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

This paper is concerned with the self-improving property for obstacle problem related to the singular porous medium equation. We establish a local higher integrability result for the spatial gradient of the $m$-th power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function.


Introduction
We are concerned in this paper with the self-improving property for the gradient of nonnegative weak solutions to the obstacle problems related to the porous medium equation. The porous medium equation is an important prototype of nonlinear diffusion equation. This kind of equation can be derived from modelling the flow of isentropic gas through a porous medium, models for groundwater infiltration or heat radiation in plasmas (see for instance [11,Chapter 2]). Bögelein, Lukkari and Scheven [3] introduced the concept of obstacle problem related to the porous medium equation. This kind of obstacle problem is a variational inequality subject to a constraint that the solution should lie above a given obstacle function. In [3] the authors established the existence and uniqueness results for the strong and weak solutions to the obstacle problem. Subsequently, the same authors [4] obtained a local Hölder continuity result of nonnegative weak solutions in the degenerate case m ≥ 1. In the fast diffusion range (n−2) + n+2 < m < 1, Cho and Scheven [5] established the local Hölder continuity result for the nonnegative weak solutions. Recently, Cho and Scheven [6] proved the higher integrability of signed weak solutions to the obstacle problems in the degenerate range m ≥ 1. Motivated by this work, we will study the higher integrability of nonnegative weak solutions to the obstacle problems in the fast diffusion range (n−2) + n+2 < m < 1. This problem is at present far from being solved.
The higher integrability for the solutions of parabolic systems was first studied by Kinnunen and Lewis [9,10]. The treatment of the porous medium type equations is much more difficult. The higher integrability result for porous medium equations has been established by Gianazza and Schwarzacher [7,8]; see also [1,2] for the case of porous medium systems. For the treatment of obstacle problem related to the singular porous medium equation, our proof closely follows the scheme of [8]. We shall work with the sub-intrinsic cylinders constructed in [8]. In order to obtain gradient estimates on intrinsic cylinders, we will distinguish between the degenerate case and the non-degenerate case. Combining energy estimates, gluing lemma and the parabolic Sobolev inequality, we establish a reverse Hölder inequality for the gradient of the m-th power of solutions on each intrinsic cylinder. The main difficulty in our proof is the treatment of the obstacle function. In order to obtain a suitable L ∞ bound for the solution, we have to impose a condition that ψ m is locally Lipschitz continuous, where ψ is an obstacle function. We also assume that the time derivative ∂ t ψ 1−m , that appears in the gluing lemma, is locally bounded. Furthermore, we use a certain stopping time argument for the covering of the superlevel set of the gradient. Contrary to the argument in [8, section 7], we use a localized maximal function instead of the strong maximal function, since the localized version can be adapted to address obstacle problems.
The present paper is built up as follows. In §2, we set up notations and state the main result. §3 presents some preliminaries and we explain the construction of the sub-intrinsic cylinders. In §4, we establish the energy estimates, while in §5 we prove a gluing lemma which describes the difference of two spatial averages. In §6, we establish the intrinsic reverse Hölder inequalities for the gradient on intrinsic cylinders. Finally the proof of the main result is presented in §7.

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 Ω is a bounded domain in R n with n ≥ 2. For T > 0, let Ω T denote the space-time cylinder Ω × (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 : |x − x 0 | < r}, Λ s (t 0 ) = (t 0 − s, t 0 + s) and Q r,s (z 0 ) = B r (x 0 ) × Λ s (t 0 ). If the reference point z 0 is the origin, then we simply write B r , Λ s and Q r,s for B r (0), Λ s (0) and Q r,s (0). In this work we study obstacle problems related to the quasilinear parabolic equations of the form ∂ t u − div A(x, t, u, Du m ) = 0. (2.1) Here, the vector field A is only assumed to be measurable and satisfies A(x, t, u, ζ) · ζ ≥ ν 0 |ζ| 2 , |A(x, t, u, ζ)| ≤ ν 1 |ζ|, (2.2) 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 : Ω T → R + with Dψ m ∈ L 2 (Ω T ) and ∂ t ψ m ∈ L m+1 m (Ω T ), we define the function classes K ψ = v ∈ C 0 ([0, T ]; L m+1 (Ω)) : v m ∈ L 2 (0, T ; H 1 (Ω)), v ≥ ψ a.e. in Ω T and K ′ ψ = v ∈ K ψ : ∂ t v m ∈ L m+1 m (Ω T ) . Let α ∈ W 1,∞ 0 ([0, T ], R + ) be a cut-off function in time and η ∈ W 1,∞ 0 (Ω, 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, Lukkari and Scheven [3]. Cho and Scheven [5] later extended the definition to the general quasilinear structure. In this paper, we adopt the definition from [5].
holds true for any v ∈ K ′ ψ , any cut-off function in time α ∈ W 1,∞ 0 ([0, T ], R + ) and any cut-off function in space η ∈ W 1,∞ 0 (Ω, R + ). 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 Ω T , (2) The time derivative ∂ t ψ 1−m is locally bounded in Ω T . The first assumption will be needed for the proof of Lemma 6.2 in §6, and the second assumption will be used to simplify estimating the weighted spatial averages from §5. We emphasize that the second assumption can be improved to an integrability condition, but the proof is too long to give here.
According to [5], the assumption (1) implies that the weak solution u is locally bounded and Hölder continuous in Ω T . There is no loss of generality in assuming for all (x, t) ∈ Ω T . For simplicity of notation, we write We are now in a position to state our main theorem.
Theorem 2.2. Let z 0 ∈ Ω T be a fixed point, and let R < 1 be a fixed positive number such that Q 8R,64R 2 (z 0 ) ⊂ Ω 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 [8,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. Our proof makes no appeal to address the Calderón-Zygmund type estimate. 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 (Ω T ). Some basic properties of the time mollification are listed in the following lemma (see for instance [3, Lemma 3.1]).
We remark that Lemma 3.1 (3) applies to the weak solution u, since the weak solution to the obstacle problem is locally Hölder continuous. Next, we recall the inequalities which was obtained from [8, Proposition 2.1].
We note that Lemma 3.2 will be used to derive the energy estimates in §4. This lemma also plays a crucial role in the proof of Lemma 5.1 in §5. Furthermore, we recall the definitions of intrinsic and sub-intrinsic cylinders which was introduced from [8, section 3]. Definition 3.3. [8] Let z 0 ∈ Ω T be a fixed point, and let r, θ > 0 such that Q r,θr 2 (z 0 ) ⊂ Ω 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 [8] to construct the sub-intrinsic cylinders which will be used in the covering argument in §7. Let z 0 = (x 0 , t 0 ) ∈ Ω T be a point such that Q R,R 2 (z 0 ) ⊂ Ω 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 . 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 [8] for this kind of cylinder as follows. [8] Fix a point z 0 ∈ Ω T and assume that Q R,R 2 (z 0 ) ⊂ Ω T . Let s ∈ (0, R 2 ] and r(s) be the radius constructed via (3.3)- (3.4). Then, the cylinder Q s (z 0 ) is sub-intrinsic and satisfies the following property: 1−m . For s, σ ∈ (0, R 2 ] and s < σ, we have the properties for the concentric cylinders Q s (z 0 ) and Q σ (z 0 ) as follows: (2) r(s) ≤ s σ b r(σ) and r(s) → 0 as s ↓ 0.
In the applications, we can use the assumption (2.4) to deduce that This enables us to take K = 1 when we apply Lemma 3.4 (6) and (7). As indicated in [8], the properties (4) and (7) imply the following Vitali-type covering property.
The Vitali-type covering Lemma will be used only in §7. This kind of covering plays a crucial role in the proof of weak type estimate for the maximal function in Lemma 7.1. On the other hand, the Vitali-type covering will be used to estimate the measure of superlevel sets of the 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 Ψ is locally integrable in Ω T . This condition is weaker than the Lipschitz condition (2.5). Our main result in this section states as follows.
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 It is easy to check that v ∈ K ′ ψ . We first remark that since u ≥ ψ, two superlevel sets {u ≥ c} and {u ≥ ψ c } are equal. More precisely, the relation holds true for any t ∈ Λ 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 In view of (4.3), we deduce 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 ∈ Λ 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 ∈ Λ s 1 (t 0 ), there holds with the obvious meaning of VI, VII and VIII. To estimate VI, we note that u − ψ c ≤ u − c 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 where we have used Young's inequality for the last estimate. Since 0 < m < 1, we have where the constant γ depends only upon m. Our next aim is to find a lower bound for VIII. We fix t 1 ∈ Λ s 1 (t 0 ) and consider the superlevel set {B r 2 (x 0 ) : u(x, t 1 ) ≥ ψ c (x, t 1 )}. On this set, u ≥ c and there holds 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 (ψ − c) 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 holds for any x ∈ {B r 2 (x 0 ) : u(x, t 1 ) ≥ ψ c (x, t 1 )}. 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 with the obvious meaning of VIII 1 and VIII 2 . We first observe that where the constant γ depends only upon m. Combining the estimates obtained for VIII 1 and VIII 2 , we deduce from (4.17) the estimate From (4.9)-(4.11) and (4.18), we are led to the conclusion that there exists a constant Another step in the proof of (4.1) is to find an estimate for diffusion term in (2.3). We first note that where 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 ∈ Ω T : c < u(z) ≤ ψ c }. This implies that h↓0 This estimate together with (4.19) yield for any t 1 ∈ Λ s 1 (t 0 ). This proves the desired estimate (4.1) by taking the supremum over t 1 ∈ Λ s 1 (t 0 ) in the first term and t 1 = t 0 + s 1 in the second one. Finally, we come to the proof of (4.2). This result will be proved if we can show that the estimate holds for any t 1 ∈ Λ 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 [5, 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 [8, page 26-28] and [5, page 12]) and we omit the details. The proof of the lemma is now complete.

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 Ψ and ∂ t ψ 1−m are locally integrable. Let B be an open ball in Ω ⊂ 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.
Proof. Our proof is in the spirit of [6, Lemma 3.2, Lemma 4.1]. Without loss of generality, we may assume that t 1 < t 2 . In the variational inequality (2.3) we choose η = ξ as a cut-off function in space and, motivated by the proof of of [6, Lemma 3.2], we choose for t ∈ (t 2 + ε, T ), as a cut-off function in time, where 0 < ε ≪ 1. Next, we distinguish between the cases (u(t 1 )) ξ B r 2 (x 0 ) ≥ (u(t 2 )) ξ B r 2 (x 0 ) and (u(t 1 )) ξ B r 2 (x 0 ) < (u(t 2 )) ξ B r 2 (x 0 ) . In the first case, the argument in [6, page 19] actually shows that 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 [ where we abbreviated and the term I h tends to zero as h ↓ 0. To estimate L, we use integration by parts to obtain 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 . Noting that ∂ ∂t Moreover, we note that Next, we consider the estimate for To estimate the diffusion term, we infer from the argument in [6, page 17] that lim sup ε↓0 lim sup Combining the estimates above, we conclude that where the constant γ depends only upon ν 0 , ν 1 and m. Applying Young's inequality, we estimate the third and fourth terms on the right-hand side as follows: where we have used Hölder's inequality for the last estimate. This implies that the following inequality holds for any µ > 0. At this stage, we set 0 < δ ≪ 1. In the estimate (5.4) we choose 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.
where the constant γ depends only on ν 0 , ν 1 and m.
This corollary is a direct consequence of Lemma 5.1 and the proof is omitted.

Reverse Hölder-type inequalities
The proof of the reverse Hölder inequalities on intrinsic cylinders follows from the analysis of two complementary cases. Following [8], 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.
6.1. The degenerate alternative. This subsection deals with the degenerate case. We first establish a boundedness result analogue to [8,Proposition 5.2]. The local boundedness for weak solutions to the singular parabolic obstacle problems was first proved by Cho and Scheven [5]. Here, we present a mean value type estimate and our proof is in the spirit of [8,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 ∈ Ω T and suppose that Q R,R 2 (z 0 ) ⊂ Ω 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 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 3.4 (5) that all the cylinders Q j are intrinsic. Moreover, from Lemma 3.4 (4) and the assumption (6.3), we deduce Then, we follow the argument in [8, page 33-34] to impose a condition k ≥ θ 1 1−m 2s and obtain ess sup Consequently, we can apply the parabolic Sobolev inequality to (u m −k 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 [8, page 34]. According to the argument in [8, 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 ∈ Ω T and suppose that Q R,R 2 (z 0 ) ⊂ Ω 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 1 s ess sup From Lemma 3.4 (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 s , the assumptions of Lemma 6.2 are fulfilled. Applying (6.4) and Hölder's inequality, we obtain similar as in [8,Corollary 5.4] that By Sobolev inequality and Lemma 3.4 (4), we deduce Using the similar argument as in the proof of [8, 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 (5.5) from Corollary 5.2 to deduce where we have used Lemma 3.4 (2), (4) for the last estimate. From this, we conclude that since s = θ s r(s) 2 . We insert this inequality in (6.11) and this implies that 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 3.4 (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

QIFAN LI
Observe that we can reabsorb the second term 1 2 θ m+1 1−m s s on the right-hand side into the left. It follows that At this point, we claim that 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 therefore the proof of Proposition 6.3 is complete.
6.2. The non-degenerate alternative. In this subsection, we prove the reverse Hölder inequality analogue to (6.6) for the non-degenerate regime. The treatment for non-degenerate case is different from the degenerate case.
Proof. Our first goal is to establish the following weak type estimate where the constant c ′ depends only on n and m. In order to prove (7.2), we note that for Moreover, the collection {Q s z (z) : z ∈Q ∩ T * ( f )(z) > λ } forms a covering of the set Q ∩ T * ( f )(z) > λ . From Lemma 3.5, we find a countable subfamily {Q s z i (z i )} ∞ i=1 of pairwise disjoint cylinders, such that holds for all j ≥ N 0 . It follows from (7.2) that for any j ≥ N 0 there holds 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, there holds which completes the proof.
Furthermore, for σ 1 , σ 2 ∈ [1, 2] and σ 1 < σ 2 , we define two concentric cylinderŝ We are interested in getting estimates on such concentric cylinders. To this end, we first need the following lemma.
and the constant γ depends only on n, m, ν 0 , ν 1 and L 1 .

4)
where the constant c 2 depends only on n, m, ν 0 and ν 1 .
where β = 1 − 2b and m+1 1−m β − 1 > 0. Next, we invoke Lemma 4.1, takes the form In the case σ z ∈ [ 1 12L R 2 , R 2 ], we apply Lemma 3.4 (4), (6) to obtain θ σ z (z) Combining this with (7.5) and (7.6) we find that which proves the estimate (7.3). Furthermore, we consider the case σ z ∈ (L 2 s z , 1 12L R 2 ). In this case, the cylinder Q σ z (z) is intrinsic and satisfies From [8,Lemma 2.4], we find that the cylinder Q σ z (z) is degenerate. This enables us to use (6.13) from the proof of Proposition 6.1. Then, there exists a constantγ =γ(n, m, ν 0 , ν 1 ) such that Combining this with (7.5) and (7.6) we finally arrive at Therefore, we can reabsorb the first term on the right-hand side of (7.7) into the left and this proves the estimate (7.3).