequality and (4.4), we conclude that We now come to the estimate of VII. We first observe that . From this inequality, we conclude that Our next aim is to find a lower bound for VIII.
We fix t 1 ∈ s 1 (t 0 ) and consider the superlevel set To estimate L 1 , we first consider the easy case (ψ mc m ) + ≤ 4 -1 (u mc m ). In this case, we get where the constant γ depends only on m. Combining this estimate with (4.13), we obtain (4.14) Next, we consider the estimate of L 2 . In the case (ψc) + ≤ 4 -1 (uc), we have In the case (ψc) + > 4 -1 (uc), we see that ψ ≥ c and u < 4ψ -3c. Furthermore, we conclude that there exists γ = γ (m) such that Combining this estimate with (4.15), we have shown that the estimate holds in any case. Therefore, we conclude from (4.12), (4.14) and (4.16) that the inequality We now turn our attention to the estimate of VIII. It follows from (4.4) that since φ ≤ 1. It remains to treat the second term on the right-hand side of (4.17). For t 1 ∈ s 1 (t 0 ), we obtain Furthermore, we deduce from (4.17) the estimate Another step in the proof of (4.1) is to find an estimate for diffusion term in (2.3). We first note that where for t ∈ (t 0s 1 , t 1 ].
By Young's inequality and the growth assumption of the vector field A, we obtain the estimate for the first term on the right-hand side where the constant γ depends only upon ν 0 and ν 1 . Next, we consider the second term on the right-hand side of (4.20). Using Young's inequality and the ellipticity assumption of the vector field A, we deduce Furthermore, we need to consider the estimate of the gradient on the superlevel set {u > c}.
Since u ≥ ψ, we have and therefore Du m = Dψ m a.e. on {z ∈ T : c < u(z) ≤ ψ c }. This implies that Combining the estimates (4.20)-(4.23), we conclude that This estimate together with (4.19) yield for any t 1 ∈ s 1 (t 0 ). This proves the desired estimate (4.1) by taking the supremum over t 1 ∈ s 1 (t 0 ) in the first term and t 1 = t 0 + s 1 in the second one. Finally, we address the proof of (4.2). This result will be proved if we can show that the estimate holds for any t 1 ∈ s 1 (t 0 ). In order to prove this estimate, we will work on the sublevel set {u < c} and the argument is similar in spirit to [4, as a comparison map and obtain where the cut-off function α is defined in (4.8) and η = φ 2 . To estimate the third term on the right-hand side, we infer from (3.2) that At this point, the desired estimate (4.24) follows from a standard argument (see for instance [10, page 26-28] and [4, page 12]) and we omit the details. The proof of the lemma is now complete. The main novelty with respect to [6, Lemma 4.1] is that we have also established an energy estimate (4.1) for the truncated function (uc) + (u mc m ) + by using (4.3) as a comparison map. This inequality will be used to prove a boundedness result for the nonnegative weak solutions in Sect. 6. Our proof of (4.2) also encompasses the use of the boundary term (1.5) which is the basic tool in the proof of [6, Lemma 4.1].

Estimates on the spatial average
This section is devoted to the study of a gluing Lemma, which concerns weighted mean values of the weak solution on different time slices. We first state and prove the gluing lemma on the condition that the functions and ∂ t ψ 1-m are locally integrable. Let B be an open ball in ⊂ R n and let η ≥ 0 be a smooth function supported in the compact setB.
Here and subsequently, we define The following lemma is our main result in this section.

Lemma 5.1 Let u be a nonnegative weak solution to the obstacle problem in the sense of
Definition 2.1. Fix a point z 0 = (x 0 , t 0 ) ∈ T and assume that Q r 1 ,s (z 0 ) ⊂ Q r 2 ,s (z 0 ) ⊂ T . Let ξ ∈ C ∞ 0 (B r 2 (x 0 )), 0 ≤ ξ ≤ 1 in B r 2 (x 0 ), ξ ≡ 1 in B r 1 (x 0 ) and |Dξ | ≤ 2(r 2r 1 ) -1 . Let 1 be the quantity Then, for any t 1 , t 2 ∈ s (t 0 ), we have as a cut-off function in time, where 0 < ε 1. According to the argument in [5, page 19], it suffices to prove the lemma in the case (u(t 1 )) ξ B r 2 (x 0 ) < (u(t 2 )) ξ B r 2 (x 0 ) . Let μ be a fixed positive constant, which will be determined later. We follow the argument in [5, page 14-15] to deduce with the obvious meaning of L 1 and L 2 . By Lebesgue's dominated convergence theorem, we see that L 1 tends to zero as h ↓ 0. Next, we consider the estimate for L 2 . To this end, we use integration by parts to obtain Moreover, we note that and this implies Next, we consider the estimate for Combining the estimates above, we apply Young's inequality to conclude that Dψ m 2 dx dt (5.5) holds for any μ > 0. At this stage, we set 0 < δ 1. In the estimate (5.5) we choose . (5.6) This concludes the estimate (5.1) by passing to the limit δ ↓ 0. Finally, if we choose then the desired estimate (5.2) follows by passing to the limit δ ↓ 0. This finishes the proof of the lemma.
Moreover, if ψ m is locally Lipschitz continuous and ∂ t ψ 1-m is locally bounded, then we can rewrite the estimates (5.1) and (5.2) in the following ready-to-use form.

Corollary 5.3 Suppose that
for some M 0 > 1. Then, for any t 1 , t 2 ∈ s (t 0 ), we have This corollary is a direct consequence of Lemma 5.1 and the proof is omitted. For the applications, we shall use (5.8) and (5.9) to the analysis of degenerate and non-degenerate regimes in Sect. 6, respectively. The purpose of [6,Lemma 4.2] is to establish a Sobolev-Poincaré type inequality [6,Lemma 4.3] for the obstacle problem which is similar to [2, Lemma 5.1]. The quantity μ in [6] is also determined in two alternatives, which are [6, (4.35)] and [6, (4.36)]. It is easy to check that our choice of μ in (5.7) is coincident with the choice of μ in [6, page 27] but (5.6) is different from [6, page 30]. Compared with [6,Lemma 4.3], our concepts of degenerate and non-degenerate regimes have no concern with the spatial gradient.
Remark 5.4 Motivated by the proof of [6,Lemma 4.2], we can remove the assumption that sup Q 8R,64R 2 (z 0 ) |∂ t ψ 1-m | < +∞. To this end, it is sufficient to improve the estimate (5.4). In In the case m > 1 3 , we use Hölder's inequality and the energy estimate (4.1) to deduce In the analysis of the non-degenerate regime, we could use (5.10) for the choice of μ as in (5.7). For the treatment of the degenerate regime, we use Young's inequality to obtain , and the quantity μ can be determined by (5.6). The proofs are left to the reader.

Reverse Hölder-type inequalities
The proof of the reverse Hölder inequalities on intrinsic cylinders follows from the analysis of two complementary cases. Following [10], we give the definitions of degenerate and non-degenerate regimes.
holds true. Moreover, we call a cylinder Q s (z 0 ) non-degenerate if and only if the following inequality holds: Next, we consider separately the degenerate and non-degenerate case. In contrast to [2, Sect. 6], our assumptions (6.1)-(6.2) show no information for the spatial gradient of the solution and the intrinsic cylinder under consideration takes the form (1.2). Our approach relies heavily on the regularity of the solution.

The degenerate alternative
This subsection deals with the degenerate case. We first establish a boundedness result analogue to [10,Proposition 5.2]. The local boundedness for weak solutions to the singular parabolic obstacle problems was first proved by Cho and Scheven [4]. Here, we present a mean value type estimate and our proof is in the spirit of [10, Proposition 5.2].

Lemma 6.2 Let u be a nonnegative weak solution to the obstacle problem in the sense of
Definition 2.1. Fix a point z 0 ∈ T and suppose that Q R,R 2 (z 0 ) ⊂ T . Let 0 < s ≤ 1 2 R 2 and r(2s) makes sense. Assume that the cylinder Q s (z 0 ) is intrinsic and Then there exists a constant γ = γ (n, m, ν 0 , ν 1 ) such that Proof There is no loss of generality in assuming z 0 = (x 0 , t 0 ) = (0, 0). For j = 0, 1, 2, . . . , set s j = s + 2 -j s, r j = r(s j ), B j = B r j and Q j = Q r j ,s j . We define a sequence of numbers k m j = k m -2 -j k m , where k > 0 is to be determined. Let ζ j = ζ j (x) be a smooth function such that ζ j ∈ C ∞ 0 (B j ), 0 ≤ ζ j ≤ 1, ζ j ≡ 1 in B j+1 and |Dζ j | ≤ 2(r jr j+1 ) -1 . We now apply the Caccioppoli estimate (4.1) with (c, φ, Q r 1 ,s 1 , Q r 2 ,s 2 ) replaced by (k j+1 , ζ j , Q j+1 , Q j ) to obtain ess sup We first observe from Lemma 3.3(5) that all the cylinders Q j are intrinsic. Moreover, from Lemma 3.3(4) and the assumption (6.3), we deduce Then we follow the argument in [10, page 33-34] to impose a condition k ≥ θ Consequently, we can apply the parabolic Sobolev inequality to (u mk m j+1 ) + ζ j on the cylinder B j × (-t j+1 , t j+1 ), which gives For more details on the proof of (6.5), we refer the reader to [10, page 34]. According to the argument in [10, page 34], we obtain Y j → 0 as j → ∞, provided that where γ > 1 depends only upon n, ν 0 , ν 1 and m. This proves (6.4) and the proof of Lemma 6.2 is complete.
We remark that the intrinsic condition for Q s (z 0 ) is necessary in the proof of Lemma 6.2. This restricts us to work with the intrinsic cylinders in the degenerate regime. With the help of Lemma 6.2, we can now establish the reverse Hölder inequality for the degenerate regime. Proposition 6.3 Let u be a nonnegative weak solution to the obstacle problem in the sense of Definition 2.1. Fix a point z 0 ∈ T and suppose that Q R,R 2 (z 0 ) ⊂ T . Let 0 < s ≤ 1 3 R 2 and r(3s) makes sense. Assume that the cylinder Q s (z 0 ) is intrinsic and satisfies (6.1). Moreover, assume that ψ m is locally Lipschitz continuous and for some M 0 > 0. Then there exists q 1 ∈ ( 1 2 , 1), depending only upon n and m, such that the following holds: Proof For abbreviation, we assume that z 0 = (x 0 , t 0 ) = (0, 0). Initially, we use (4.2) from Lemma 4.1 to obtain From Lemma 3.3(1), (2), (4) and Hölder's inequality, we obtain Before proceeding further, we distinguish between two cases: Observe that the desired estimate (6.6) holds immediately in the first case. It remains to treat the second case. We first note that Our next aim is to find an upper bound for s -1 θ m+1 1-m s . Let η ∈ C ∞ 0 (B r(3s) ), 0 ≤ η ≤ 1 in B r(3s) , η ≡ 1 in B r(2s) and |Dη| ≤ 2(r(3s)r(2s)) -1 . We denote by λ 0 the constant Using a similar argument to the proof of [10, Proposition 6.2], we infer from (6.1), (6.7), (6.8) and (6.10) that where α = 2q 1 m m+1 . To estimate the second term on the right-hand side, we apply the estimate (5.8) from Corollary 5.3 to deduce with the obvious meaning of L 1 , L 2 and L 3 . We first consider the estimate for L 2 . Since u ≥ ψ, we apply Lemma 3.3(1), (4) and Hölder's inequality to deduce -- This implies that Next, we rewrite L 3 as follows: Combining the estimates above, we arrive at Observe that we can reabsorb the second term 1 In the case s -1 θ m+1 1-m s ≤ 1, it is easy to see that (6.13) holds trivially. In the case s -1 θ m+1 1-m s > 1, the desired estimate (6.13) directly follows from (6.12). This proves (6.13) and the proof of Proposition 6.3 is complete.

The non-degenerate alternative
In this subsection, we prove the reverse Hölder inequality analogue to (6.6) for the nondegenerate regime. The treatment for non-degenerate case is different from the degenerate case. Proposition 6.4 Let u be a nonnegative weak solution to the obstacle problem in the sense of Definition 2.1. Fix a point z 0 ∈ T and suppose that Q R,R 2 (z 0 ) ⊂ T . Let 0 < s ≤ R 2 and suppose that the cylinder Q s (z 0 ) is intrinsic and satisfies (6.2). Moreover, assume that ψ m is locally Lipschitz continuous and for some M 0 > 0. Then there exists q 1 ∈ ( 1 2 , 1), depending only upon n and m, such that the following holds: Proof For simplicity of presentation, we assume that z 0 = (0, 0). Let us first construct a smooth function η ∈ C ∞ 0 (B r(s) ) satisfying 0 ≤ η ≤ 1 in B r(s) , η ≡ 1 in B r(s/2) and |Dη| ≤ 2(r(s)r(s/2)) -1 . Define where t ∈ s . Let σ 1 , σ 2 ∈ [ 1 2 , 1] and σ 1 < σ 2 . Applying a similar argument to the proof of [10, Proposition 6.3], we infer from (6.2) that (6.15) Furthermore, we apply the Caccioppoli estimate (4.2) with (c, Q r 1 ,s 1 , Q r 2 ,s 2 ) replaced by (λ, Q σ 1 s , Q σ 2 s ) to obtain 1 s ess sup For any σ ∈ [ 1 2 , 1], we set We now choose q 1 ∈ ( 1 2 , 1) satisfying (6.9). According to the proof of [10, Proposition 6.3], we infer from (6.16) that where α = 2q 1 m m+1 . It remains to treat the third term on the right-hand side of (6.17). We first consider the case r(s) Next, we turn our attention to the case r(s) 1 m ≤ s r(s) . We apply the inequality (5.9) from Corollary 5.3 to obtain with the obvious meaning of L 1 , L 2 and L 3 . We first consider the estimate for L 1 . From (6.15), we get To estimate L 2 , we recall that u ≥ ψ. From (6.15) and Hölder's inequality, we obtain Finally, we address the estimate of L 3 . Recalling that r(s) 1 m ≤ s r(s) , we have This proves that T j (f ) converges in measure to f . Then there exists a subsequence T j k (f ) converging to f almost everywhere. It follows that, for almost every z ∈Q, which completes the proof.
On the other hand, for such a choice of L, we see immediately that 3χs z ≤ 6χs z ≤ 6χL -1 R 2 ≤ R 2 and Q 3χs z (z) ⊂ Q 6χs z (z) ⊂ Q L 1 s z (z) ⊂Q σ 2 .