On the H\"older regularity of signed solutions to a doubly nonlinear equation. Part II

We demonstrate two proofs for the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad p>2,\quad 0<q<p-1. \] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.


INTRODUCTION AND MAIN RESULTS
Initiated in [1], we continue our study on the Hölder regularity of weak solutions to a class of doubly nonlinear parabolic equations whose model case is (1.1) ∂ t |u| q−1 u − div |Du| p−2 Du = 0 weakly in E T .
Here E T := E × (0, T ] for some T > 0 and some E open in R N . In [1] we have studied the borderline case, i.e., p > 1 and q = p − 1, and in this note we will take on the doubly degenerate case, i.e., p > 2 and 0 < q < p − 1. Our main result states that locally bounded, weak solutions to (1.1) are Hölder continuous in the interior, and up to the parabolic boundary of E T , if proper assumptions on the boundary are imposed. Two proofs will be presented, both of which are entirely local and structural.
As a matter of fact, we shall consider parabolic partial differential equations of the general form (1.2) ∂ t |u| q−1 u − div A(x, t, u, Du) = 0 weakly in E T where A(x, t, u, ζ) : E T × R N +1 → R N is a Carathéodory function. Namely, it is measurable with respect to (x, t) ∈ E T for all (u, ζ) ∈ R × R N , and continuous with respect to (u, ζ) for a.e. (x, t) ∈ E T . Moreover, we assume the structure conditions |A(x, t, u, ζ)| ≤ C 1 |ζ| p−1 , for a.e. (x, t) ∈ E T , ∀ u ∈ R, ∀ ζ ∈ R N , where C o and C 1 are given positive constants.
In the sequel, we will refer to the set of parameters {p, q, N, C o , C 1 } as the structural data. We also write γ as a generic positive constant that can be quantitatively determined a priori only in terms of the data and that can change from line to line.
Postponing the formal definitions of weak solution, we will proceed to present the main results on the interior regularity in Section 1.1 and the boundary regularity in Section 1.2. . Now we state our main result concerning the interior Hölder continuity of weak solutions to (1.2), subject to the structure conditions (1.3). Throughout this note, we assume that p > 2 and 0 < q < p − 1 unless otherwise stated.
Theorem 1.1. Let u be a bounded, local, weak solution to (1.2) -(1.3) in E T . Then u is locally Hölder continuous in E T . More precisely, there exist constants γ > 1 and β ∈ (0, 1) that can be determined a priori only in terms of the data, such that for every compact set K ⊂ E T , for every pair of points (x 1 , t 1 ), (x 2 , t 2 ) ∈ K.
Remark 1.1. Local boundedness is sufficient for Theorem 1.1 to hold. In fact, local boundedness is inherent in the notion of weak solutions, cf. Appendix A. Moreover, the method also applies to equations with lower order terms like in [3, Chapters II -IV] and in [6,Appendix C]. However we will not pursue generality in this direction. Instead, concentration will be made on the actual novelty.
Remark 1.2. Theorem 1.1 implies a Liouville type theorem; the argument is the same as [1, Corollary 1.1] which we refer to for details.

Boundary Regularity.
Results on the boundary regularity will be stated in this section. Let us first consider the following initial-boundary value problem of Dirichlet type: g ∈ L p 0, T ; W 1,p (E) , and g is continuous on S T with modulus of continuity ω g (·).

(D)
As for the geometry of the boundary ∂E, we introduce the property of positive geometric density there exists α * ∈ (0, 1) and o > 0, such that for all x o ∈ ∂E, for every cube K (x o ) and 0 < ≤ o , there holds Here for > 0 we have set K (x o ) to be the cube with center at x o ∈ R N and edge 2 , whose faces are parallel with the coordinate planes. When x o = 0 we simply write K . Intuitively, condition (G) means that there is an exterior cone whose vertex is attached to x o and whose angle is quantified by α * . Next, we consider the Neumann problem. The boundary ∂E is assumed to be of class C 1 , such that the outward unit normal, which we denote by n, is defined on ∂E. The initial-boundary value problem of Neumann type is formulated as where the structure conditions (1.3) and assumption (I) for the initial data are still in force. For the Neumann datum ψ we assume for simplicity that, for some absolute constant C 2 , there holds Although more general conditions should also work (cf. [3, Section 2, Chapter II]), we however will not pursue generality in this direction. The formal definitions of weak solutions to (1.4) and (1.5) will be given in Section 1.4. Now we are ready to present the results concerning regularity of solutions to (1.4)  . Assume (I) holds. Then u is continuous in K × [0, T ] for any compact set K ⊂ E. More precisely, there is a modulus of continuity ω(·), determined by the data, dist(K, ∂E), M and ω o (·), such that for every pair of points (x 1 , t 1 ), (x 2 , t 2 ) ∈ K × [0, T ]. In particular, if u o is Hölder continuous with exponent β o , then ω(r) = γM r β with some γ > 0 and β ∈ (0, β o ] depending on the data, dist(K, ∂E) and β o . Remark 1.3. As we shall see in the proof of Theorem 1.2, the estimate on the modulus of continuity actually holds true for all p > 1 and q > 0, if t 1 = 0 or t 2 = 0.

Near S T -Neumann Type Data.
Theorem 1.4. Let u be a bounded weak solution to the Neumann problem (1.5). Assume ∂E is of class C 1 and (N) holds. Then u is Hölder continuous in any compact set K ⊂ E T . More precisely, there exist constants γ > 1 and β ∈ (0, 1) determined by the data, C 2 , dist(K; {t = 0}) and the structure of ∂E, such that for every pair of points (x 1 , t 1 ), (x 2 , t 2 ) ∈ K.
Remark 1.4. The proofs of Theorems 1.2 -1.4 are local in nature. As a result, it suffices to require the boundary data in the Dirichlet problem (1.4) or the Neumann problem (1.5) to be taken just on a portion of the parabolic boundary.
1.3. Novelty and Significance. The doubly nonlinear parabolic equation (1.2) accounts for many physical models, including dynamics of glaciers, shallow water flows and friction dominated flows in a gas network. We refer to [1] for a source of physical motivations. The mathematical interest of this equation lies in the degeneracy or the singularity or both it possesses, and a broader class of parabolic equations it generates, which include the parabolic p-Laplacian and the porous medium equation as particular instances. The issue of local Hölder regularity for this equation has been investigated by a number of authors, in various forms and with different notions of solution, cf. [8,9,13,15]. However, all of them assume that p > 2 and 0 < q < 1. The main novelty of our results consists in extending the known range to a larger one, that is, p > 2 and 0 < q < p − 1, cf. Figure 1.3. On the other hand, even in the case p > 2 and 0 < q < 1, our results are not covered by the previous works, as they either use different notions of solution [9,13,15], or assume non-negativity of the solution [8].
One of our main technical advances from the previous works lies in that we dispense with any kind of logarithmic type energy estimates. As such our arguments should have further implications in the context of the so-called Q-minima from the calculus of variations, cf. [15].
The expansion of positivity for the degenerate parabolic equations has been established in [5] as a key tool to study Harnack's inequalty. Roughly speaking, it asserts that the measure of the positivity set of a non-negative, super-solution translates into pointwise positivity at later times. Using it to handle the Hölder regularity seems new in the doubly degenerate setting. Similar ideas have appeared in [7,11] in different forms, either for the parabolic p-Laplacian or for the porous medium equation. The virtual advantage of this important property lies in the simplification it brings and a geometric character it offers. On the other hand, the proof of this property is not easy, and meanwhile it is only known to hold in the context of partial differential equations. This latter point unfortunately results in certain restrictions for its application near the boundary. In particular, when we deal with the boundary regularity for Neumann problems, the original approach of DiBenedetto [4] has to be evoked and adapted.
Our arguments can be adapted to the borderline cases. In particular, when q = 1, the arguments deal with the degenerate, parabolic p-Laplacian; when p = 2, the porous medium equation can be treated; when q = p − 1, we are back to our first work [1]; see also [10] for nonnegative solutions. Beyond these borderline cases, it will be a subject of our next investigations.
. This guarantees that all the integrals in (1.7) are convergent.
A function u that is both a local weak sub-solution and a local weak super-solution to (1.2) -(1.3) is a local weak solution.

Notion of Solution to the Dirichlet Problem. A function
for all non-negative test functions ζ ∈ W 1,q+1 loc 0, T ; L q+1 (E) ∩ L p loc 0, T ; W 1,p o (E) . Moreover, settingq := min{2, q+1}, the initial datum is taken in the sense that for any compact The Dirichlet datum g is attained under u ≤ (≥)g on ∂E in the sense that the traces of (u − g) ± vanish as functions in Notice that no a priori information is assumed on the smoothness of ∂E.
A function u that is both a weak sub-solution and a weak super-solution to (1.4) is a weak solution.

Notion of Solution to the Neumann Problem. A function
is a weak sub(super)-solution to (1.5), if for every compact set K ⊂ R N and every sub-interval ψ(x, t, u)ζ dσdt for all non-negative test functions Here dσ denotes the surface measure on ∂E. The Neumann datum ψ is reflected in the boundary integral on the right-hand side. Moreover, the initial datum is taken as in the Dirichlet problem.
A function u that is both a weak sub-solution and a weak super-solution to (1.5) is a weak solution.

ENERGY ESTIMATES
In this section we present certain energy estimates for weak sub(super)-solutions to (1.2) -(1.3). They are analogs of the energy estimates derived in [1], which will be referred to for details. Moreover, it is noteworthy that they actually hold true for all p > 1 and q > 0.
The different roles played by sub-solutions and super-solutions are emphasized. When we state "u is a sub(super)-solution..." and use " ± " or " ∓ " in what follows, we mean the subsolution corresponds to the upper sign and the super-solution corresponds to the lower sign in the statement.
For any k ∈ R, we denote the truncated functions For w, k ∈ R we define two non-negative quantities For b ∈ R and α > 0, we will embolden b α to denote the signed α-power of b as Throughout the rest of this note, we will use the symbols to denote (backward) cylinders with the indicated positive parameters; when the context is unambiguous, we will omit the vertex (x o , t o ) from the symbols for simplicity. First of all, we present energy estimates for local weak sub(super)-solutions defined in Section 1.4.1. The proof is similar to [1, Proposition 3.1], which we refer to for details. The only difference is that in the present situation, u p−1 must be replaced by u q in terms related to the time derivative and g ± has to be defined as above. Since the testing functions and the treatment of the term containing the vector-field A remain unchanged, the constant γ on the right-hand side of the estimates is independent of q.
Next, we consider the situation near the initial level t = 0 when a continuous datum u o is prescribed. Suppose the level k satisfies The following energy estimate can be obtained as in [1,Proposition 3.2].
and every non-negative, piecewise smooth cutoff function ζ independent of t and vanishing on ∂K R (x o ), there holds ess sup Next, we turn our attention to the energy estimates near S T . When dealing with Dirichlet data we need to assume the following restrictions on the level k The following energy estimate can be obtained as in [1,Proposition 3.3].
2), and every non-negative, piecewise smooth cutoff function ζ vanishing on Finally, we deal with the energy estimates for the Neumann problem (1.5). The following can be obtained as in [1,Proposition 3.4].
There exists a constant γ > 0 depending on C o , C 1 , p and the structure of ∂E, such that for all cylinders Q R,S with the vertex (x o , t o ) ∈ S T , every k ∈ R, and every non-negative, piecewise smooth cutoff function ζ vanishing on

PRELIMINARY TOOLS
For a compact set K ⊂ R N and a cylinder Q := K × (T 1 , T 2 ] ⊂ E T we introduce numbers µ ± and ω satisfying In this section, we collect some lemmas, which will be the main ingredients in the proof of Theorem 1.1. The first one is a De Giorgi type lemma, which actually holds for all p > 1 and q > 0. There exists a constant ν ∈ (0, 1) depending only on the data, such that if The De Giorgi iteration has been performed in [12, Lemma 2.2] for super-solutions, whereas the proof for sub-solutions is analogous. In order to obtain the present formulation, choose a = 1 2 and replace M by ξω. If |µ ± | > 8ξω, there is nothing to prove. In the opposite case, the assumption |µ ± | ≤ 8ξω allows us to estimate max{L q−1 , M q−1 } by max{9 q−1 , 1}M q−1 . Therefore, the critical number ν depends only on the data. The next lemma is a variant of the previous one, involving quantitative initial data. Again, it actually holds for all p > 1 and q > 0 Proof. After enforcing that |µ − | ≤ 8ξω, this is essentially the content of [12, Lemma 3.1] for super-solutions, the case of sub-solutions being similar. More precisely, one has to choose a = 1 2 , replace M by ξω and note that the constant max{L q−1 , M q−1 } is controlled by max{1, 9 q−1 }(ξω) q−1 whenever |µ − | ≤ 8ξω. This allows to choose the parameter θ in [12, Lemma 3.1] in the form ν o (ξω) q+1−p for some ν o depending only on the data.
The previous lemma propagates pointwise information in a smaller cube, without a time lag. The next lemma translates measure theoretical information into a pointwise estimate over an expanded cube of later times. This is essentially the expansion of positivity for the degenerate, parabolic p-Laplacian established in [5]; see also [6, Chapter 4, Proposition 4.1]. As such it actually holds for p > 2 and q > 0.
Introduce the parameters Λ, c > 0 and α ∈ (0, 1). Suppose that There exist constants b > 0 and η ∈ (0, 1) depending only on the data, Λ, c, a and α, such that Proof. We may assume (x o , t o ) = (0, 0) and prove the case of super-solutions only as the other case is similar.
Here the symbol u q k has been emboldened to denote the signed power of u k defined in Section 2. To proceed, we define

HereĀ is defined bȳ
where y denotes the truncation y := min max{y, 0}, 1 − 1 2 q (cω) q . Meanwhile, one verifies that the structure conditions In order to eliminate the dependence on ω in the structure conditions ofĀ, we consider the which satisfies Here A is defined by Thus, an easy calculation shows that A satisfies the conditions In other words, the function v is a non-negative, local, weak super-solution to the parabolic p-Laplacian type equation (3.1) in Q. This allows us to apply the expansion of positivity in [6, Chapter 4, Proposition 4.1]. The measure theoretical information for u implies a similar inequality for u k ; in fact we have Taking into account −Λω ≤ u k ≤ − 1 2 cω, the information that u k (·, 0) ≥ µ − + aω can be converted into an estimate from below for v. Indeed, by the mean value theorem, we estimate An application of [6, Chapter 4, Proposition 4.1] to v (with C ≡ 0 and M = aω q ) yields that for some positive constants η, δ ∈ (0, 1) and b > 1 depending only on the data C o , C 1 , p, N and on α, For v this means that v(·, t) ≥ η aω q a.e. in K 2 for all t in the interval We revert to the original function u with the aid of the mean value theorem. More precisely, we estimate for all t in the above interval. Redefining η a/ γ as η and γ 2−p b p−2 δ as b, the claim follows.
1. An inspection of the above proof shows that η = γa for some positive γ depending only on the data, α, c and Λ. The conclusion of Lemma 3.3 holds true for a smaller η by properly making a smaller.
The following lemma examines the situation when pointwise information is given at the initial level. It actually holds for p > 2 and q > 0. There exists a positive constant ν 1 depending only on the data, c and Λ, such that if , provided the cylinders are included in Q.
Proof. Suppose u is a local, weak super-solution as the other case is similar. Moreover, we may assume (x o , t o ) = (0, 0). Introduce v like in the proof of Lemma 3.3, which turns out to be a non-negative, local, weak super-solution to the parabolic p-Laplacian type equation (3.1) in Q. Using the mean value theorem, the information that u(·, 0) ≥ µ − + ηω in K yields that v(·, 0) ≥ γηω q in K for some positive γ = γ(q, c, Λ). Consequently, we may apply [6, Chapter 3, Lemma 4.1] or [12,Lemma 3.2] to v. For a ∈ (0, 1) at our disposal we have v ≥ aγηω q a.e. on K 1 for some constantc ∈ (0, 1) depending on C o , C 1 , p, N . As in the proof of Lemma 3.3 we convert this into an estimate for u. First, the scaling in time gives Note that ν 1 depends on C o , C 1 , p, q, N, c, Λ and a. As in the proof of Lemma 3.3, we may apply the mean value theorem to estimate for some positive γ = γ(q, c, Λ) and therefore on K 1 Finally, choosing the free parameter a such that aγ/ γ = 1/2 on the one hand determines the value of ν 1 in dependence on the data, c and Λ, and on the other hand implies the desired bound from below.

THE FIRST PROOF OF THEOREM 1.1
The proof of Theorem 1.1 in this section relies on the expansion of positivity from Lemma 3.3. This important tool simplifies our arguments, though the attainment of it is difficult and turned out to be a major achievement in the recent theory, cf. [5,6]. As such the same simplification can be carried out in [1]. On the other hand, the argument of this section does not seem applicable directly to the boundary regularity for the Neumann problem. For this reason, we will give a second proof of Theorem 1.1 in Section 5, referring back to our previous arguments in [1] that are modeled on [4].
E T with a radius ≤ 1 and set For some A > 1 to be determined in terms of the data, we may assume that Our proof unfolds along two main cases, namely for some ξ ∈ (0, 1) to be determined, (4.3) when u is near zero: µ − ≤ ξω and µ + ≥ −ξω; when u is away from zero: µ − > ξω or µ + < −ξω.
Note that (4.3) 1 implies that |µ ± | ≤ 2ω. We deal with this case in Sections 4.2 -4.4; the other case will be treated in Section 4.5.

4.2.
Reduction of Oscillation Near Zero-Part I. In this section, we will assume that (4.3) 1 holds true. We work with u as a super-solution near its infimum. To proceed further, we assume 2 ω, will be considered later. Observe that (4.4) implies (4.5) Let us consider for instance the first case, i.e. (4.5) 1 , as the other one can be treated analogously. Hence we have 1 4 ω ≤ µ + ≤ 2ω and Lemma 3.3 is at our disposal with c = 1 4 and Λ = 2. Suppose A is a large number, and consider the "bottom" sub-cylinder of Q (Aθ), that is, One of the following two alternatives must hold true: Here the number ν ∈ (0, 1) is determined in Lemma 3.1 in terms of the data. First suppose (4.6) 1 holds true. An application of Lemma 3.1 (with ξ = 1 4 ) gives us that, recalling |µ − | ≤ 2ω due to (4.3) 1 , (4.7) u ≥ µ − + 1 8 ω a.e. in 1 2 Q. Here the notation 1 2 Q should be self-explanatory in view of Lemma 3.1. In particular, the above pointwise lower bound of u holds at the time level t o = −(A − 1)θ p for a.e. x ∈ K 1 2 , which serves as the initial datum for an application of Lemma 3.2. Indeed, we fix ν o in Lemma 3.2 depending on the data and choose ξ ∈ 0, 1 8 so small that i.e. we choose Thus, enforcing |µ − | ≤ ξω, we obtain that which in turn yields a reduction of oscillation (4.9) ess osc Here in (4.8) we have tacitly used the fact that q < p − 1 in the determination of ξ. Keep also in mind that A > 1 is yet to be determined in terms of the data. The case µ − > ξω will be treated in Section 4.5; whereas if −2ω < µ − < −ξω, we may apply Lemma 3.4 with c = ξ, Λ = 2 and η = η o ∈ 0, 1 8 . Indeed, fixing ν 1 in Lemma 3.4 depending on the data and ξ, we choose η o to satisfy Here we have tacitly used the fact that p > 2 in the determination of η o . In this way, Lemma 3.4 asserts that which yields the reduction of oscillation (4.10) ess osc

Reduction of Oscillation Near Zero-Part II.
In this section, we still assume that (4.3) 1 and (4.4) hold true. However, we turn our attention to the second alternative (4.6) 2 . We work with u as a sub-solution near its supremum. Since under our assumptions there holds µ + − 1 4 ω ≥ µ − + 1 4 ω, we may rephrase (4.6) 2 as µ + − u ≥ 1 4 ω ∩ Q > ν| Q|. Then it is not hard to see that there exists Indeed, if the above inequality does not hold for any s in the given interval, then implying a contradiction to the above measure theoretical information. Recall that due to (4.5) 1 we actually have 1 4 ω ≤ µ + ≤ 2ω. Then, based on the above measure theoretical information at t * , an application of Lemma 3.3 with c = 1 4 , Λ = 2 and a fixed constant a = 1 8 = 1 2 c yields constants b > 0 and η 1 ∈ (0, 1) depending only on the data, such that µ + − u(·, t) ≥ η 1 ω a.e. in K 1 2 for all times t * + 1 2 bω q−1 (η 1 ω) 2−p p ≤ t ≤ t * + bω q−1 (η 1 ω) 2−p p . Now, we determine A such that the set inclusion − θ 1 4 p , 0 ⊂ t * + 1 2 bω q−1 (η 1 ω) 2−p p , t * + bω q−1 (η 1 ω) 2−p p is satisfied. To this end, we first consider the requirement 0 ≤ t * + bω q−1 (η 1 ω) 2−p p , which follows if the stronger condition is fulfilled. This leads to the choice Note that we may assume A > 1, since we could choose a smaller constant η 1 in the definition of A by Remark 3.1 and use the fact that p > 2. The second requirement −θ 1 4 p ≥ t * + 1 2 bω q−1 (η 1 ω) 2−p p is satisfied if we are able to verify the stronger condition which is eqivalent to 1 − 1 2 ν + 1 4 p ≤ 1 2 A. Since ν ∈ (0, 1), the last inequality holds true if A ≥ 4. However, as mentioned above, we may assume it by making η 1 smaller. Altogether, the above analysis determines A through η 1 and yields a reduction of oscillation (4.11) ess osc To summarize, let us define where 1 2 ξ is as in (4.9), 1 2 η o is as in (4.10) and η 1 is as in (4.11). Combining (4.9) -(4.11) gives the reduction of oscillation (4.12) ess osc In order to iterate the above argument, we introduce we need to choose 1 = λ for some λ ∈ (0, 1), such that To this end, we first let

Reduction of Oscillation Near Zero
Concluded. Now we may proceed by induction. Suppose that, up to i = 1, 2, · · · j − 1, we have built For all the indices i = 1, 2, · · · j − 1, we alway assume that (4.3) 1 holds true, i.e., µ − i ≤ ξω i and µ + i ≥ −ξω i . By this means the previous arguments can be repeated and we have for all i = 1, 2, · · · j, Consequently, iterating the above recursive inequality we obtain for all i = 1, 2, · · · j,

Reduction of Oscillation Away From Zero.
In this section, let us suppose j is the first index satisfying the second case in (4.3), i.e. either µ − j > ξω j or µ + j < −ξω j . Let us treat for instance µ − j > ξω j , for the other case is analogous. We observe that since j is the first index for this to happen, one should have Here, we assume that there exists an index j − 1 such that the first case in (4.3) is fulfilled. This can be justified by choosing ω = 1 ξ u L ∞ (E T ) in Section 4.1. Moreover, since Q j ⊂ Q j−1 , by the definition of the essential supremum one estimates As a result, we have (4.14) The bound (4.14) indicates that starting from j the equation (1.2) resembles the parabolic p-Laplacian type equation in Q j . We drop the suffix j from our notation for simplicity, and It is straightforward to verify that v belongs to the function space (1.6) defined on Q and satisfies Moreover, since ω/µ − ≤ 1/ξ, we have that To proceed, it turns out to be more convenient to consider w := v q , which because of (4.15) belongs to the function space (1.6) q=1 defined on Q and satisfies where we have defined the vector-field A by for a.e. (x, t) ∈ Q, any y ∈ R and any ζ ∈ R N . This time y is defined by Employing (4.15) again, we verify that there exist positive constants C o = γ o (p, q, ξ)C o and C 1 = γ 1 (p, q, ξ)C 1 , such that Note that ξ is already fixed in (4.8) in terms of the data. To proceed, we introduce the function which satisfies and belongs to the function space (1.6) q=1 defined on Q. Here the function A is defined by and subject to the structure conditions This shows that w is a local weak solution to the parabolic p-Laplacian type equation in Q.
First proved in [4] the power-like oscillation decay for solutions to this kind of degenerate parabolic equation is well known by now. We state the conclusion in the following proposition in a form that favors our application, and refer to the monographs [3,14] for a comprehensive treatment of this issue.  According to (4.14) we find that Further, by definition of the corresponding cylinders, we obtain that Q σ ( θ) ⊂ Q, provided holds true. This can be achieved by choosing σ small enough, i.e.
In view of the lower and upper bound on the ratio ω/µ − , the number σ can be chosen only in terms of the data, such that ess osc w ≤ γ ω r β1 ≤γ r β1 forγ = γ γ/ξ and for any 0 < r ≤ , with some β 1 ∈ (0, 1) depending only on the data. Since p > 2, we may estimate θ > θ o := γ ξ 2−p and conclude that ess osc Reverting to w and using the fact that q + 1 < p and (4.14) in order to estimate we obtain that ess osc where depends only on the data. Recalling the definition of w, by the mean value theorem and (4.15) one easily estimates that for some positive γ = γ(q, ξ), ess osc w.
Finally, we revert to u and the suffix j, and use (4.14) to estimate µ − j ≤ 1+ξ 1−η ω j , which leads to , whenever 0 < r < j . Since ≤ 1, we have that ω j ≤ ω 1 ≤ max{ω, L} =: ω L and therefore we obtain that Q r ( ). Combining this with (4.13) and (4.19), we arrive at the following: for all 0 < r < , ess osc Without loss of generality, we may assume the above oscillation estimate holds with replaced by some ∈ (r, ). Then taking = (r ) 1 2 and properly adjusting the Hölder exponent, we obtain the power-like decay of oscillation ess osc .
At this stage, the proof of Theorem 1.1 can be completed by a standard covering argument.

THE SECOND PROOF OF THEOREM 1.1
The purpose of this section is to present another proof of Theorem 1.1 without using the expansion of positivity (Lemma 3.3). As we shall see, the arguments in Section 5.2 are similar to that of Section 4.2. The main difference appears in Section 5.3. To avoid using Lemma 3.3 as done in Section 4.3, we perform an argument of DiBenedetto [4], adapted in [1]. The virtual advantage of this section is that the proof relies solely on the energy estimates in Proposition 2.1. As such it offers an amenable adaption near the boundary given Neumann data, cf. Section 6.3.

The Proof
Begins. The set-up is the same as in Section 4.1. Namely, we introduce the quantities {µ ± , ω, θ, L, A} and the cylinders Q (Aθ) ⊂ Q o . Moreover, they are connected by the intrinsic relation (4.1). For a positive ξ to be determined, the proof unfolds along two main cases, as in (4.3).

5.2.
Reduction of Oscillation Near Zero-Part I. Like in Section 4.2, we assume that (4.3) 1 holds and work with u as a super-solution near its infimum. Then we proceed with the assumption (4.4), which implies one of (4.5) holds. We may take (4.5) 1 , such that 1 4 ω ≤ µ + ≤ 2ω. The second proof departs from here. Suppose that for somet ∈ − (A − 1)θ p , 0 , where ν is the constant determined in Lemma 3.1 in terms of the data. According to Lemma 3.1 applied with ξ = 1 4 , we have u ≥ µ − + 1 8 ω a.e. in (0,t) + Q 1 2 (θ), since the other alternative, i.e., |µ − | ≥ 2ω, does not hold due to (4.3) 1 . This pointwise information parallels (4.7) in Section 4.2. Similar arguments can be reproduced as in Section 4.2 to obtain the reduction of oscillation as in (4.9) -(4.10). In particular, only Lemma 3.1, Lemma 3.2 and Lemma 3.4 are used. In this process we fix the constant ξ as in (4.8) depending on the data and A, which will be chosen next in terms of the data.
Recall that due to (4.5) 1 we have 1 4 ω ≤ µ + ≤ 2ω. Thus our assumptions for the following Proof. For ease of notation, we set s = 0. Further, for δ > 0 and 0 < ε ≤ 1 8 to be determined by the data and ν, we consider Q := K × (0, δε 2−p θ p ] and k = µ + − εω ≥ 1 8 ω. Applying the energy estimate in Proposition 2.1 with a standard non-negative time independent cutoff function ζ(x, t) ≡ ζ(x) that equals 1 on K (1−σ) for some σ ∈ (0, 1) to be fixed later, vanishes on ∂K and satisfies |Dζ| ≤ (σ ) −1 , we obtain for all 0 < t < δε 2−p θ p that Defining kε = µ + −εεω for someε ∈ (0, 1 2 ), we estimate the term on the left-hand side bŷ Further, note that by the mean value theorem and the restriction 1 Next, by (5.3) we obtain for the first term on the right-hand side of the energy estimate that and by the choice of ζ and u ≤ µ + for the second term on the right-hand side thaẗ Combining the preceding estimates leads to Rewriting the fractional number of integrals on the right-hand side and using the mean value theorem as well as the restrictions 1 4 ω ≤ µ + ≤ 2ω and k ≥ 1 8 ω yields the bound where γ depends only on q. Inserting this into the previous inequality, we conclude that 16N . Then, we choose δ small enough that γδ σ p ≤ 1 16 ν and ε small enough that δε 2−p ≥ 1, where we take into account that p > 2. Redefiningεε as ε, we finish the proof of the lemma.
Proof. Consider the cylinder K 2 × (−(A − 1)θ p , 0] and a time independent cutoff function ζ(x, t) ≡ ζ(x) vanishing on ∂K 2 and equal to 1 in K such that |Dζ| ≤ 2 −1 . Applying the energy estimate from Proposition 2.1 with levels k j = µ + − 2 −j−1 εω for j = 0, · · · , j * − 1, we obtain thaẗ Taking into account the choice of ζ, by an application of the Sobolev imbedding [3, Chapter I, Proposition 3.1] and the preceding estimate we conclude that Hence, for the quantity Y n = |A n |/|Q n | we deduce the recursive inequality and γ only depends on the data. Thus, the lemma on fast geometric convergence, i.e. [3, Chapter I, Lemma 4.1], ensures the existence of a constant ν 1 ∈ (0, 1) depending only on the data such that Y n → 0 if we assume that the smallness condition Y o ≤ ν 1 holds true.
At this stage, we conclude the reduction of oscillation in the remaining case where (5.2) and (5.3) hold. To this end, denote by ε ∈ (0, 1), γ > 0 and ν 1 ∈ (0, 1) the corresponding constants from Lemmas 5.1, 5.2 and 5.3. Choose a positive integer j * large enough that γ j p−1 p * ≤ ν 1 and Q 1 2 ((A − 1)θ) ⊃ Q 1 4 (θ), where A = 2 j * (p−2) + 1. Hence, applying in turn Lemmas 5.2 and 5.3, we arrive at µ + − u ≥ εω 2 j * +1 a.e. in Q 1 4 (θ). This gives the reduction of oscillation ess osc Recall the reduction of oscillation achieved in Section 5.2 via arguments of Section 4.2. Namely, 1 2 ξ is chosen in the reduction of oscillation (4.9) and 1 2 η o is chosen in the reduction of oscillation (4.10). Combining all cases gives the reduction of oscillation exactly as in (4.12) with the choice from which the rest of the proof can be reproduced just like in Section 4.

PROOF OF BOUNDARY REGULARITY
Since Theorems 1.2 -1.4 can be proved in a similar way as interior Hölder continuity, we will only give sketchy proofs, where we keep reference to the tools and strategies used in the interior case and highlight the main differences. Let θ = ( 1 4 ω) q+1−p . We may assume that Like in the proof of interior regularity, we start by distinguishing between the main cases when u is near zero: µ − ≤ ω and µ + ≥ −ω; when u is away from zero: µ − > ω or µ + < −ω.
The second case reduces to the corresponding estimate for weak solutions to parabolic p-Laplacian equations; see [3, Chapter III, Lemma 11.1]. In the first case, which implies |µ ± | ≤ 2ω, we proceed by a comparison to the initial datum u o . More precisely, we assume that since otherwise, we would obtain the bound ess osc Qo u ≤ 2 ess osc As both cases can be treated analogously, we consider only the second inequality with µ − and work with u as a super-solution. Using |µ − | ≤ 2ω, Lemma 3.2 (with ξ = 1 4 ) yields a constant ν o ∈ (0, 1) depending only on the data, such that Thus, we arrive at the reduction of oscillation ess osc Q1 u ≤ 7 8 ω.
Finally, taking the initial datum into account, we conclude that ess osc Q1 u ≤ max 7 8 ω, 2ω uo ( ) . Now we may proceed by an iteration argument as in [1, Section 7.1] to conclude the proof.
6.2. Proof of Theorem 1.3. Consider the cylinder is the constant from the geometric condition (G). Further, we assume that (x o , t o ) = (0, 0) for ease of notation and define For some A > 1 to be determined in terms of the data, we may assume that Q (Aθ) ⊂ Q o , such that ess osc otherwise we would have Our current hypothesis to proceed consists of the measure information (6.2) and −2ω < µ − < −ξω as we have assumed µ + ≥ −ξω in (6.1) 1 . Departing from this, we have two ways to proceed: one is to use the expansion of positivity (Lemma 3.3); the other is to follow the arguments in Sections 5.3.2 -5.3.3. We only describe the first option.
In fact, by Lemma 3.3, the measure information (6.2) translates into the pointwise estimate u ≥ µ − + ηω a.e. in Q 1 2 (A 2 θ) for some η ∈ (0, 1) depending on the data and ξ. This gives us a reduction of oscillation as usual, and hence finishes the reduction of oscillation under the condition (6.1) 1 The constant A 2 is determined by the data in this step, through b and η of Lemma 3.3. The final choice of A is given by the larger one of A 1 and A 2 .
As in the interior case, we repeat the arguments inductively until the second case of (6.1) is satisfied for some index j for the first time. Starting from j, the equation behaves like the parabolic p-Laplacian type equation within Q j ∩ E T . In order to render this point technically, we adapt the proof for interior regularity, where we use in particular the boundary regularity result [1, Proposition 7.2] for the parabolic p-Laplacian near the lateral boundary.
6.3. Proof of Theorem 1.4. First of all, we observe that the second proof of interior regularity (Theorem 1.1) in Section 5 is based solely on the energy estimates in Proposition 2.1 and a corresponding Hölder estimate for solutions to the parabolic p-Laplacian.
A key ingredient -the Sobolev imbedding (cf. [3, Chapter I, Proposition 3.1]) -was used in order to establish Lemma 3.1, Lemma 3.2, Lemma 3.4, Lemma 5.1 and Lemma 5.3, assuming the functions (u − k) ± ζ p vanish on the lateral boundary of the domain of integration. This assumption in turn is fulfilled by choosing a proper cutoff function ζ. In the boundary situation similar arguments have been employed in Section 6.2 or in Section 6.1 by restricting the value of the level k according to the Dirichlet data as in (2.2) or the initial data as in (2.1).
However, in the current situation of Neumann data the functions (u−k) ± ζ p under conditions of Proposition 2.4 do not vanish on S T and therefore such a Sobolev imbedding cannot be used in general. On the other hand, a similar Sobolev imbedding (cf. [3, Chapter. I, Proposition 3.2]) that does not require functions to vanish on the boundary still holds for the functional space u ∈ C 0, T ; L p (E) ∩ L p 0, T ; W 1,p (E) .
The appearing constant now depends on N , the structure of ∂E and the ratio T /|E| p N , which is invariant for cylinders of the type Q = K × (− p , 0] and Q ∩ E T as well, provided ∂E is smooth enough. In particular, Lemma 3.1, Lemma 3.2, Lemma 3.4, Lemma 5.1 and Lemma 5.3 can be proved in this boundary setting.
Finally, we remark that the use of De Giorgi's isoperimetric inequality (cf. [ is attached to S T and is so small that t o − p−1 > 0. According to the preceding considerations we proceed exactly as in the second proof of interior regularity in Section 5. Obviously, in the present situation all cylinders have to be intersected with E T . In this way, we conclude a reduction of oscillation for the lateral boundary point (x o , t o ). The analysis has been carried out in [1, Appendix A] for q = p − 1. However, the same proof actually works for all p > 1 and q > 0 after minor changes.
In particular, when u is a local weak solution, u + and u − are non-negative, local weak subsolutions to (1.2) -(1.3). By [2,Theorem 4.1], they are locally bounded and hence u is also.
In order to formulate an analog of Lemma A.1 near the lateral boundary S T for a sub(super)solution u to (1.4), consider the cylinder Q R,S = K R (x o ) × (t o − S, t o ) whose vertex (x o , t o ) is attached to S T . Further, for a level k satisfying (2.2), we are concerned with the following truncated extension of u in Q R,S :

Moreover, the extension of A defined by
A(x, t, u, ζ) := A(x, t, u, ζ) in Q R,S ∩ E T , is a Carathéodory function satisfying (1.3) with structure constants C o and C 1 replaced by min{1, C o } and max{1, C 1 }, respectively. In this situation, the following lemma holds.