On the existence and Hölder regularity of solutions to some nonlinear Cauchy–Neumann problems

We prove uniform parabolic Hölder estimates of De Giorgi–Nash–Moser type for sequences of minimizers of the functionals Eε(W)=∫0∞e-t/εε{∫R+N+1yaε|∂tW|2+|∇W|2dX+∫RN×{0}Φ(w)dx}dt,ε∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {E}}_\varepsilon (W) = \int _0^\infty \frac{e^{- t/\varepsilon }}{\varepsilon } \bigg \{ \int _{\mathbb {R}_+^{N+1}} y^a \left( \varepsilon |\partial _t W|^2 + |\nabla W|^2 \right) \textrm{d}X + \int _{\mathbb {R}^N \times \{0\}} \Phi (w) \,\textrm{d}x\bigg \}\,\textrm{d}t, \qquad \varepsilon \in (0,1) \end{aligned}$$\end{document}where a∈(-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in (-1,1)$$\end{document} is a fixed parameter, R+N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}_+^{N+1}$$\end{document} is the upper half-space and dX=dxdy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{d}X = \textrm{d}x \textrm{d}y$$\end{document}. As a consequence, we deduce the existence and Hölder regularity of weak solutions to a class of weighted nonlinear Cauchy–Neumann problems arising in combustion theory and fractional diffusion.


Introduction
In this paper we construct Hölder continuous weak solutions to the weighted nonlinear Cauchy-Neumann problem where N ≥ 1, a ∈ (−1, 1), R N +1 + := {X = (x, y) : x ∈ R N , y > 0}, ∇ and ∇• stand for the gradient and the divergence operators w.r.t.X , respectively, and The weight y a belongs to the Muckenhoupt A 2 -class (cf.[16,20]), the function U 0 is a given initial data and β ∈ C(R; R) is of combustion type, satisfying Problem (1.1) is related to the localized/extended version of the reaction-diffusion equation where s := 1−a 2 ∈ (0, 1) (see [35,45] and [5,Section 2]), and the diffusion process is governed by the fractional power of the heat operator, which is nonlocal both in space and time: where G N is the fundamental solution to the heat equation and is the gamma function (cf.[39,Section 28]).For smooth functions u depending only on the space variables x ∈ R N , it reduces to the fracional laplacian (− ) s , while if u = u(t), it is the Marchaud derivative (∂ t ) s (cf.[28,45]).Such operator appears in a wide range of applications such as biology, physics and finance (see e.g.[2,17] and the monograph [23]) and has notable interpretations in Continuous Time Random Walks theory (see [29] and the references therein).In recent years this class of equations has been the subject of intensive research: we quote [12,13] for traveling wave analysis, [2,[5][6][7]17] for unique continuation and obstacle problems, and [4] for nodal set analysis.In our context, solutions to (1.3) may be employed to approximate some free boundary problems arising in combustion theory and flame propagation, in the singular limit β → 1  2 δ 0 (cf.[14,37] for the nonlocal elliptic framework and [15] for the local parabolic one).
In this work, weak solutions to (1.1) will be obtained through a variational approximation procedure known in the literature as the Weighted Inertia-Energy-Dissipation method, introduced in the works of Lions [27] and Oleinik [36] (see also the paper of De Giorgi [19] in the context of nonlinear wave equations).Later, it has been investigated by many authors: we quote the works of Akagi and Stefanelli [1], Mielke and Stefanelli [30], Bögelein et al. [8,9], Rossi et al. [38] and the references therein.However, the variational techniques we use are inspired by the methods developed by Serra and Tilli in [40,41] (see also the more recent [3]).We introduce below the main ideas in a rather informal way, and postpone the formal definitions and statements in subsequent sections.
We set and, for every fixed ε ∈ (0, 1), we introduce the functional (w) dx dt. (1.4) If E ε has a minimizer U ε (or a minimizer pair (U ε , u ε := U ε | y=0 )) in some suitable space U 0 (see (1.9)) with in the weak sense (see Lemma 2.2).The above problem is nothing more than (1.1) with the extra term −εy a ∂ tt U ε : it is thus reasonable to conjecture that under suitable boundedness and compactness properties, one may pass to the limit as ε → 0 along some subsequence and obtain a limit weak solution U to (1.1).We stress that for each ε ∈ (0, 1) the approximating problem (1.5) is elliptic in space-time and the drift y a ∂ t U ε is a lower order term, while, as ε → 0, it degenerates along the time direction and, in the limit, the problem completely changes nature, becoming parabolic.
As already mentioned, our main goal is to establish uniform estimates for families of minimizers of the functional E ε and then pass to the limit by compactness.We will deal with two types of uniform bounds: global energy estimates and Hölder estimates.The former are obtained adapting the techniques of [40,41], while the latter will follow from a De Giorgi-Nash-Moser type result ( [18,32,34]) for weak solutions to where F ε and f ε belong to suitable classes of spaces (see Appendix A and (3.2)).This is our main contribution: we prove parabolic Hölder estimates "up to {y = 0}" for weak solutions to problem (1.6), that we transfer to sequences of minimizers of (1.4) and, in turn, to the limit function U , as ε → 0. We anticipate that even though these Hölder estimates have a local nature (we work directly with the local weak formulation of (1.6)), we will need an extra compactness assumption guaranteed by the global energy estimates, and depending on the initial data U 0 (see Proposition 2.1).
Since problem (1.6) is elliptic for every ε > 0 but becomes parabolic in the limit ε = 0, we cannot expect to prove uniform elliptic Hölder estimates (i.e.elliptic in the (X, t) variables), but "only" parabolic ones.To do this, we will combine uniform local energy estimates, uniform local L 2 → L ∞ bounds and a uniform oscillation decay lemma.It is important to stress that the proofs of these intermediate steps are not just the mere adaptation of the parabolic theory (see for instance [32,46]), but are tailored to the degeneracy of the problem along the time direction.We refer the reader to Sect.1.2 for further discussions and connections with the existing literature.
Finally, it is important to mention that our strategy seems to be quite flexible and different parabolic problems may be attacked with similar techniques.For instance, one may fix s ∈ (0, 1) and try to approximate weak solutions to with a sequence of minimizers of over a suitable functional space (notice that here we work in the purely nonlocal framework, without making use of the extension theory for the fractional laplacian).
In view of [1,41], the techniques we use to prove the energy estimates should easily be adapted to this setting too, whilst the general strategy we follow to obtain the Hölder bounds seems to be more difficult to repeat and must be adapted depending on the different nature of the problem.

Functional setting
To simplify the notation, we work with the functional and then we will transfer the information to the minimizers of (1.4), using standard even reflections w.r.t.y (see Lemma 2.4 and Remark 2.5).Indeed, notice that F ε is nothing more than E ε but the integration is on the whole R N +1 and, as always, u := U | y=0 .We consider the space , for every R > 0 (the definitions of the L p,a spaces are given in Appendix A).In particular, by [33], each function U ∈ U has a trace on the hyperplane {y = 0} and an "initial" trace we denote with Since each term in (1.7) is nonnegative, we will view F ε as a functional defined on U taking values in [0, +∞] and we will minimize it on the space U, subject to the initial condition The choice of U 0 ∈ H 1,a (R N +1 ) is quite important for our approach: we have U 0 ∈ U (and thus U is not empty) and, by [33] again, we also have that is well-defined.To prove our main estimates, it will be crucial to assume where L N denotes the N -dimensional Lebesgue measure.This is not a very restrictive assumption: in the majority of the applications the function u 0 is assumed to be smooth and compactly supported in R N .To simplify the notations, it is convenient to introduce the set of initial traces and the (non-empty) closed convex linear space Notice that the assumptions on the initial trace guarantee that any minimizer U ∈ U 0 is nontrivial.

Main results
Our main result is the following theorem.
2) and U 0 ∈ T 0 as in (1.8).Then there exist α ∈ (0, 1), a sequence ε k → 0 + and a sequence of minimizers {U ε k } k∈N of (1.4) in U 0 depending only on N , a, β L ∞ (R) and U 0 , such that for every open and bounded set K ⊂ R N +1 × (0, ∞), there exists C > 0 independent of k, such that This is the first result concerning uniform Hölder bounds for minimizers of (1.4): to the best of our knowledge, the existing literature treats exclusively uniform bounds of energy type (see the already mentioned [1,3,8,9,30,40,41]).Theorem 1.1 has an interesting corollary, that we state after giving the definition of weak solutions to problem (1.1).

Definition 1.2. Let
2) and U 0 ∈ T 0 as in (1.8).Then there exist α ∈ (0, 1), a weak solution (U, u) to (1.1) satisfying a sequence ε k → 0 + and a sequence of minimizers {U ε k } k∈N of (1.4) in U 0 such that, as k → +∞, Some comments are in order.Our approach allows to treat both the existence and the Hölder regularity of weak solutions using the same approximating sequence, in contrast with the classical theory where the two issues are often unrelated.Indeed, the existence and Hölder regularity for weak solutions to (1.1) can be proved separately using more classical methods.For the existence, we believe that the approximation scheme used in [24, Section 2] can be easily adapted to our framework.It is also important to notice that both methods allow to construct weak solutions with bounded H 1 energy (locally in time), depending on the H 1 energy of the initial data: this automatically excludes "pathological" solutions such as Jones' solution [26], in the case a = 0 and β ≡ 0.
On the other hand, the results concerning the Hölder regularity of weak solutions to parabolic weighted equations like (1.1) are obtained working in the pure parabolic setting, and are based on the validity of some Harnack inequality in the spirit of Moser [32], see e.g.[5,10,16,21].In our framework neither an elliptic nor a parabolic Harnack inequality for weak solutions of (1.5) can hold, with constants independent of ε ∈ (0, 1).This is due to the different nature of the elliptic and parabolic Harnack inequality (see [31,Theorem 1] and [32,Theorem 1], respectively) and the drastic loss of ellipticity in the limit ε → 0 (see also the counter-example in [32, pp. 103] in the case a = 0 and β ≡ 0).On the contrary, parabolic Hölder regularity is preserved under the limit.
We end this paragraph with a few words about the reaction function β.It is worth to mention that other kind of reactions can be considered but, for simplicity, we decided to focus on the class defined in (1.2): the positivity of β guarantees that the functional F ε is nonnegative (uniformly in ε), while the fact that (u) ≤ χ {u>0} allows us to prove the crucial level estimate (2.6).The assumption suppβ = [0, 1] gives some additional properties such as the weak maximum principle stated in Lemma 2. 4. For what concerns the uniform Hölder bounds, the only information we use is that β ∈ L ∞ (R).

Structure of the paper
The paper is organized as follows.In Sect. 2 we prove the existence of minimizers of F ε in U 0 (for every fixed ε ∈ (0, 1)) and we establish the main global uniform energy estimates (2.8) and (2.9), which play a key role in the proof of Proposition 2.1.This is the first main step in our analysis: we show the existence of a sequence of minimizers U ε j weakly converging to some function U which is also a weak solution to (1.1).
In Sect. 3 we prove a L 2,a → L ∞ local uniform bound for weak solutions to (1.6) (see Proposition 3.1).The main difficulty here is to derive a uniform energy estimate: since problem (1.6) is elliptic for every fixed ε > 0 but becomes parabolic in the limit as ε → 0, the best we can expect is to obtain a uniform energy estimate of parabolic type.We anticipate that the standard parabolic techniques do not work in this framework (see Remark 3.7) and new methods that exploit the degeneracy of the equation are used.
In Sect. 4 we show Proposition 4.1: under the additional compactness assumption (4.1), there is a sequence of weak solutions to (1.6) having locally bounded C α,α/2 seminorm.As explained in Remark 4.4, this compactness assumption is required in order to prove a parabolic version of the so-called "De Giorgi isoperimetric lemma" (cf.Lemma 4.2).
In Sect. 5 we show Theorem 1.1 and Corollary 1.3: the proofs are easy consequences of Propositions 2.1, 4.1 and a standard covering argument.
Finally, in the appendices (Appendix A, B and C) we recall some technical tools and results we exploit through the paper, and the full list of notations.

Global uniform energy estimates
This section is devoted to the proof of the following proposition.Proposition 2.1.Let {U ε } ε∈(0,1) ∈ U 0 be a family of minimizers of F ε .Then there exist U ∈ U 0 and a sequence and, furthermore, the pair (U, u) satisfies ) In particular, (U, u) is a weak solution to (1.1).
Before addressing to the proof of the above statement, we show some basic properties of minimizers of the functional F ε .
Notice that, since ϕ ∈ C ∞ 0 (Q ∞ ), the last integral converges to zero as h → 0, while as h → 0. Consequently, using that β is bounded, ϕ| y=0 is compactly supported and the minimality of U , we can pass to the limit as h → 0, to deduce and take ϕ = e t/ε η.Noticing ∂ t ϕ = e t/ε 1 ε η + ∂ t η and rearranging terms, (2.3) follows.Now, we show that for every ε ∈ (0, 1) the functional F ε has a minimizer in U 0 .To simplify the notations, we introduce the functional This functional is related to (1.7) through the following relations and U 0 is convex and invariant under time transformations, the minimization of J ε on U 0 is equivalent to the minimization of F ε on the same space.In other words, U ∈ U 0 is a minimizer of F ε if and only if V ∈ U 0 is a minimizer of J ε .
We are left to prove the existence of a minimizer.Since V 0 ∈ U 0 and J ε (V 0 ) < +∞, there exists a minimizing sequence {V j } j∈N ⊂ U 0 : (2.7) In particular, for every fixed R > 0, we have for some C R > 0 independent of j and so, since ), up to passing to a suitable subsequence, still denoted with V j .Similar, setting v j := V j | y=0 , we have that {v j } j∈N is uniformly bounded in ) by the trace theorem (see for instance [33]) and so, up to a subsequence, up to passing to an additional subsequence.Notice that since U 0 is closed and convex, we have V ∈ U 0 .Further, by continuity, we have (v j ) → (v) a.e. in Q and so, by lower semicontinuity and Fatou's lemma, it follows Lemma 2.4.Let ε ∈ (0, 1) be fixed.Then: • If V 0 is even w.r.t.y, then there exists a minimizer of J ε in U 0 which is even w.r.t.y.
Proof.Let V := V ε ∈ U 0 be a minimizer of J ε and assume V 0 ≥ 0 a.e.. Then V + is an admissible competitor, with J ε (V + ) < J ε (V ), unless V ≥ 0 a.e.Similar if V 0 ≤ 1 a.e., W := min{V, 1} is an admissible competitor and, since Remark 2.5.The last point of the above statement and Lemma 2.3 tell us that if the initial data is even w.r.t.y, then we may assume that F ε has a minimizer U ε which is even w.r.t.y.Such minimizer satisfy where

Proof of Proposition 2.1
Proposition 2.1 will be obtained as a consequence of the following energy estimates.
Proposition 2.6.(Global uniform energy estimates) There exists C > 0 depending only on N , a and U 0 such that for every family and, for every R ≥ ε, The above statement is the key result of this section and will be proved by combining Lemma 2.7 and Corollary 2.8 that we show below.As in the above subsection, we consider a minimizer V of J ε and we write where (2.11) Notice that, since V is a minimizer, we have The main idea of the following lemma is to find a different expression for the derivative of the function E defined in (2.12).The new formulation for E is crucial to prove our main estimates.
Lemma 2.7.Let V ∈ U 0 be a minimizer of J ε .Then Proof.We follow the proof of [40, Proposition 3.1] (see also [8,Lemma 4.5] and [3, It is easily seen that, if |λ| ≤ λ 0 for some λ 0 > 0 small enough, then ϕ is strictly increasing with ϕ(0) = 0.In particular, the inverse ψ = ϕ −1 exists, it is smooth and, by (2.15), satisfies The key idea of the proof is to use the function ϕ to construct a competitor W .It is obtained as an inner variation of V : Since ϕ(0) = 0, we have W = V when t = 0 and so W ∈ U 0 .Further, by (2.15), W = V when λ = 0 (by sake of simplicity, the dependence on λ is omitted in the notations for ϕ, ψ and W ). Now, from the formulation of J ε introduced in (2.10) and the change of variable t = ψ(τ ), we have In view of (2.15) and ( 2 and thus J ε (W ) < +∞.In particular, we deduce that, for any small λ (|λ| ≤ λ 0 ), W ∈ U 0 is an admissible competitor.Actually, recalling that W = V when λ = 0 and V is a minimizer, it must be lim Proceeding exactly as in [3, Lemma 4.2], we compute Consequently, recalling that t → e −t {I(t) + R(t)} ∈ L 1 (R + ) and using the dominated convergence theorem, we can pass to the limit in (2.17) and, making use of (2.18), we can write (2.17) explicitly: where we have used ψ(τ ) = τ , and ϕ = ψ = 1 when λ = 0 (see (2.15) (2.20) Further, using the definition of E given in (2.12) and integrating by parts, it follows (2.21) The "boundary terms" in the integration by parts disappear since ζ(0) = 0 and e −t E(t) → 0 as t → +∞.Finally, plugging (2.20) and (2.21) into (2.19), it follows and, for every t ≥ 0, for some constant C > 0 depending only on N , a and U 0 .
To prove the second part of the statement, we fix t ≥ 0 and we notice that (2.24) implies The thesis follows by the second definition in (2.11).
Proof of Proposition 2.6.Let U be a minimizer of As a first consequence, we immediately see that (2.8) follows changing variable (t = ετ ) in (2.22).Similar, the same change of variable in (2.23) for all t ≥ 0. Now, let R ≥ ε and define k = R/ε .In view of the arbitrariness of t, we can apply the above estimate for t = j and sum over j = 0, . . ., k − 1 to obtain Since ε ≤ R ≤ kε, (2.9) follows.
Proof of Proposition 2.1.In view of (2.8) and (2.9), {U ε } ε∈(0,1) is equibounded in H 1,a (Q + R ) for every fixed R > 0. Consequently, the usual diagonal procedure shows the existence of a sequence ε j → 0 and U ∈ U 0 such that the first and the third limit in (2.1) are satisfied (here we may also use Sobolev embedding and trace theorems as in Lemma 2.3).The second limit in (2.1) follows by [42,Corollary 8], up to passing to another subsequence, since for every Finally, (2.2) follows by passing into the limit as j → +∞ into (2.3)(with ε = ε j ) and using (2.1).

Uniform L 2,a → L ∞ estimates
This section is devoted to the proof of some local and uniform L 2,a → L ∞ estimates for weak solutions to the linear problem (1.6) (the notion of weak solution is introduced in Definition 3.2 below).and ( p, q) satisfying (3.2).Then there exists a constant C > 0 depending only on N , a and q such that every family {U ε } ε∈(0,1) of weak solutions to problem (1.6) for every ε ∈ (0, 1).
We divide the proof of Proposition 3.1 in two main steps: we establish an energy estimate for solutions and nonnegative subsolutions (Lemma 3.5) and then we exploit it to prove a "no-spikes" estimate (see Lemma 3.8).
Before moving forward, we fix some important notations and give the definition of weak solution to problem (1.6).Let N ≥ 1 and a ∈ (−1, 1).We will always consider exponents p, q ∈ R satisfying We anticipate that the assumption p > 2 is needed only when N = 1 and a ∈ (−1, 0] (for all other values of N and a we have p = (N + 3 + a)/2): this is due to the fact that this range of parameters is critical for the Sobolev inequality (cf.Theorem A.2 and Theorem A.3).Let us proceed with the notion of weak solutions to (1.6).It is related to Definition 1.2, for problem (1.1): let R > 0, ε ∈ (0, 1), ( p, q) satisfying (3.2), 1 .We say that W ε is a weak solution to (1.6) Even in the setting of problem (1.6), it is convenient to simplify the notation as follows.If W ε is a weak solution, we notice that the even extension where Fε is the even extension of F ε and fε := 2 f ε .For this reason, we will always work with weak solutions defined in the whole cylinder Q R and, to recover the information on W ε , we restrict them to the upper-half cylinder for every η ∈ C ∞ 0 (Q R ).Remark 3.3.In the whole section, even if not mentioned, we will always work with weak solutions (or subsolutions) in the sense of Definition 3.2.Further, to simplify the reading, we drop the notations Fε and fε , writing F ε and f ε instead.We stress that this does not change our estimates, since the extra factor 2 can be easily reabsorbed through a dilation of the variables (X, t).Remark 3.4.(Scaling) In the proof of Proposition 4.1 we will use that if R > 0 and U ε is a weak solution in Q R , then the function We are ready to prove the energy estimate for families of nonnegative weak solutions (the same proof applies to family of weak solutions, see Remark 3.6).Lemma 3.5.(Energy estimate) Let N ≥ 1, a ∈ (−1, 1) and ( p, q) satisfying (3.2).Then there exists a constant C > 0 depending only on N , a and q such that for every r Since U is a weak subsolution, we may assume F, f ≥ 0 (up to replacing them with F + and f + , respectively) and so, by Lemma B.1 (part (ii)), we may also assume U ≥ 1 in Q 1 (up to consider the subsolution max{U, 1} instead of U ).
Let ψ be a Lipschitz function vanishing on ∂Q 1 , which will be chosen later.Testing the differential inequality of U with η = U ψ 2 , we easily obtain the differential inequality where we recall that u = U | y=0 in the sense of traces and ψ| y=0 (x, t) = ψ(x, 0, t).The energy inequality (3.4) will be obtained combing two different bounds that we prove in two separate steps.
Step 1.We prove that ess sup for some c > 0 depending only on N , a and q.
Taking into account the relation we rewrite (3.5) neglecting the nonnegative term in the l.h.s.involving ∂ t U to deduce for some c 0 > 0 depending only on N , and φ = φ(t) be defined by Furthermore, since 1 e z −1 ≤ 2 z for every z > 0, and t * ≤ 2 , we have Now, choose ψ(X, t) = ϕ(X )φ(t) and write Using the definitions of φ and φ in [−1, t * ], that ϕ ∈ C ∞ 0 (B r ) with 0 ≤ ϕ ≤ 1 and integrating by parts, we find and Further, by (3.10) and (3.11), and thus by (3.7) and (3.9) (3.12) Now let us estimate the terms into the r.h.s. of (3.8).First, we have thanks to (3.9) and the fact that φ = 0 in . Second, by Hölder's inequality since F L p,a (Q 1 ) ≤ 1 (and F ≥ 0) by assumption.
We are left to estimate the trace term, that we reabsorb using the second term in the l.h.s. of (3.8).Using that u ≥ 1 and f ≥ 0, applying Hölder's inequality and recalling that f L q ∞ (Q 1 ) ≤ 1, we obtain , where we have set v := uψ y=0 , and q is the conjugate of q.Since q > N 1−a , we have 2 ≤ 2q ≤ 2 σ , where σ := N N −1+a (cf.Theorem A.1) and so, by interpolation and Young's inequality, we obtain for every δ > 0 and a suitable c δ > 0, satisfying c δ → +∞ as δ → 0 (ϑ ∈ (0, 1) is given in the interpolation inequality and depends on N , a and q).Now, by (A.2), the definition of v and Cauchy-Schwartz's inequality, we have for some c > 0 depending only in N and a, while, by (A.1) and the Cauchy-Schwartz's inequality again, for every A > 1.Now, we fix δ ∈ (0, 1), such that 2cδ ≤ 1 2 and A > 1 such that 2cc δ A − 1−a 2 ≤ 1 2 (notice that both δ and A depend only on N , a and q).Combing the last four inequalities with (3.8), we obtain for some new c > 0 depending only on N , a and q, and thus (3.6) follows in light of (3.12), (3.13) and (3.14).
Remark 3.7.The energy estimate (3.4) is the main step for proving the L 2 → L ∞ bound (3.1).In light of what comes next, we believe it is important to compare the proof w.r.t. the classical parabolic framework (formally, the limit case ε = 0).Assume for simplicity F = 0 and f = 0 and consider a weak solution to (3.18) Testing the weak formulation with η := U ψ 2 , we obtain which is (3.5) "with ε = 0".For every fixed −1 < s < τ < 1, one may choose ψ 2 (X, t) = ϕ 2 (X )χ [s,τ ] (t) and so, integrating by parts w.r.t.time and using Young's inequality, it is not difficult to find for some C > 0 (depending only on N and a).The bound (3.6) immediately follows from (3.19), choosing ϕ as in (3.9) and neglecting the nonpositive term in the l.h.s.The L 2,a bound for ∇U is obtained similar to (3.15).In our setting testing with ψ 2 = ϕ 2 (X )χ [s,τ ] (t) is not admissible since the weak derivative of the function t → χ [s,τ ] (t) is not L 2 (−1, 1).However, we notice that our proof is somehow more elementary: the time factor in the test function ψ = ϕ(X )φ(t) we use to prove the bound for B 1 |y| a U 2 (X, τ ) dX is obtained by solving an easy first order ODE.In this way, we bypass the approximation procedure of χ [s,τ ] needed in the purely parabolic framework.Lemma 3.8.(No-spikes estimate) Let N ≥ 1, a ∈ (−1, 1) and ( p, q) satisfying (3.2).Then there exists a constant δ > 0 depending only on N , a and q such that for every family {U ε } ε∈(0,1) of subsolutions in Q 1 such that for every ε ∈ (0, 1), then for every ε ∈ (0, 1).
Proof.Let us set U = U ε , F ε = F, f ε = f and assume either N ≥ 2 and a ∈ (−1, 1), or N = 1 and a ∈ [0, 1).For every integer j ≥ 0, define the nonnegative subsolutions (with F + and f + , see Lemma B.1) and the quantity Applying the energy inequality (3.4) to V j+1 with = r j+1 and r = r j , we have for some C > 0 depending only on N , a and q and thus, by Sobolev inequality (cf.Theorem A.3, formula (A.3)) and the definition of V j , we deduce for some new C > 0 and γ > 1 (depending only on N , a and q).On the other hand, by Hölder's inequality where γ is the conjugate exponent of γ , and |A| a := A |y| a dX dt for every measurable set A ⊂ R N +2 .Further, using the definition of V j , it follows and, by Hölder's inequality again, for every j ≥ 1 and for some new C > 0 depending only on N , a and p, where γ := 1 γ − 1 p > 0 (in view of (3.2) and the definition of γ , cf.Theorem A.3).To complete the argument, it is sufficient to notice that since γ < 1 γ , we have for some new C > 0 (depending only on N and a), whenever E j ≤ 1.Thus, choosing δ such that Cδ < 1, we deduce E j ≤ 1 for every j and E j → 0 as j → +∞.Finally, since C j → 1 and r j → 1 2 as j → +∞, we obtain by definition of V j which implies (3.21) and our statement follows.
The very same argument works when N = 1 and a ∈ (−1, 0), by taking γ = 2 and using inequality (A.4) instead of (A.3).Notice that in this range of parameters we have p > p = 2: this implies p ∈ (1, 2) and thus the chain of inequalities in (3.25) does not change.
Proof of Proposition 3.1.Set U = U ε and define , where δ > 0 is as in Lemma 3.8.Since V is a nonnegative subsolution (with λ + F + and λ + f + ) and satisfies (3.20), then (3.21) gives Repeating the same argument with , we obtain and our thesis follows choosing C = 2 √ δ , recalling that δ depends only on N , a and q.Remark 3.9.Thanks to [16, Remark 2.1, Lemma 2.1], our proofs can be easily adapted to treat the case of weak solutions/sub-solutions in Q 1 (X 0 , t 0 ), with constants independent of (X 0 , t 0 ) ∈ R N +2 .

Uniform Hölder estimates
This section is devoted to the proof of a uniform Hölder estimate for families of solutions to (1.6) (in the sense of Definition 3.2).As mentioned in the introduction, we will consider sequences of weak solutions satisfying the following compactness assumption for some r > 0 and some U ∈ C([−r 2 , r 2 ] : L 2,a (B r )).As pointed out in Remark 4.4, in the classical parabolic framework this property can be deduced working directly with the equation and exploiting the parabolic energy estimate.Even though we do not have any counter-example, this approach seems not work in our setting.Anyway, in light of Proposition 2.1, (4.1) is always guaranteed when working with sequences {U ε j } j∈N of minimizers of the functional (1.7).
The following proposition is the main result of this section.
Proposition 4.1.Let N ≥ 1, a ∈ (−1, 1) and ( p, q) satisfying (3.2).There exist α ∈ (0, 1) and C > 0 depending only on N , a, p and q such that for every U , every sequence ε j → 0 and every sequence {U ε j } j∈N of weak solutions in Q 8 satisfying (4.1) with r = 4 and for every j ∈ N and some C 0 > 0, then there exist a subsequence j k → +∞ such that The estimate (4.3) will follow as a consequence of an oscillation decay type result which, in turn, is obtained by combining the L 2,a → L ∞ bound for nonnegative subsolutions (see Proposition 3.1) and a parabolic version of the "De Giorgi isoperimetric lemma" (see Lemma 4.3).Lemma 4.3 is the key step: our approach is based on the ideas of [46,Section 3] and relies on the validity of its "elliptic" counterpart, that we state in Lemma 4.2 (the proof is a modification of [46,Lemma 10]).

Lemma 4.2. ("De Giorgi isoperimetric lemma")
for some C 0 > 0 and Then for every p ∈ (1, 2), there exists a constant c > 0 depending only on N , a, C 0 and p such that Proof.We consider a new function V defined as dX and writing X = (x, y) and Z = (z, ζ ), we have where, in the last inequality, we have used Hölder's inequality.By [20, Theorem 1.5] and Hölder's inequality again, we have for some c > 0 depending on p. Combining the two inequalities, our statement follows.
Lemma 4.3.Let N ≥ 1, a ∈ (−1, 1), ( p, q) satisfying (3.2), and Assume there exist δ > 0, a sequence ε j → 0 and a family {U and for every j ∈ N. Then there exists σ > 0 depending only on N , a, p, q and δ such that for every U and every ε j k → 0 satisfying (4.1) (with j = j k and r = 2), we have for every k ∈ N.
Proof.We assume by contradiction there exist sequences ) and a sequence of weak solutions for every k ∈ N.
Let us set V k := (U k ) + and V := U + .By assumption, we have ).Further, by Lemma B.1, each V k is a nonnegative weak subsolution and thus, combining the assumption U k ≤ 1 and (3.4), we may assume that for some C > 0 depending only on N , a, p and q.Notice that by C([−4, 4] : L 2,a (B 2 )) convergence, we also have convergence in measure, that is, for every > 0, Step 1.We prove that To see this, we notice that given > 0, the set { ≤ V ≤ 1 2 − } can be trivially written as the disjoint union of we may pass to the limit as k → +∞ and notice that the last relation in (4.5) and (4.6 0 and thus, by the arbitrariness of > 0, (4.7) follows.
Step 3.This is the most delicate part of the proof.We prove that We test the differential inequality of V k with η k = V k ψ 2 k , for a sequence of test functions ψ k we will choose in a moment.Following the proof of Lemma 3.5, we obtain (cf.(3.5)) Now, using Young's inequality, we reabsorb the gradient part of the first term in the r.h.s. with the second term in the l.h.s., and we apply Hölder's inequality to the last three terms in the r.h.s. to obtain for some numerical constant C 0 > 0 (the last inequality easily follows in light of (4.4) and that 0 while χ k is a Lipschitz approximation of χ [s,τ ] as k → +∞, defined as follows for some positive δ k → 0 as k → +∞.Notice that, by dominated convergence, as k → +∞, for some C > 0 depending only on N and a.Similar as k → +∞, for some new C > 0 depending also on p and q.Furthermore, by taking C > 0 larger and using (3.4), we have by choosing Finally, using the definition of χ k and integrating by parts in time, we find as k → +∞.To check the limit, we use that V k , V and ϕ are bounded in Q 2 , and so as k → +∞ and thus the limit in (4.16) follows by triangular inequality.Consequently, passing to the limit as k → +∞ in (4.11) and using (4.13), (4.14), (4.15) and (4.16), we deduce for some constant C > 0 depending only N , a, p, q and ϕ.Now, by (4.8), we may choose s ∈ (−2, −1) such that V (•, s) = 0 in B 1 , to obtain On the other hand, by (4.9), we know that either 2 in B 1 , the above inequality and (4.12) yield for some c 0 > 0 depending only N , a, p, q and ϕ, and thus it must be V (•, τ ) = 0 in B 1 for a.e.τ ≤ s + c 1/ p 0 .Iterating this procedure, (4.10) follows.
Remark 4.4.To understand the role played by the assumption (4.1), it is useful to compare with the classical parabolic framework.The main difference here is that the parabolic equation (combined with the parabolic energy estimate) gives enough compactness for carrying out the contradiction argument.Indeed, let U k be a sequence of weak solutions to (3.18) satisfying U k L 2,a (Q 2 ) ≤ 1.For every k ∈ N, we have ).Thus, noticing that the sequence {U k } k is uniformly bounded in L 2 (−1, 1 : H 1,a (B 1 )) (by the parabolic energy estimate, see Remark 3.7) and using the equation of U k above, we obtain that {∂ t U k } k is uniformly bounded in L 2 (−1, 1 : H −1,a (B 1 )) and so, by the Aubin-Lions lemma, we have U k → U in L 2,a (Q 1 ), up to passing to a suitable subsequence.This is enough to show that U satisfies (4.7) and (4.8).To prove (4.10), it suffices to notice that, since {U k (t)} k is uniformly bounded in H 1,a (B 1 ) for a.e.t ∈ (−1, 1), we may also assume ) for a finite increasing sequence of times t n ∈ (−1, 1), up to passing to another subsequence.This allows us to pass to the limit into (3.19)(with τ = t n+1 and s = t n ) and complete the argument of Step 3.
On the contrary, in our "approximating setting", the weak formulation (3.3) (with , and the energy estimate (3.4) gives uniform bounds for for sequences of solutions uniformly bounded in L 2,a (Q 2 ).It is not clear to us if these ingredients can be combined to obtain strong L 2,a (Q 1 ) compactness, or not.For this reason we require C([−1, 1] : L 2,a (B 1 )) strong compactness in (4.1) which, as already mentioned, is guaranteed for families of minimizers of (1.7) by Proposition 2.1.Lemma 4.5.Let N ≥ 1, a ∈ (−1, 1), ( p, q) satisfying (3.2), and Q := B 1 × (−2, −1).There exist δ 0 , θ0 ∈ (0, 1) depending only on N , a, p and q such that for every U , every sequence ε j → 0 and every sequence {U ε j } j∈N of weak solutions in Q 2 satisfying (4.1) with r = 2, such that and for every j ∈ N, then for every j ∈ N.
Proof.Let σ > 0 be as in Lemma 4.3, δ 0 := 2 −n 0 √ δ, where n 0 is the largest integer such that and δ ∈ (0, 1) will be chosen later.Let us set U j = U ε j , F j = F ε j , f j = f ε j and define The assumptions on U j imply that V j,n is a solution in Q 2 with F j,n := 2 n F j and f j,n := 2 n f j , satisfying for every j, n ∈ N and 4,4] : L 2,a (B 2 )) as j → +∞.Further, notice that by (4.17) and the definition of δ 0 , we have for every n ≤ n 0 and every j.Now, fix δ ∈ (0, 1) and assume Then, using the definition of V j,n (and V j,n+1 ) and that V j,n ≤ 1, we easily see that Consequently, if (4.18) holds true for every n ≤ n 0 and some j ∈ N, V j,n satisfies the assumptions of Lemma 4.3, and so for every n ≤ n 0 .However, since {0 < V j,n < 1/2}∩{0 < V j,m < 1/2} = ∅ for every m = n and every j, the above inequality implies |Q 1 ∪ Q| a ≥ n 0 σ , in contradiction with the definition of n 0 .Consequently, (4.18) fails for n = n 0 and every j ∈ N, that is for every j ∈ N. Let us set k 0 := n 0 + 1.Since (V j,k 0 ) + is a nonnegative subsolution (with (F j,k 0 ) + and ( f j,k 0 ) + ), we obtain by (3.1) and the definition of δ 0 for some C > 0 depending only on N , a, p and q, and every j ∈ N. Now, taking δ such that Cδ ≤ 1 2 , the above inequality gives V j,k 0 ≤ 1 2 in Q 1/2 for every j and thus for every j, which is our thesis, choosing θ0 := 2 −k 0 −1 .
Proof.Let δ 0 , θ0 > 0 be as in Lemma 4.5, and let U j = U ε j , F j = F ε j and f j = f ε j .We fix δ > 0, and define We have K j ≥ δ for every j ∈ N and, further, each V j is a weak solution in Q 4 (and thus in Q 2 ) with Fj := 2 K j F j , f j := 2 K j f j satisfying for every j, by definition of K j .Now, in view of (4.1), we have that {U j } j∈N is uniformly bounded in L 2,a (Q 4 ) and thus, by (3.1) and (4.2), it is bounded in L ∞ (Q 2 ).This implies the existence of a subsequence j k → +∞, K ∈ [δ, +∞) and l ∈ R (both K and l are finite depending on δ 0 and U L 2,a (Q 2 ) ) such that as k → +∞, where U k := U j k and K k := K j k .As a consequence, one can easily verify that for a finite number of indexes for an infinite number of indexes, and thus, up to passing to an additional subsequence and eventually considering −V k instead of V k , we may assume for every k ∈ N.Then, Lemma 4.5 yields for every k ∈ N, that is Taking the sup Q 1/2 and subtracting inf for every k ∈ N. Our thesis follows by passing to the limit as δ → 0 and choosing θ 0 = θ0 2 .
Proof of Proposition 4.1.Let δ 0 and θ 0 as in Corollary 4.6, and δ > 0. We set U j := U ε j , F j := F ε j , f j := f ε j and define for every j ∈ N. Notice that thanks to (4.2), (4.1) and (3.1), {K j } j∈N is uniformly bounded.Thus, similar to the proof above, we have K j → K ≥ δ, as j → +∞, up to passing to a subsequence.Consequently, as j → +∞.Now, let r n := 4 −n , n ∈ N. We show that there exist n 0 ∈ N, C > 0 and α ∈ (0, 1) depending only on N , a, p and q such that, up to passing to a subsequence ε ) shows that it is enough to prove (4.20) for points (X 0 , t 0 ) with y 0 = 0: basically, if Hölder regularity (or oscillation decay) fails at some point, then such point must belong to the region where the weight |y| a is degenerate or singular).As a consequence, since the equation of V k is invariant under translations w.r.t.x and t, it is enough to consider the case (X 0 , t 0 ) = 0.
With this goal in mind, let us define V j,n (X, t) := 1 K j V j (r n X, r 2 n t), j, n ∈ N.
By Remark 3.4 each V j,n satisfies By definition and scaling, we easily see that V j,n L ∞ (Q 4 ) ≤ 1 and for every j, n ∈ N, where we have set The positivity of ν follows from (3.2).In a moment we will choose n 0 ∈ N such that r ν n ≤ δ 0 , for every n ≥ n 0 and thus we may assume F j,n L p,a (Q 4 ) + f j,n L q ∞ (Q 4 ) ≤ δ 0 , for every n ≥ n 0 and j ∈ N. Further, exploiting (4.19), it is not difficult to check that for every fixed n ∈ N, we have as j → +∞, where Ṽn (X, t) := V (r n X, r 2 n t).Then, for every fixed n ≥ n 0 the sequence {V j,n } j∈N satisfies the assumptions of Corollary 4.6 and thus there exist subsequences j k → +∞ and ε k := ε j k → 0 such that for every fixed n ≥ n 0 and every k such that ε k ≤ r 2 n , we have osc where V k,n := V j k ,n .Re-writing such inequality in terms of V k , it follows osc for every k such that ε k ≤ r 2 n .Let us take α ∈ (0, 1), n 0 ∈ N and C > 0 satisfying Then, if n = n 0 , (4.20) holds true by definition of C, recalling that V k L ∞ (Q 4 ) ≤ 1, for every k.Now, assume that (4.20) holds true for some n ≥ n 0 .Then, by definition of n 0 , α, C and (4.21) and the inductive assumption, we obtain osc for every k such that ε k ≤ r 2 n+1 , and so (4.20) follows.Notice that, since ε k → 0, we may extract a decreasing subsequence ε n := ε k n such that ε n ≤ r 2 n , for every n ∈ N and thus, by (4.20), we have osc for every (X 0 , t 0 ) ∈ Q 1 and every n ≥ n 0 , where V n := V k n .
To complete the proof, we check that for every (X, t), (Z , τ ) ∈ Q 1 , n ∈ N, taking eventually new constants C > 0 and α ∈ (0, 1) depending only on N , a, p and q.Since the constants α, C and n 0 in (4.22) are independent of (X 0 , t where we have used that V n L ∞ (Q 1 ) ≤ 1.When n ≥ n 0 we directly apply (4.22) to deduce and (4.23) follows.Now, re-writing (4.23) in terms of U n := U k n , we obtain ∞ (Q 4 ) (X − Z , t − τ ) α , for every n ∈ N and every (X, t), (Z , τ ) ∈ Q 1 , where F n := F j kn and f n := f j kn .The bound (4.3) follows by letting δ → 0. Remark 4.7.As pointed out in Remark 3.9, the proofs of this section work for weak solutions in Q r (X 0 , t 0 ) too, with minor modifications and constants independent of (X 0 , t 0 ) ∈ R N +2 .
By Lemma 2.2 and Proposition 2.1, there exist U ∈ U 0 and a sequence ε j → 0 such that the pair (U j , u j ) := (U ε j , U ε j | y=0 ) satisfies for every η ∈ C ∞ 0 (Q ∞ ) and every j ∈ N and, further, U j → U in C loc ([0, ∞) : L 2,a (R N +1 )) (5.1) as j → +∞.Now, let us fix R > 0, t 0 := (8R) 2 and consider the sequence V j (X, t) := U j (R X, R 2 t + t 0 ).Setting f j := β(u j ) (with for every j ∈ N) and f j,R (x, t) := R 1−a f j (Rx, R 2 t), we have by Remark 3.4 for every η ∈ C ∞ 0 (Q 8 ) and every j ∈ N. Consequently, by (5.1), we may apply Proposition 4.1 and Proposition 3.1 to the sequence {V j } j∈N to deduce for every k ∈ N, some C > 0 and α ∈ (0, 1) depending only on N and a, and some sequence j k → +∞ (depending also on R).Re-writing this uniform bound in terms of U j k and recalling the definition of f j k , we find for every k ∈ N, where QR := B R × (63R 2 , 65R 2 ).Finally, since uniformly in k by Proposition 2.6 and Poincaré inequality (up to taking C larger depending on U 0 ), we can combine Remark 3.9, Remark 4.7 and a standard covering argument to complete the proof of (1.10).
Proof of Corollary 1.3.The thesis follows by combining Proposition 2.1, Theorem 1.1 and the Arzelà-Ascoli theorem.
[ 16,20,22,25].The symbols I , B and Q denote a generic interval in R, a generic ball in R N +1 , and a generic parabolic cylinder in R N +2 , respectively.For p ∈ (1, ∞) and a ∈ (−1, 1), we define As it is well-known, these spaces enjoy some notable properties that we resume below.
where we have used (3.3) with test p j (U )η and that η, p j , p j ≥ 0. Since |y| a Fη dX dt f η| y=0 dxdt, as j → +∞, thanks to the Lebesgue dominated convergence theorem (and trace theorem), we deduce that U + is a weak subsolution in Q R with F + and f + by passing to the limit as j → +∞.
To complete the proof, it is enough to notice that −U is a weak solution in Q R with −F and − f .Then U − = (−U ) + is a weak subsolution in Q R with F − = (−F) + and f − = (− f ) + .
Remark B.2.The same proof shows that if U ε is a weak subsolution in Q R , then for every l ∈ R, the function (U ε − l) + is a weak subsolution in Q R , with F ε and f ε replaced by (F ε ) + and ( f ε ) + , respectively.

Appendix C
We report below the list of notations we use in the paper.