CONTINUITY UP TO THE BOUNDARY FOR OBSTACLE PROBLEMS TO POROUS MEDIUM TYPE EQUATIONS

. We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This result seems to be new even for signed solutions to the (obstacle free) Cauchy-Dirichlet problem to the singular porous medium equation, which is retrieved as a special case.


Introduction
In the present paper, we consider the obstacle problem to partial differential equations whose prototype is the porous medium equation (PME) where q ∈ (0, ∞).We refer to 0 < q < 1 as the degenerate case, and to q > 1 as the singular case.In the literature, the singular porous medium equation is often called the fast diffusion equation.For the standard theory of equations of this type, see e.g.[12,15,28,29,30].More specifically, we consider problems of the form          ∂ t |u| q−1 u − div A(x, t, |u| q−1 u, ∇u) = 0 in Ω T , u ≥ ψ in Ω T , u = g on S T , u = g o on Ω × {0}, where A is a vector field satisfying the assumptions specified in Section 2.2, and S T = ∂Ω × (0, T ) denotes the lateral boundary of Ω T .
In nonlinear potential theory, the obstacle problem is used as a standard tool.Supersolutions, which are defined in potential theory via a parabolic comparison principle, are connected to weak supersolutions in certain ways.In particular, they are typically approximated by weak supersolutions.Such approximants can be constructed via successive obstacle problems, in which regularity up to the boundary plays an important role, see e.g.[17] for the elliptic case and [18] for the porous medium equation.
Local Hölder continuity for nonnegative solutions to the (obstacle free) porous medium equation was proven by DiBenedetto and Friedman in [14] in the degenerate case.For the corresponding result in the singular case see [15].Signed solutions to the porous medium type equations were treated in [22].
In presence of an obstacle, local Hölder continuity was shown for quasilinear equations in [27], and problems of quadratic growth were treated in [11].For the porous medium equation, an analogous result was shown in [8] in the degenerate regime in case of nonnegative obstacles.The corresponding result in the singular case was proven in [10], where also more general equations of porous medium type were included.The local Hölder continuity in case of signed obstacles has been treated in [24], in which the result covered the full range of the parameter q.
In the obstacle free case continuity up to the boundary for solutions to degenerate porous medium type problems of the form (1.1) (1,3,4) was proven in [13].Boundary regularity for nonnegative solutions to the degenerate model equation in terms of Perron's method was developed in [20].The results were extended in [4], where the authors proved a barrier characterization of regular boundary points, including also domains that are non-cylindrical.Furthermore, it is worth to mention that existence of a (unique) solution that is continuous up to the boundary has been shown in [1,2], in which the author considers nonnegative very weak solutions to the model equation in non-cylindrical domains for the full parameter range.Boundary regularity for doubly nonlinear equations was shown in [6] in the doubly degenerate case.Several tools in the aforementioned paper are also adapted and applied in this paper.
Questions on existence related to obstacle problems for porous medium type equations are addressed in [3,9,25,26].
At this stage, we state our main result.
Theorem 1.1.Let q ∈ (0, ∞) and Ω ⊂ R n be a bounded open set which satisfies the geometric density condition (2.9) for some α * , ̺ o > 0. Furthermore, suppose that the obstacle function ψ satisfies (2.6), the lateral boundary datum g satisfies (2.7) and the initial datum g o (2.8).Let u be a weak solution to the obstacle problem (1.1) according to Definition 2.1.Then u can be extended as a function that is continuous up to the boundaries S T and Ω× {0}.More precisely, if K ⊂ Ω× (0, T ) is a compact set, there exists a modulus of continuity ω determined by C o , C 1 , n, q, α * , ̺ o , u ∞ , ω ψ , ω g and dist(K, Ω × {0}) such that for every (x o , t o ) ∈ K ∩ S T and (x 1 , t 1 ) ∈ K. Furthermore, if K ⊂ Ω is a compact set, there exists a modulus of continuity ω determined by C o , C 1 , n, q, u ∞ , ω ψ , ω go and dist(K, ∂Ω) such that for every x o ∈ K and (x 1 , t 1 ) ∈ K × [0, T ).If ψ satisfies (2.6), g ∈ C(Ω T ∪ ∂ p Ω T ), where ∂ p Ω T = (Ω × {0}) ∪ S T , satisfies (2.7) and the initial datum g o satisfies (2.8) with g o ≡ g(•, 0), then u can be extended as a function in Ω T ∪ ∂ p Ω T that is continuous up to ∂ p Ω T .Moreover, there exists a modulus of continuity ω determined by for every (x o , t o ) ∈ ∂ p Ω T and (x 1 , t 1 ) ∈ Ω T ∪ ∂ p Ω T .
1.1.Strategy.The strategy of the proof can roughly be divided into two parts.In Section 2, we show that with appropriate truncation levels the truncated solutions to the obstacle problem are weak sub-or supersolutions to the obstacle free problem.Together with a suitable extension argument we are able to treat the truncated solutions as local sub-or supersolutions to the obstacle free problem also in cylinders that intersect the complement of the domain.In this case, admissibility of the truncation levels depends on the extrema of both the obstacle and the boundary datum.
In the second part we show continuity up to the parabolic boundary, which we prove separately up to the lateral and initial boundaries.Near the lateral boundary we consider backward cylinders whose vertices are attached to the lateral boundary, while near the initial boundary we use forward cylinders attached to the initial boundary.The proofs are based on estimates for sub-and supersolutions to the obstacle free problem in both cases.
We separate the cases where the solution is near zero and where it is away from zero.In the former case, extrema of the solution are controlled by the oscillation of the solution, obstacle and the boundary data in a given cylinder, while the latter is complementary to the former case.In both of these cases, we need to split the proof into further alternatives.Very roughly speaking, we distinguish whether quantities related to the extrema or oscillation of the solution are large or small compared to quantities related to the extrema or oscillation of the boundary values and obstacle function.By choosing said quantities appropriately, we are able to truncate the solution, extend it to the complement, and treat the extended truncations as subor supersolutions to the obstacle free problem by results in Section 2.3.Then, the reduction in oscillation is shown by using tools from the obstacle free case, which are described in Section 3. At this point, boundary conditions together with (2.9) obviously play an important role.
The tools in Section 3 for reduction in oscillation are divided into the local case and the case near the initial boundary, which are further divided into tools used in the cases near zero and away from zero.In the cases away from zero, we apply a De Giorgi type iteration scheme, which is essentially based on energy estimates given in the beginning of Sections 3.1 and 3.2.In the case near zero on the initial boundary, we use a result on propagation of positivity, see Lemma 3.12.In the case near zero on the lateral boundary we use separate tools in the singular and degenerate cases.In the degenerate case we are able to exploit a standard De Giorgi type iteration argument, while in the singular case the same approach is not applicable as such.In that case, we are required to rely on a heavier tool; a certain formulation of expansion of positivity, see [22], which is adapted to our setting in Proposition 3.5.
1.2.On the notion of solution.In this paper we consider a notion of solution to the obstacle problem, in which the variational inequality holds in a local sense.The boundary conditions are imposed separately and they are attained in the standard Sobolev sense slice-wise on the lateral boundary and in the L q+1 -sense on the initial boundary.The reason for using the local notion mainly is that it allows comparison maps whose boundary values do not necessarily coincide with the boundary values of the solution, which makes the proofs in Section 2.3 attainable without unnecessary complications.Furthermore, this allows us to relax the assumptions on the boundary data.It was shown in [9] that the local inequality implies the global one (which was used e.g. in [9,25,26]), provided that the domain and boundary data satisfy appropriate regularity assumptions and the boundary data is attained in the aforementioned sense.The reverse direction also holds under similar assumptions, which is discussed in Appendix B. 1.3.Novelty and significance.To the authors' knowledge, the issue of boundary regularity has not been addressed in the literature even in the obstacle free case for signed solutions to the singular porous medium equation.Our result already covers this as a special case.Furthermore, in contrast to the results in the local case ([8, 10, 24]), we are able to treat an obstacle and boundary datum with general moduli of continuity.
Indeed, the approach we use has certain differences compared to the proofs for local Hölder regularity to the obstacle problem in [8,10,24].In the aforementioned cases, the obstacle was involved in all parts of the proof.Furthermore, the construction of a shrinking sequence of cylinders was strongly affected by continuity properties of the obstacle.Namely, the first step of the construction and oscillation decay estimate (which was eventually proved in non-intrinsic cylinders) relied heavily on the assumption that the obstacle is Hölder continuous.In our proof of boundary regularity the assumptions on the obstacle can be relaxed.Due to the conditions on the boundary of the domain and chosen alternatives, the treatment of reduction in oscillation can be reduced entirely to a problem in the obstacle free case provided that the boundary satisfies rather mild regularity assumptions.Furthermore, this allows us to construct cylinders and prove an oscillation decay estimate in a manner which is not affected by any specific continuity properties of the obstacle or the boundary datum.
Moreover, instead of assuming that the solution is bounded in the first place, the proof is included in Section 2.4 based on a maximum principle for subsolutions to the obstacle free problem.
Acknowledgments.K. Moring has been supported by the Magnus Ehrnrooth Foundation.Both authors have also been supported by the FWF-Project P31956-N32 "Doubly nonlinear evolution equations".The authors would like to thank Christoph Scheven for useful discussions and hospitality during the second author's visit at the University of Duisburg-Essen.
2. Definition and some properties of solutions 2.2.Definition of solutions.In (1.1) we assume that A : , it is measurable with respect to (x, t) ∈ Ω T for all (u, ξ) ∈ R × R n and continuous with respect to (u, ξ) ∈ R × R n for a.e.(x, t) ∈ Ω T .Moreover, we suppose that A satisfies the structure conditions where C o , C 1 > 0 are given constants.For a given obstacle function ψ : Ω × [0, T ) → R we define classes of functions . Furthermore, we denote For the sake of generality, we define weak solutions to the obstacle problem as local solutions that attain the lateral boundary values g or the initial boundary values g o in the following way.For a discussion of the connection to global weak solutions we refer to Appendix B.
).Here, we define the time term by ( Throughout the paper, we will use the following notion when referring to a solution to the partial differential equation (1.1) 1 without obstacle.
is a local weak sub(super)solution to (1.1) holds for all test functions ϕ ∈ C ∞ 0 (Ω T , R ≥0 ).(2) Moreover, we say that u is a weak sub(super)solution to (1.1) 1 with initial values g o ∈ L q+1 loc (Ω) if it is a local weak sub(super)solution and holds true for every compact set K ⊂ Ω.
In the proof of boundary regularity, we will make following assumptions on the obstacle ψ and boundary data g and g o : (2.6) , with modulus of continuity ω g (•) on S T ; (2.7) Furthermore, we will assume that the domain Ω satisfies the geometric density condition, i.e. there exist α * ∈ (0, 1) and ̺ o > 0, such that for all x o ∈ ∂Ω and ̺ ∈ (0, (2.9) 2.3.Truncation of solutions to the obstacle problem.In this section, we show that truncations of weak solutions to the obstacle problem to (1.1) with suitable truncation levels are weak sub(super)solutions to (1.1) in the obstacle-free setting in the intersection of the space-time cylinder Ω T with cylinders with vertex in Ω T or on its lateral boundary S T .
To this end, for v ∈ L 1 (Ω T ), v o ∈ L 1 (Ω) and h > 0 we define a mollification in time by The properties of the this mollification procedure are collected in the following lemma.We refer to [19,Lemma 2.9] and [7,Appendix B].
] h be defined by (2.10) with v o ∈ X.Then the following statements hold true: (1) We have that With this mollification in time at hand, we are able to proceed as follows.

With these choices, let us denote [[u]]
At this point, suppose that k > sup Q∩ΩT ψ.We define a comparison map Further, if [[u]] i > k, note that by our choice of ϕ we have that Moreover, by choosing i large enough, there holds ψ − Now for all such i we have Fix σ > 0 and ε = 1 ϕ ∞ min 1 2 (k − sup Q∩ΩT ψ), σ and choose compactly supported α and η in Q R,S ∩ Ω T such that αη ≡ 1 in spt(ϕ).Using this together with (2.11), the fact that and integration by parts, we derive the following estimate for the parabolic part of (2.2) By integration by parts, for the term on the third line of the right-hand side of the preceding inequality we obtain that ¨ΩT Further, note that the terms on the first, second and fourth line of the right-hand side of the penultimate inequality vanish when i → ∞.Thus, by combining the estimates and passing to the limit i → ∞, we get that lim sup , for the diffusion part of the variational inequality (2.2) we compute that Further, using (2.1) 1 , for the term on right-hand side we find that By combining all the estimates and dividing by ε we infer that Thus, finally passing to the limit σ ↓ 0, we obtain that we use the dominated convergence theorem to pass to the limit k ↓ sup Q ψ, which completes the proof of (2).
In order to show that (1) holds, we define a comparison map where ) together with small enough ε > 0. Since we have that Proceeding similar as in the proof of (2), we conclude the proof of (1).
For a cylinder Q R,S (x o , t o ) with a vertex (x o , t o ) ∈ S T such that 0 < S < t o , we define an extension of A(x, t, u, ξ) by Observe that A is a Carathéodory function satisfying (2.1) with C o and C 1 replaced by min{1, C o } and max{1, C 1 } respectively.
In the following we show that we can extend min{u, k} and max{u, k} by constant k from Q R,S ∩ Ω T into Q R,S as super-and subsolution, respectively, to the obstacle free porous medium type equation with vector field A.
In the proof we will exploit the following Hardy's inequality.
Lemma 2.5.Let Ω be a bounded open set in R n which satisfies (2.9) and u ∈ H 1 0 (Ω).Then, there exists a constant c depending only on n and α * such that Remark 2.6.Observe that the positive geometric density condition is not the weakest possible assumption in Lemma 2.5.The result holds true under the assumption that R n \ Ω is uniformly 2-thick, see e.g.[21].Thus, Lemma 2.7 is also valid under the same assumption.
Lemma 2.7.Suppose that Ω ⊂ R n is a bounded open set and satisfies (2.9).Let (x o , t o ) ∈ S T and consider the cylinder Q R,S (x o , t o ) with R > 0 and 0 < S < t o .Let u be a weak solution to the obstacle problem according to Definition 2.1 with obstacle ψ satisfying (2.6) and lateral boundary values g satisfying (2.7).
(1) If k ≤ inf ST ∩QR,S g, then is a weak supersolution in Q R,S in the sense of Definition 2.2 with A replaced by the vector field A.
is a weak subsolution in Q R,S in the sense of Definition 2.2 with A replaced by the vector field A.
Proof.We start with the case (2).For the proof of the property that 4, which does not necessarily vanish on the boundary of Ω.As a starting point, we take inequality (2.12), which reads as

Denote the inner parallel set by Ω
λ with a numerical constant c > 0. Now we plug ϕη λ in the place of ϕ, which gives us that We estimate the second term above by where we used Hölder's inequality in the third line and Hardy's inequality slicewise in the sixth line.Observe that Hardy's inequality is applicable, since k ≥ sup QR,S ∩ST g and thus ϕ(•, t)(u(•, t) − k) + ∈ H 1 0 (Ω) for a.e.t ∈ (0, T ).As η λ → χ Ω pointwise when λ → 0, we obtain that I and the term that is left in II converge to the corresponding integral over Ω T .
By passing to the limit σ ↓ 0, we obtain that By summing up the two (in)equalities above it follows that ), which completes the proof for case (2).The case (1) can be treated analogously.

2.4.
Boundedness of solutions to the obstacle problem.Next, we show that weak solutions to the obstacle problem are bounded if the boundary data are bounded and the obstacle is continuous up to the boundary.First, we state a maximum principle for weak subsolutions to (obstacle free) porous medium type equations.
Lemma 2.8.Let u ∈ L 2 (0, T ; H 1 (Ω)) ∩ L q (Ω T ) be a weak subsolution to (1.1) Proof.Fix ε > 0 and let t 1 = ε/2 and t 2 ∈ (4ε, T ).We can write the mollified weak formulation of (2.4) for weak subsolutions as , where σ > 0 and α = α(t) ≤ 1 is a piecewise affine approximation of χ (ε,τ ) (t) with τ ∈ (4ε, t 2 ).Observe that ϕ is an admissible test function, since ϕ ≤ 1 and ϕ ∈ L 2 (0, T ; H 1 0 (Ω)).When passing to the limit h → ∞, the right hand side vanishes.Further, by structure condition (2.1) 1 for the divergence part on the left-hand side of the preceding inequality we obtain that Moreover, by the fact that the map s → (s−k)+ (s−k)++σ is increasing, and by integration by parts, we estimate the parabolic part on the left-hand side of the penultimate inequality by By collecting all the estimates above, we obtain that By passing to the limit ε → 0 and using initial condition (2.13), we have u(x, τ ) ≤ k for a.e.x ∈ Ω.Since this holds for a.e.τ ∈ (0, t 2 ) and t 2 ∈ (0, T ) is arbitrary, we have that u ≤ k a.e. in Ω T .
Lemma 2.9.Let u be a weak solution to the obstacle problem according to Definition 2.1 with obstacle ψ ∈ C(Ω T ), lateral boundary values g ∈ L 2 (0, T ; H 1 (Ω)) ∩ L ∞ (Ω T ) with g ≥ ψ a.e. in Ω T and initial boundary values Proof.Observe that u is bounded from below, since u ≥ ψ and ψ ∈ C(Ω T ).

Auxiliary results
In this section, we collect further tools that we will need in the proof of Theorem 1.1.For w, k ∈ R we define The following estimates follow from the definition above, see e.g.[5, Lemma 2.2].
Lemma 3.1.There exists a constant c = c(q) > 0 such that for all w, k ∈ R and q > 0, the inequality 3.1.Tools for local sub(super)solutions.In this section, assume that u is a local weak sub(super)solution to (1.1) 1 in some cylinder We start with the standard energy estimate.The following statement is retrieved as a special case of [6, Proposition 2.1] by setting p = 2.For a detailed proof see also [5,Proposition 3.1].
(2) If u is a local weak supersolution in Q to (1.1) 1 according to Definition 2.2, then there exists c = c(C o , C 1 ) > 0 such that Throughout Section 3.1, we will use parameters µ − , µ + ∈ R and ω > 0 satisfying 3.1.1.Tools for the case near zero.First we state a shrinking lemma in the degenerate case.The proof is an adaptation of [5, Lemma 4.2] to the porous medium setting.
Lemma 3.3.Let 0 < q < 1, j * ∈ N be an arbitrary positive integer and ε ∈ (0, 1).Denote θ = (2 −j * εω) q−1 and suppose that Q 2̺,θ̺ 2 (x o , t o ) ⋐ Q.Let u be a locally bounded local weak sub(super)solution to (1.1) 1 in Q. Suppose that for some α ∈ (0, 1) Then we state a De Giorgi type lemma in the local, obstacle free case.See [6, Lemma 3.1] in connection with [23, Lemma 2.2].Lemma 3.4.Let q > 0 and u be a locally bounded local weak sub(super)solution to (1.1) 1 in Q.Let θ = (ξω) q−1 for some ξ ∈ (0, 1) and Q ̺,θ̺ 2 (x o , t o ) ⋐ Q.There exists a constant ν ∈ (0, 1) depending only on C o , C 1 , n and q such that if In the singular case, we exploit the result on expansion of positivity in [22, Proposition 2.2].Since it is stated and proved for a different formulation of (1.1) 1 , which is equivalent in the case of locally bounded solutions, we have to add the assumption that max{|µ − |, |µ + |} ≤ 16M in the following proposition.This is due to the fact that the expansion of positivity in [22, Proposition 2.2] is shown for the quantity ±((µ ± ) q − u q ) in our notation, but we use the result for ±(µ ± − u).Proposition 3.5.Let q > 1 and assume that u is a locally bounded local weak sub(super)solution to (1.1) 1 according to Definition 2.2 in a cylinder Q ⊃ B 16̺ (y)× (s, s + δM q−1 ̺ 2 ] for some M > 0 and δ ∈ (0, 1) determined below.Suppose that for α ∈ (0, 1), and that max{|µ + |, |µ − |} ≤ 16M .Then, there exist constants ξ, η, ε, δ ∈ (0, 1) depending only on C o , C 1 , n, q and α such that or the inequality ± µ ± − u(•, t) ≥ ηM a.e. in B 2̺ (y) holds true for all times Proof.We only give the proof for a local weak subsolution, since the proof for supersolutions is analogous.For v := u q and m = 1 q we have that Therefore, ∇v is the weak gradient of v. Further, since ∇v m is zero a.e. in the set {v = 0}, we have that ∇v m = m|v| m−1 ∇vχ {v =0} , a.e. in Ω T .Now, we define the vector field Then A satisfies the structure conditions for a.e.(x, t) ∈ Ω T and every (v, ξ) ∈ R × R n .In this setting, we are able to apply the result on expansion of positivity in the singular case in [22].Defining Since there exists a constant Then, by using [22, Proposition 2.2] with M replaced by c 1 M , we infer that there exist constants ξ, ε, δ, η ∈ (0, 1) depending only on C o , C 1 , n, m and α such that a.e. in B 2̺ (y) At this point, we use the fact that 0 < m < 1 and the assumption that max{|µ Combining the preceding inequalities, reverting to u = v m and defining ξ := 1 8 c 1 ξm , δ, this translates to the conclusion that Remark 3.6.Replacing c 1 M by κc 1 M with κ ∈ [κ o , 1] for some κ o ∈ (0, 1) in the proof of Proposition 3.5, we obtain that under the hypotheses of Proposition 3.5 there holds µ ± > 8ξκ o M or Thus, redefining ξ as ξκ o and η as ηκ o , pointwise positivity can be claimed as close to s as we need.
The next result is an analogue to [6, Lemma Lemma 3.7.Let q > 0 and introduce the parameters Λ, c > 0 and α ∈ (0, 1).Assume that u is a locally bounded local weak sub(super)solution in Q to (1.1) 1 .Suppose that cω ≤ ±µ ± ≤ Λω and for some 0 < a ≤ 1 2 c there holds Then, there exist constants η, b ∈ (0, 1) depending only on C o , C 1 , n, q, Λ, c, a and α such that 3.1.2.Tools for the case away from zero.Next we will state and prove tools for the obstacle free problem in the case where u is bounded away from zero.This is expressed in terms of µ ± and ω by assuming that holds true for some constant ξ ∈ (0, 1).First, we state an auxiliary lemma concerning the truncation levels used in this subsection.
Then we give a version of De Giorgi type lemma.Lemma 3.9.Let Q ̺, 1  2 θ̺ 2 (z o ) ⋐ Q be a parabolic cylinder such that either (3.1) or (3.2) holds true.Let η ∈ 0, ξ 2 .Assume that u is a locally bounded, local weak sub(super)solution to (1.1) 1 in the sense of Definition 2.2 in Q.Then there exists a constant ν 1 = ν 1 (n, q, C o , C 1 , ξ) ∈ (0, 1) such that if We prove the result for supersolutions in case (3.1),where u is above zero.The considerations for the remaining three cases, i.e. for subsolutions in case (3.1) and for super-and subsolutions in case (3.2) are analogous.We omit z o for simplicity and start the proof by defining Observe that clearly k j > 0 for all j ∈ N 0 .Let 0 ≤ ϕ ≤ 1 be a cut-off function that equals 1 in Q j+1 and vanishes on the parabolic boundary of From the fact that 0 < µ − ≤ u < k j in the set A j := {u < k j } ∩ Q j and the energy estimate in Lemma 3.2 (2), we obtain with a constant c = c(C o , C 1 , n, q, ξ), where we used Lemma 3.8.With these estimates at hand, we infer in particular sup Dividing by |Q j+1 | and denoting Y j = |A j |/|Q j |, we conclude that In the case where u is away from zero, the next lemma transfers positivity from a measure condition at a single time slice to a pointwise estimate in a whole cylinder.The proof applies Lemma 3.9.Lemma 3.10.Let Q ̺,θ̺ 2 (z o ) ⋐ Q be a parabolic cylinder such that either (3.1) or (3.2) holds true.Assume that u is a locally bounded, local weak sub(super)solution in Q according to Definition 2.2.Then for any ν ∈ (0, 1) there exists a constant Proof.We prove the result for supersolutions in case (3.1).The remaining three cases are analogous.For simplicity, we let z o = (0, 0) and omit it from the notation.
By applying Lemma 3.2 (2) we get for c = c(C o , C 1 , q, ξ) > 0 by using Lemma 3.8.By combining the two estimates above and using k j − k j+1 = ξ2 −(j+1) ω we have By summing this over j = 3, ..., s o + 1 for some s o ∈ N ≥3 we obtain At this stage we apply Lemma 3.9 to conclude that 3.2.Tools for sub(super)solutions near the initial boundary.Throughout this section, we are concerned with weak sub(super)solutions to (1.1) 1 in a subset for every nonnegative, time-independent piecewise smooth cutoff function η vanishing on ∂B ̺ (x o ).
Proof.The proof follows the idea from [5, Proposition 3.2].Applying Lemma 3.2 with t o = s, s − t 1 in place of s for some t 1 > 0 and ϕ ≡ η in B ̺ (x o ) × (t 1 , s), integrating over t 1 ∈ (0, h) for some h > 0, dividing by h and using that all terms are non-negative yields that sup h<t<s ˆB̺(xo)×{t} In the degenerate case 0 < q < 1, by Lemma 3.1 and the triangle inequality we estimate In the singular case q > 1, by Lemma 3.1 and Hölder's inequality we find that . Therefore, recalling the choice of the level k and using that u takes the initial condition in the sense of Definition 2.2, we conclude that the second term on the right-hand side of (3.3) vanishes in the limit h ↓ 0 and we obtain the claimed energy estimate.
In this section, we will use parameters µ − , µ + ∈ R and ω > 0 satisfying

3.2.1.
Tools for the case near zero.We recall a result on propagation of positivity with pointwise information given at the initial time slice, see [6,Lemma 3.2].It will be used in reduction in oscillation up to the initial boundary.Lemma 3.12.Let u be a locally bounded weak sub(super)solution to (1.1) 1 in Q + .Set θ = (ξω) q−1 for some ξ ∈ (0, 1) and suppose that Q + ̺,θ̺ 2 (x o , t o ) ⋐ Q + .There exists a positive constant ν o depending only on C o , C 1 , n and q such that if (3.4) Further, we suppose that there either holds First, we state an analogue to Lemma 3.8.
Since µ + ≥ u > k j > 0 in the set A j = {u > k j } ∩ Q j , from the energy estimate, Lemma 3.11, we obtain that Observe that by Lemma 3.13 we have that sup and using that ν ∈ (0, 1), we have that If 0 < q < 1, let where A > 1 and ξ ∈ (0, 1) are constants determined by C o , C 1 , n, q and α * later on.
If q > 1, we use the rescaling argument in Appendix A with M = 2 ξ u ∞ .We let ̺o > 0 be so small that 4 max{osc Q∩ΩT ψ, osc Q∩ST g} ≤ 1.This is possible since ψ is uniformly continuous in Ω T and g in Q ∩ S T .Now we let In the whole range 0 < q < ∞ we let θ o = ω q−1 o and ̺ o = ̺o 32 and define cylinders In this section, we say that u is near zero if and that u is away from zero if 4.1.Reduction in oscillation near zero.Observe that with our choice of ω o we start in case (4.2) where u is near zero.In particular, we have that ) Together with (4.1), we conclude that max osc or one of the following cases must hold: At this stage, assume that the first case of (4.5) holds and set k := µ + o − 1 8 ω o .Now u k := max{u, k} is a local weak subsolution to the obstacle free porous medium equation (1.1) 1 in Q o ∩ Ω T by Lemma 2.4 (2).Note that by (4.5) 1 and Lemma 2.7 (2), u k can be extended from Q o ∩ Ω T to Q o by k such that the resulting function is a weak subsolution to the porous medium equation in Q o .Thus, we can work with Q o as an interior cylinder.In the following, we omit (x o , t o ) to simplify our notation.
Observe that the definitions of k and u k and the positive geometric density condition (2.9) imply , and j * ≥ 2 in Lemma 3.3 be so large that Lemma 3.4 is applicable, and fix A = 2 j * +3 and ξ = 2 −j * .Then, the aforementioned lemmas together with (4.6) yield that 2 .
Let η := min{2 −(j * +4) , η 1 } and δ = 1 − η.By similar considerations for u − µ − o if (4.5) 2 holds, we deduce osc By taking the case that both conditions in (4.5) are violated or (4.4) does not hold into account, we obtain that osc In the case q > 1, by taking into account Remark 3.6, we assume that 1 − ε ≤ 3 4 in Proposition 3.5 and let ξ denote the according constant.Thus, by means of Proposition 3.5 from (4.6) we deduce that In the case, we use Lemma 3.7 with a = 1 2 ξ and recall that b ∈ (0, 1) to conclude that By choosing η = min 1 8 η 1 , η 2 and δ = 1 − η we obtain that osc By similar arguments for (4.5) 2 and by taking the case that both conditions in (4.5) are violated or (4.4) does not hold into account, we find that osc Now, consider the whole range 0 < q < ∞.For the corresponding parameters η and δ in the degenerate and singular cases, respectively, let , and let If (4.2) holds, we may again use alternatives analogous to (4.5) in the cylinder Q ′ 1 , and proceed iteratively.In this way we can build a sequence with indices i = 1, ..., j − 1 in which (4.2) holds true up to some j ∈ N.For each i = 1, 2, ..., j, we define 2 , µ − i := inf Qi u and µ + i := µ − i + ω i and deduce that for each i = 1, 2, ..., j, we have that 4.2.Reduction in oscillation above zero.Suppose that j is the first index for which (4.3) holds.In this section, we assume that we are in the case (4.3) 2 , whereas the case (4.3) 1 will be treated in the next section.Observe that µ Now, define θ * = µ + j q−1 .If q < 1, we immediately have that θ * ≤ ω q−1 j .On the other hand, since u is near zero for j − 1, we have that By (4.7) we conclude that At least one of the following must hold: max{osc Qj ∩ΩT ψ, osc Qj ∩ST g} = 1 4 ω j or Suppose that (4.9) 1 holds.By setting k := µ + j − 3 8 ω j , we have that u k = max{u, k} is a weak subsolution to (1.1) 1 in Q j .Then (2.9) implies that If (4.9) 2 true, then we have that u k = min{u, k} with k = µ − j + 3 8 ω j is a weak supersolution to (1.1) 1 in Q j .Again, we have that In this case, using Lemma 3.10 shows that On the other hand, if (4.9) fails, we have that osc Again, define µ − j+1 = inf Qj+1∩ΩT u and µ + j+1 = µ − j+1 + ω j+1 .Observe that since ω j+1 ≤ ω j we have that ξω j+1 ≤ ξω j < µ − j ≤ µ − j+1 , which implies that we are again in case (4.3) 2 for j + 1, and also that Now, we can iterate with the choices ̺ j and λ := 1 8 , which yields that for any i > j, there holds osc Qi∩ΩT u ≤ ω i .

4.3.
Reduction in oscillation below zero.Suppose that (4.3) 1 holds, which implies that analogously as in the case where u is above zero.If q < 1 we have θ * ≤ θ.If q > 1, it follows that At this stage, for i > j we iterate the arguments in this section with the choices ̺ j , λ := 1  8 .This shows that for any i > j, we have that osc Qi∩ΩT u ≤ ω i .
Collecting the results from the three preceding sections, we infer that osc where Q Suppose that 0 < q < 1. Recalling the definition of r i , we find that If q > 1 we have in a similar fashion that In the following, by the short notation λ (s) we mean λ if 0 < q < 1, and λ s if q > 1.
Hence, for γ 1 = log δ log λ (s) we find that osc where c = c(C o , C 1 , n, q, α * ).Let r ∈ (0, r o ).Then there exists i ∈ N 0 such that r i+1 < r ≤ r i .Since r i /r i+1 = λ −1 (s) , it immediately follows that osc where c = c(C o , C 1 , n, q, α * ).In a similar fashion, for r ∈ [r o , ̺ o ) we obtain that osc Thus, altogether we have that osc for every r ∈ (0, ̺ o ).Without loss of generality, we can replace ̺ o by any ̺ ∈ (r, ̺ o ).By choosing ̺ = √ r̺ o and by observing that ̺o = 32̺ o , for any r ∈ (0, ̺ o ) we obtain the desired oscillation decay estimate osc This concludes the proof of the claim for (x o , t o ) ∈ S T in Theorem 1.1.
Remark 4.1.Observe that ̺o depends on the instance t o and the estimate in Theorem 1.1 will depend on the distance to the boundary Ω × {0} in this case.This is due to the fact that lateral and initial boundary data may not be compatible (i.e., continuous at the corner points).If we suppose g ∈ C(Ω T ∪ ∂ p Ω T ) and g o ≡ g(•, 0), we do not need this restriction.Indeed, by replacing oscillation and extrema of g over Q ∩ S T by Q ∩ ∂ p Ω T in the argument, we can extend u k by k to the negative times in the proof, which will be again a weak sub(super)solution to the obstacle free problem in the whole cylinder Q.

Continuity up to the initial boundary
We set x o = 0 and define If 0 < q < 1, let If q > 1, we use the rescaling argument in Appendix A with M = 2 u ∞ and let Again, by uniform continuity of ψ and g o we suppose that ̺ o is so small that 4 osc Q ψ ≤ 1 and 4 osc Then, in the whole range 0 < q < ∞ we have that In the following, we will omit (x o , 0) in order to simplify our notation.Further, we say that u is near zero if taking into account that (5.5) 3 may hold or (5.4) is violated and combining the results, we have that osc We define θ 1 := 1 4 ω 1 q−1 and ̺ 1 := λ̺ o , where λ := 1 2 7 8 o .Now, we set At this stage, we iterate the preceding arguments.More precisely, for all indices i = 1, ..., j − 1 for which (5.2) is satisfied we define Then, for all i = 1, ..., j we obtain that osc Qi u ≤ ω i .5.2.Reduction in oscillation above zero.Consider the first index j for which (5.2) is false.In this section, we assume that (5.3) 1 holds, i.e. u is above zero.For (5.3) 2 we refer to the next section.Since (5.2) holds true for j − 1, for δ = 7   8   we have that If 0 < q < 1, then θ * ≤ θ j , and if q > 1, then θ * < 16 δ q−1 θ j .

5.3.
Reduction in oscillation below zero.Suppose that j ∈ N is the first index such that (5.3) 2 holds true.This implies that where δ = 7 8 .By combining the estimates above we obtain that −µ − j ≤ 4 δ ω j .
Using Lemmas 3.14 and 3.15 for alternatives (5.6) 1,2 as in Section 5.2, defining , where ̺j+1 = λ̺ j with λ := j+1 |.Now we can repeat the arguments above.Let B j = B ̺j and for any i > j define i := sup Qi u and µ − i := µ + i − ω i and conclude that for any i > j there holds osc Qi u ≤ ω i .
In order to treat the divergence part in the variational inequality, we claim that Estimating the integral on the right-hand side of the preceding inequality by Young's inequality for convolutions, we conclude that Combining the preceding two inequalities and recalling the definition of δ i , we obtain that ¨ΩT ∇ e Furthermore, using that α(ϕ ε − η) u q − u q i ∂ t u i = 1 hi α(ϕ ε − η) u q − u q i (u − u i ) ≥ 0 by construction of the mollification and that u q i ∂ t u i = ∂ t |u i | q+1 , integrating by parts yields the estimate Furthermore, we have Finally, by inserting the preceding estimates into the time term and passing to the limit ε ↓ 0 by means of the dominated convergence theorem, we conclude the claim of the lemma.
For b ∈ R and α > 0 we denote the signed α-power of b by b
we conclude the proof by using fast geometric convergence lemma [15, Lemma 5.1, Chapter 2].

Qj+1∩ΩT u ≤ ω j = 4 u
≤ max δω j , 4 osc in Definition 2.1.Lemma B.2.A global weak solution to the obstacle problem according to Definition B.1 attains initial values in the sense of (2.3) in Definition 2.1.
We say that u ∈ K ψ,g (Ω T ) is a weak solution to the obstacle problem with lateral boundary values g ∈ K ψ (Ω T ) if (2.2) holds.(3) Moreover, we say that u is a weak solution to the obstacle problem with initial values g o ∈ I ψo (Ω) if (2.2) holds and ) ηω in the set {u ≤ k j+1 }, by Hölder's and Sobolev's inequalities [15, Proposition 4.1, Chapter 2] we obtain that Let u be a weak sub(super)solution to (1.1) 1 in Ω T with an initial datum g o ∈ L ∞ loc (Ω).There exists a constant c = c(C o , C 1 ) > 0 such that for all cylinders Q + ̺,s (x o , 0) ⊂ Ω T and every k ∈ R satisfying k ≥ sup B̺(xo) g o for weak subsolutions, k ≤ inf B̺(xo) g o for weak supersolutions, 3.2.2.Tools for the case away from zero.Next we prove propagation in measure and a De Giorgi type lemma near the initial boundary.In the following, we suppose that µ + − 1 4 ω ≥ sup B̺(xo) g o for weak subsolutions, µ − + 1 4 ω ≤ inf B̺(xo) g o for weak supersolutions.
By combining Lemmas B.2 and B.3 we arrive at the following result.Lemma B.4.Let Ω ⊂ R n be a bounded open set which satisfies (2.9).Suppose that ψ ∈ C(Ω T ) and that g and g o satisfy the assumptions in Definition B.1.Let u be a global weak solution according to the aforementioned definition.Then u is a weak solution to the obstacle problem according to Definition 2.1 satisfying (1), (2) and (3).Remark B.5. Observe that the assumption (2.9) is not the weakest possible for Ω in Lemmas B.3 and B.4.Both hold true under the assumption that R n \ Ω is uniformly 2-thick, see also Remark 2.6.