Elliptic regularization of some semilinear parabolic free boundary problems

We prove existence of strong solutions to a family of some semilinear parabolic free boundary problems by means of elliptic regularization. Existence of solutions is obtained in two steps: we first show some uniform energy estimates and then we pass to the weak limit. To carry out this second step, we establish uniform non-degeneracy estimates for the approximating sequence as well as parabolic non-degeneracy and optimal regularity for the limit solution. To the best of our knowledge, this is the first time the elliptic regularization approach is used in the context of parabolic obstacle problems.

Such parabolic free boundary problems appear in Chemical engineering (see [29]) and transport of thermal energy in plasma (see [24]).The mathematical study of its solutions and their free interfaces was initiated by Caffarelli in [8,9] (see also [10]) in the case γ = 1, in which (1.1) reduces to a version of the Stefan problem (see e.g.[20,10,19,4]), and later extended in the elliptic setting to the range γ ∈ (0, 2) by Alt and Phillips [2] (see also [27,28]).In the parabolic setting, we mention the works of Weiss [35,36] (see also [12], by Choe and Weiss), where the classification of blow-ups and fine regularity properties of the free boundary were established, together with sharp bounds on the (parabolic) Hausdorff dimension of ∂{u > 0}.Notice that in [35] the singular range γ ∈ [0, 1) was considered too.To the best of our knowledge, there are not further results in the range γ < 1, except the very recent papers [14,15], [16] and [17], where the authors have studied the range γ < 0 in the elliptic framework.
Existence of weak/strong solutions to (1.1) is well-known (see e.g.[20]).However, the elliptic regularization approach is, to the best of our knowledge, new in the context of parabolic obstacle problems, and presents some interesting features with respect to the classical approach.It is a variational approximation procedure (also known as WIED method), introduced in the works of Lions [23] and Oleinik [26] (see also the article of De Giorgi [13]).Due to its intrinsic flexibility, elliptic regularization has been applied in many contexts, see for instance [21,25,6,1,7,30] in the parabolic setting, and also [31,32] in the context of nonlinear wave equations (see also the more recent works [5,3] concerning systems with strong competition and nonlocal parabolic problems, respectively).
The main idea of this approach is to approximate solutions to the parabolic problem (1.1) by using (absolute) minimizers of the functional where ε ∈ (0, 1) is a free parameter and w + is the standard notation for max{w, 0}.Indeed, it is not difficult to check that, when γ ∈ (1, 2), any critical point u ε satisfies (1.4) in the weak sense (the case γ = 1 is more involved, see the comments below).Problem (1.4) is exactly (1.1) "up to" the extra term −ε∂ tt u ε , which makes the equation elliptic for every ε > 0, but degenerate as ε → 0: it is thus reasonable to expect that, if minimizers of (1.3) enjoy some uniform boundedness properties, we may pass to the weak limit into (1.4) (along a suitable subsequence ε j → 0) and obtain a solution u to (1.1).This plan has two main steps: 1. Prove uniform energy estimates in the spirit of [31,32] and deduce compactness of families of minimizers in suitable Sobolev spaces.2. Pass rigorously to the limit into (1.4) to obtain (1.1).The main difficulties arising from the fact that we are dealing with a free boundary problem appear precisely in the second step and when γ = 1 (the case γ ∈ (1, 2) is quite standard since f γ is continuous).Indeed, there are two issues that must be treated with care, and are the core of this paper.First, we prove that minimizers u ε of E ε satisfy (1.4).This is not obvious, as mentioned above, since the function w → w + is not differentiable at w = 0; see Lemma 3.1 below.Second, once we have the limit of the minimizers, denoted by u (which is obtained by compactness and after passing to a subsequence), we show that we can take the limit ε → 0 in (1.4) and obtain (1.1).This requires to prove that χ {uε>0} → χ {u>0} and this is also non-obvious.Indeed, to prove it we establish a uniform non-degeneracy property for the family u ε close to free boundary points, as well as optimal regularity and parabolic non-degeneracy estimates for the limit u.All these ingredients and fine estimates for the measure of ∂{u > 0} are crucially exploited to pass to the limit as ε → 0.
It is important to stress that our approach is not only aimed to construct solutions in an alternative way with respect to the existing literature (regularizing χ {u>0} in the nonlinearity f γ (u) and using classical tools, see [20]).Indeed, we expect some features of the elliptic problem (with ε > 0) being inherited by solutions to (1.1), and we hope this could lead to a different approach to study parabolic free boundary problems in the future.For this, the first step is to establish the appropriate convergence of approximating minimizers towards solutions to (1.1), as we do in this paper.
The techniques we use in this article allow us to prove existence of solutions for every γ ∈ [1, 2), and some of them do not work for γ ∈ (0, 1) (see Remark 2.6 below).The main reason is that we use in a crucial way the weak formulation of the problem, obtained by considering competitors of the form u + δϕ with δ > 0 and ϕ ∈ C ∞ c (Q), and the weak convergence in H 1 .Since for γ ∈ (0, 1) the function u → f γ (u) has a strong singularity at u = 0, we cannot use the standard weak formulation, but we would need to consider competitors constructed through domain deformations, in order to avoid differentiating the function u → u γ + .This approach is much more delicate and will be treated in a forthcoming article.
Remark 1.1.As mentioned above, the elliptic regularization approach is quite flexible and more general/complex equations can be attacked with similar techniques (see for instance [1,32]).Here we just mention that our methods can be slightly modified to prove existence of strong solutions to where Ω ⊂ R n is a bounded domain and homogeneous Dirichlet or Neumann conditions are posed on ∂Ω × (0, ∞).The proof of such fact is postponed to Section 4; see Corollary 4.1.
In the next subsection, we introduce the functional setting and we state our main result.
1.1.Functional setting and main result.We will work with the space . ., n), for every r > 0. The functional (1.3) is well-defined on U with values in [0, +∞]: we will seek for minimizers u ∈ U , subject to the initial condition u| t=0 = u 0 in the sense of traces, where In other words, we will minimize E ε over the space We remark that the assumptions on the initial data guarantee that u 0 (seen as a function of x and t) belongs to U 0 : in particular, U 0 = ∅.This will be useful in the proof of Lemma 2.2, where we will use u 0 as a competitor and prove the elementary, yet crucial, estimate (2.3).We do not expect such assumptions to be optimal, but they have the advantage to make the arguments direct and easy-readable.We now introduce the notion of strong solutions to (1.1).
Definition 1.2.Let n ≥ 1 and γ ∈ [1, 2).We say that a function u is a strong solution to , and u| t=0 = u 0 in the sense of traces.• The integral relation Note that this definition differs from one for weak solutions in the fact that ∂ t u is an L 2 -function and the integral (1.7) involves the term ∂ t uη and not −u∂ t η.In particular, the fact that ∂ t u is in L 2 yields that ∆u is also in L 2 .
Finally, we state our main result.
2) , u 0 as in (1.6), and f γ as in (1.2).Then there exist a sequence of minimizers {u ε j } j∈N of (1.3) in U 0 and a continuous strong solution u ∈ U 0 to (1.1) such that The rest of the paper is organized as follows.In Section 2 we prove existence and uniqueness of minimizers of E ε in U 0 , and we establish the uniform energy estimates given in (2.6) and (2.7).As a corollary, we prove weak and strong convergence of minimizers (see Proposition 2.1 below).Section 3 is divided into three parts.First, in Section 3.1 we write the Euler-Lagrange equations for minimizers and show uniform non-degeneracy near free boundary points.Then, in Section 3.2 we establish optimal regularity and parabolic non-degeneracy results for solutions to parabolic obstacle problems.Finally, all these ingredients are combined crucially in Section 3.3 to prove Theorem 1.3.In the last section of the article, Section 4, we comment on the slight modifications that one must perform in our methods in order to construct strong solutions to (1.5) with homogeneous Dirichlet or Neumann conditions on the parabolic boundary.

Energy estimates and Compactness
In this section we establish the following result.Proposition 2.1.Let {u ε } ε∈(0,1) ∈ U 0 be a family of minimizers of E ε .Then there exist u ∈ U 0 and a sequence ε j → 0 such that Before addressing to the proof of the above statement, we show some basic properties of minimizers of the functional E ε .
2.1.Existence of minimizers.It is not difficult to show that for every ε ∈ (0, 1) the functional E ε has a unique minimizer in U 0 (see Lemma 2.2 below).Before presenting the proof, we introduce the functional Since v and u coincide at t = 0, minimizing E ε in U 0 is equivalent to minimizing J ε in the same space.
Working with the functional J ε allows us to keep the notations easier.
Lemma 2.2.For every ε ∈ (0, 1), the functional J ε defined in (2.2) has a unique minimizer v ε in U 0 .Such minimizer satisfies v ε ≥ 0 a.e. in Q.Moreover, there exists a constant C > 0, depending only on n, γ, and u 0 , such that Proof.By considering u 0 (as a function of x and t) we see that, thanks to (1.6), we have (2.4) and thus J ε ≡ +∞ on U 0 .To prove the existence of a minimizer, we consider a minimizing sequence Consequently, for every fixed r > 0, we have for some C > 0 independent of j.Since for every j we have v j | t=0 = u 0 , given δ > 0, for a.e.s ∈ (0, r 2 ) it holds Integrating between 0 and r 2 and taking δ := 1/(2r 2 ) we find for some new C > 0 independent of j.Therefore, it follows that {v j } j is bounded in loc (Q), up to passing to another subsequence and, by the fact that U 0 is closed and convex, we conclude v ∈ U 0 .Finally, by lower semicontinuity and Fatou's lemma, we obtain Now, uniqueness follows by the convexity of J ε .Furthermore, since u 0 ≥ 0, if v ε is a minimizer then (v ε ) + is an admissible competitor and thus, by minimality, 3) follows from the minimality of v ε and the bound (2.4).

2.2.
Proof of Proposition 2.1.Proposition 2.1 will be obtained as a consequence of the following energy estimates.

Proposition 2.3. (Energy estimates)
There exists a constant C > 0, depending only on n, γ, and u 0 , such that for every minimizer u ε of E ε we have and, for every r ≥ ε, Indeed, with these uniform bounds at hand we readily obtain the main result of this section.
Proof of Proposition 2.1.In light of (2.6) and (2.7), the family {u ε } ε∈(0,1) of minimizers of E ε is uniformly bounded in H 1 (Q + r ) for every fixed r > 0 and thus a standard diagonal procedure shows the existence of a sequence ε j → 0 and u ∈ U 0 such that the first limit in (2.1) is satisfied.By the Sobolev embedding, we immediately obtain also the second limit, up to passing to another subsequence.
In the rest of the section we establish Proposition 2.3.This is the first key result of the paper and will be obtained as the byproduct of Lemma 2.4 and Corollary 2. 5.In what follows we will use the following notations: we will denote R + := (0, +∞) and, for a minimizer v of J ε , we write where Since v is a minimizer, it is clear that the functions t → I(t), t → R(t) and t → e −t (I(t) + R(t)) are locally integrable in R + .Consequently, the function Notice that, by definition, (2.9) In the following result we give an alternative expression for E ′ which will be useful later to obtain energy estimates.
Corollary 2.5.There exists a constant C > 0, depending only on n, γ, and u 0 , such that for every minimizer v ε of J ε , we have and, for every r ≥ 0, Proof.First, since I ≥ 0, the function t → E(t) is non-increasing and so E(t) ≤ E(0) for all t ≥ 0.
Recalling that E(0) = J ε (v ε ) and the bound (2.3) in Lemma 2.2, it follows that (2.20) for all t ≥ 0, where C > 0 is a constant depending only on n, γ, and u 0 .Now, from Lemma 2.4 we have E ′ = −2I a.e. in R + .Integrating this expression and using (2.20), we obtain for all t ≥ 0 (recall that E ≥ 0 by definition) and (2.18) follows by the arbitrariness of t ≥ 0 and the definition of I.
To prove (2.19) for r ≥ 0, it is enough to use (2.20) to deduce that With the previous result at hand we can now establish the uniform bounds of Proposition 2.3.

Proof of the main theorem
This section is devoted to the proof of Theorem 1.3, which is split in three steps.We first write the Euler-Lagrange equations for minimizers u ε of E ε (see Lemma 3.1 below) and then, in Lemma 3.2, we establish a uniform non-degeneracy property.Next, we establish parabolic non-degeneracy and optimal regularity results for solutions to parabolic obstacle problems.Last, we combine all these ingredients to pass two the limit as ε → 0, along a suitable subsequence, establishing Theorem 1.3.Recall that we are using the notation Q := R n × (0, +∞) and f γ (u) := γχ {u>0} u γ−1 .
• Assume γ ∈ (1, 2).Then the function u → u γ + is everywhere differentiable, and thus as δ → 0. Consequently, taking the limit of the incremental quotient above and using the minimality of u, we obtain Choosing ϕ = e t/ε η and noticing that ∂ t ϕ = e t/ε 1 ε η + ∂ t η , (3.1) easily follows.• Assume γ = 1.In this case, the function u → u + is not differentiable at u = 0: we thus generalize the argument in [18, Proposition 5.12] to our degenerate-elliptic setting.By Lemma 2.2, we know that u ≥ 0 a.e. in Q.Consequently, one easily obtains that as δ → 0 + .Since by minimality E ε (u + δϕ) − E ε (u) ≥ 0, we may choose ϕ = e t/ε η as before and pass to the limit as δ → 0 + to find in the weak sense.Now, given r > 0, we bound , where we have used the estimate (2.6) in the last inequality (notice that here the constant C > 0 depends on r but is independent of u).Thus, by the arbitrariness of r > 0, it follows that loc (Q), and so, by W 2,2 -estimates of Calderón-Zygmund type, we obtain u ∈ W 2,2 loc (Q) (see for instance [18, Section 2]).Since W 2,2 -regularity is enough to apply Rademarcher's theorem, we deduce that ∂ t u, ∂ tt u, ∆u = 0 a.e. in {u = 0}.
Combining this with (3.4), we finally obtain We now establish a uniform non-degeneracy property for solutions to (3.1) when γ = 1.Notice that, even though non-degeneracy is quite standard in obstacle problems, it is not clear that such property can be made uniform with respect to ε ∈ (0, 1), which is exactly what we prove next.Lemma 3.2.There exists a constant c > 0, depending only on n, such that for every ε ∈ (0, 1), every (x 0 , t 0 ) ∈ Q such that B 1 (x 0 , t 0 ) ⊂ Q, every weak solution u ε to , and every r ∈ (0, 1 2 ), we have (3.6)sup Br(zε,τε) Proof.Since the equation (3.5) is invariant under translations and here the initial condition at {t = 0} plays no role, we may assume (x 0 , t 0 ) = (0, 0).We also drop the ε-subindexes to make the proof easier to read, that is, we set u := u ε , and (z, τ in view of the definition of c above.Note that u (and hence w) is smooth in {u > 0} ∩ B r (z k , τ k ) by standard elliptic estimates.By definition, we also have and thus, by the maximum principle, that is sup Passing to the limit as k → +∞, (3.6) follows thanks to the continuity of u.

3.2.
Non-degeneracy and optimal regularity: parabolic setting.We present below two technical results that we will exploit in the proof of Theorem 1.3, in the case γ = 1.The first one, Lemma 3.3, is a non-degeneracy property -it can be seen as the parabolic version of Lemma 3.2-, while the second one, Corollary 3.5, is an optimal regularity estimate, obtained by combining interior estimates and an optimal growth property established below in Lemma 3.4.The optimal regularity we prove coincides with the optimal regularity of solutions to the parabolic obstacle problem (see for instance [11]).The proof closely follows the classical one, but we work in a slightly more general framework, which is exactly what we need to carry out the limiting procedure in the proof of Theorem 1.3.
Here and in the rest of the paper, we use the usual notation for parabolic cylinders.That is, we set As customary, the center point (x 0 , t 0 ) is omitted when (x 0 , t 0 ) = (0, 0).Lemma 3.3.Let u ∈ C(Q 1 ) be a nonnegative weak solution to Then, there exists a constant c • > 0 depending only on n such that, for every (x 0 , t 0 ) ∈ {u > 0} ∩ Q 1/2 and every r ∈ (0, 1/2), it holds sup Proof.The argument follows the proof of Lemma 3.2.Given (x 0 , t 0 ) Hence, using the maximum principle and that where ∂ p Ω denotes the parabolic boundary of a set Ω ⊂ R n+1 .Since from which we obtain sup We conclude by taking the limit as k → +∞ and using that u(x 0 , t 0 ) ≥ 0 (note that u(x 0 , t 0 ) = 0 when (x 0 , t 0 ) ∈ ∂{u > 0}).
Next we establish an optimal growth bound and, as corollary, an optimal regularity estimate.The proof is quite standard, but we present it for completeness, following [4, Lemma 5.3] where it is done for the parabolic obstacle problem.
Then, there exists C 0 > 0 depending only on n and K such that ≤ K, we first notice that [35, Lemma 5.1] (or, equivalently, [4, Lemma 5.2]) yields that for every δ ∈ (0, 1) and r ∈ (0, 1  4 ), (3.9) sup where C 0 > 0 depends only on n and δ, and In light of (3.9), it is sufficient to bound u in the set Q r (x 0 , t 0 ) \ P δ r (x 0 , t 0 ) for some δ ∈ (0, 1) that will be chosen later.This will be obtained by comparison with the function . First, it is immediate to check that the choice of b makes w caloric in R n+1 .Now, on the one hand, setting δ := b 2 ∈ (0, 1), we have which allows us to deduce that u ≤ w in ∂ p Q 1/4 (x 0 , t 0 ) \ P δ 1/4 (x 0 , t 0 ), thanks to the assumption u L ∞ (Q 1 ) ≤ K and the definition of a.On the other hand, for every ρ ∈ (0, 1  4 ) we have by definition of w and sup for every ρ ∈ (0, 1 4 ), as an immediate consequence of (3.9).Combining the last two inequalities with the definition of a, we deduce that u ≤ w in ∂P δ 1/4 (x 0 , t 0 ) ∩ {t − t 0 > −δ/16} and therefore, since u is sub-caloric in Q 1 (since f ≤ 0 a.e. in Q 1 ), we obtain that u ≤ w in Q 1/4 (x 0 , t 0 ) \ P δ 1/4 (x 0 , t 0 ) as wanted.The result then follows since v ≤ Cr 2 in Q r (x 0 , t 0 ) \ P δ r (x 0 , t 0 ) for every r ∈ (0, 1  4 ) and some C > 0 depending on a and b.
To complete the proof it is enough to show that ∇u is Lipschitz in space, that is for some constant L > 0 depending only on n and K. Indeed, (3.14) implies D 2 u L ∞ (Q 1/2 ) ≤ L which, combined with the fact that u is a weak solution to (3.7) and f L ∞ (Q 1 ) ≤ K, gives the bound In the next argument, we will use the parabolic distance.For points (x, t), (y, τ ) ∈ R n+1 , it is defined as dist p ((x, t), (y, τ )) := inf{ρ > 0 : (y, τ ) ∈ Q ρ (x, t)}.For a point (x, t) ∈ R n+1 and a set A ⊂ R n+1 , we set dist p ((x, t), A) := inf (y,τ )∈A dist p ((x, t), (y, τ )).
Let us establish (3.14).To do it, let (x, t), (z, t) In this subsection, we combine all the ingredients introduced above to prove our main result.
Proof of Theorem 1.3.Let {u ε } ε∈(0,1) be the family of minimizers of E ε in U 0 .Then by Proposition 2.1, there are ε j → 0 and u ∈ U 0 such that u ε j converge to u in the sense of (2.1) and, by Lemma 3.1, each u ε j satisfies (3.1) for every j ∈ N.For simplicity, we set u j := u ε j for j ∈ N.
Step 1: local uniform convergence of the sequence of minimizers.First, up to passing to another subsequence, we may assume that for every r > 0 and every (x for some α ∈ (0, 1) and C > 0 depending only on n (the parabolic Hölder norm • C α,α/2 is quite standard, see for instance [3,Appendix C] for the definition).This easily follows by combining the (re-scaled) estimates in [3, Proposition 3.1 and Proposition 4.1] with U ε = u ε j = u j , f ε = 0, F ε = −χ {uε j >0} , p = q = ∞, and a = 0. 2 Further, the sequence {u j } j∈N is uniformly bounded in L 2 (Q + r ) for every r > 0 -see the argument leading to (2.5)-and thus for some other constant C > 0 independent of j.Therefore, combining the Arzelà-Ascoli theorem with a standard covering argument, and up to passing to another subsequence, we deduce that u j → u locally uniformly in Q, which, in particular, implies that u is continuous in Q.
Step 3: limit problem for γ = 1.In the rest of the proof, we assume γ = 1.Since χ {u j >0} are bounded in L ∞ (Q) uniformly in j ∈ N, there exists a nonnegative function χ ∈ L ∞ (Q) such that χ {u j >0} ⇀ ⋆ χ in L ∞ (Q) up to passing to a subsequence.Consequently, passing to the limit in (3.1) as above and recalling that L where Q R is any cylinder such that Q 2R ⊂ Q and R > 0.
To check (3.15), on the one hand we notice that if (x, t) ∈ {u > 0} ∩ Q R , then χ {u j >0} (x, t) = 1 for j large enough, by uniform convergence.Therefore, χ = 1 in {u > 0} ∩ Q R .On the other hand, we claim that for any δ for j large enough.Indeed, if we assume that this is not true, there exist Then, by our uniform non-degeneracy estimate (3.6), we have for some (z k , τ k ) ∈ B δ/2 (x k , t k ) and some c > 0 independent of k.Up to passing to a subsequence we have that (x k , t k ) → (x 0 , t 0 ) and (z k , τ k ) → (z, τ ) ∈ B δ/2 (x 0 , t 0 ) ⊂⊂ {u > 0} c .Thus, u(x 0 , t 0 ) = 0, but this contradicts the fact that u k (z k , τ k ) → u(x 0 , t 0 ) ≥ cδ 2 (which follows from uniform convergence).Consequently, the claim is proved and we conclude that χ = 0 in int({u = 0}) ∩ Q R , where int(A) denotes the interior of a set A.
Summing up, we have that u and χ = 0 in int({u = 0}) ∩ Q R (in particular, u fulfills the assumptions of both Lemma 3.3 and Corollary 3.5 with f = χ).
To prove (3.15), it remains to show that ∂{u > 0} ∩ Q R has zero measure, which is what we do next, in the spirit of [35,Theorem 5.1].
Once (3.16) is established, let us show that the free boundary has zero measure.By contradiction, we assume that L n+1 (∂{u > 0} ∩ Q R ) > 0. Now, since the set ∂{u > 0} is measurable, χ ∂{u>0} is integrable in Q R and thus, for almost every point (x, t) ∈ ∂{u > 0} ∩ Q R , (3.18) L n+1 (∂{u > 0} ∩ E r (x, t)) L n+1 (E r (x, t)) → 1 as r ↓ 0, where E r (x, t) := B r (x) × (t − r, t + r).Let us take one of such points (x, t) (recall that this is allowed by our assumption of L n+1 (∂{u > 0} ∩ Q R ) being positive).Up to a translation, we may assume that (x, t) = (0, 0).Let us also take r • > 0 such that, for r < r • , L n+1 (∂{u > 0} ∩ E r/2 ) ≥ L n+1 (E r/2 )/2.Now, we take a sequence r k → 0 such that 1/(2r k ) ∈ N.For each of these r k (which from now on we will denote simply by r), we decompose the cylinder E r/2 (up to a set of zero measure) into 1/r disjoint parabolic cylinders Q r/2 (0, t i ) for some {t i } i .If r < r • , then it is not difficult to see that the number of cylinders Q r/2 (0, t i ) which intersect ∂{u > 0}, which we denote by N , satisfies N ≥ 1/r.From such N cylinders, we can take another collection {Q r/2 (0, t j )} j with cardinality at least N/4 and such that the distance between each pair of cylinders in the collection is greater than r 2 .Thus, we can build another collection of parabolic cylinders {Q r/2 (z j , τ j )} j , with the same cardinality, such that (z j , τ j ) ∈ ∂{u > 0} ∩ Q r/2 (0, t j ).For each of these cylinders we can use the density property (3.16) and, adding up (using that the cylinders are pairwise disjoint), we get for some constant c > 0 depending only on n and K R .Note that in the last inequality we have used that N ≥ 1/r.As a consequence, which combined with (3.18) -recall that we assume (x, t) = (0, 0)-yields a contradiction.
• Finally, we notice that the proofs of Lemma 3.1, Lemma 3.2, Lemma 3.3, Lemma 3.4, and Corollary 3.5 are purely local and do not depend on the boundary behavior of minimizers.Therefore, minimizers of Ẽε in Ũ0 satisfy the same statements with Q replaced by Ω ∞ .
Putting such remarks together and proceeding as in the proof of Theorem 1.3, our statement follows.