Abstract
Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the p-Laplacian type diffusion is also considered.
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1 Introduction
The temperature u of a material undergoing a multi-phase change, for instance ice-water-vapor, can be described by the following nonlinear parabolic partial differential equation
Here E is an open set of \(\mathbb {R}^N\) with \(N\ge 1\) and \(E_T:=E\times (0,T]\) for some \(T>0\).
The enthalpy \(\beta (\cdot )\) is a maximal monotone graph in \(\mathbb {R}\times \mathbb {R}\) defined by (cf. Fig. 1)
where we have assumed that \(0=e_o<e_1<\dots <e_{\ell }\),
and denoted
The equation (1.1) will be understood in a proper weak sense to be made precise later.
The main result is that locally bounded, local weak solutions to (1.1) with \(p\ge 2\) are locally continuous and a modulus of continuity is explicitly quantified.
1.1 Statement of the results
From here on, we will deal with the following more general parabolic partial differential equation modeled on (1.1):
where \(\beta (\cdot )\) is defined in (1.2). The function \(\textbf {A}(x,t,u,\xi ):E_T\times \mathbb {R}^{N+1}\rightarrow \mathbb {R}^N\) is assumed to be measurable with respect to \((x, t) \in E_T\) for all \((u,\xi )\in \mathbb {R}\times \mathbb {R}^N\), and continuous with respect to \((u,\xi )\) for a.e. \((x,t)\in E_T\). Moreover, we assume the structure conditions
where \(C_o\) and \(C_1\) are given positive constants, and we take \(p\ge 2\).
In the sequel, the set of parameters \(\{d, \nu _i, p,N,C_o,C_1, \Vert u\Vert _{\infty , E_T}\}\) will be referred to as the data. A generic positive constant \(\gamma \) depending on the data will be used in the estimates.
Let \(\Gamma :=\partial E_T-\overline{E}\times \{T\}\) be the parabolic boundary of \(E_T\), and for a compact set \(\mathcal {K}\subset E_T\) introduce the parabolic p-distance from \(\mathcal {K}\) to \(\Gamma \) by
The formal definition of local weak solution to (1.3) will be given in Sect. 1.3. Now we proceed to present the main theorem, where by \(\ln ^{(k)}\) we mean the logarithmic function composed k times.
Theorem 1.1
Let u be a bounded weak solution to (1.3) in \(E_T\), under the structure condition (1.4) for \(p\ge 2\). Then for every pair of points \((x_1,t_1), (x_2,t_2)\in \mathcal {K}\), there holds that
where
for some absolute constant \(c\in (0,1)\), and for some \(C>1\) and \(\sigma \in (0,1)\) depending on the data.
Remark 1.1
All constants in Theorem 1.1 are stable as \(p\downarrow 2\).
Remark 1.2
Even though all the proofs are given for the specific \(\beta \) in (1.2), nevertheless a more general graph can be considered, namely
where \(\beta _{AC}=\beta _{AC}(s)\) denotes an absolutely continuous and hence a.e. differentiable function in \(\mathbb {R}\), such that
for two positive constants \(\alpha _o\) and \(\alpha _1\). This reflects the fact that the thermal properties of the material under consideration might change according to the temperature. The graph (1.5) can be reduced to (1.2) by a straightforward adaption of the change of variables introduced in [4, Sect. 1]. Furthermore, Theorem 1.1 continues to hold for (1.3) with lower order terms, which take into account the convection resulting from the heat transfer. Again, the modifications of the proofs can be modeled on the arguments in [4,5,6], but we refrain from pursuing generality in this direction, focusing instead on the actual novelties.
Theorem 1.1 bears global information of \(\beta \) through the range of u. However, once a modulus of continuity is obtained, we can confine the range of u by restricting space-time distance, such that u only experiences one jump of \(\beta \) at most.
Corollary 1.1
(Localization) Under the hypotheses of Theorem 1.1, the modulus improves automatically to the one for the two-phase problem.
1.2 Novelty and significance
Graphs \(\beta \) such as the one in (1.2), but exhibiting just a single jump, say at the origin, arise from a weak formulation of the classical Stefan problem, which models a liquid/solid phase transition, such as water/ice. It is quite natural to ask whether the transition of phase occurs with a continuous temperature across the water/ice interface. This question was initially raised in a 1960 paper of Oleĭnik (see [18]) and was later reported in [14, Chapter V, Sect. 9]. Since then an important research field was born, and soon new problems started to be posed, besides the one originally formulated by Oleĭnik in her 1960 paper. The interested reader can refer to [21], to have at least an overview of the huge development that the research about the Stefan problem has witnessed. In these notes the issue is the regularity of local solutions, the ultimate goal being to prove the continuity of solutions to (1.1) for a general maximal monotone graph \(\beta \). Such a result has not been achieved yet, even though it is clear that the coercivity of \(\beta \) is essential for a solution to be continuous, as pointed out by examples in [8].
Continuity results for (1.1) with \(\beta \) as in (1.2) but with a single jump, and \(p=2\), have been given in [3, 4, 19, 22]. Moreover, Ziemer proved the continuity up to the boundary for general Dirichlet boundary data. Whereas Caffarelli and Evans heavily relied on the properties of the Laplacian, and their result cannot be extended to the full quasilinear case of (1.3), DiBenedetto’s approach is flexible enough to deal with the general framework, and it also allows lower order terms, which are thoroughly justified from a physical point of view, since they describe convection phenomena.
A quantitative estimate on the modulus of continuity, still in the case of a single jump and \(p=2\), was given in [7, Remark 3.1], but without proof. Few years later, DiBenedetto quantified Ziemer’s results, and in [5] proved that solutions have a boundary modulus of continuity of the kind
A major step forward towards a full proof of the local continuity of solutions to (1.3) with \(p=2\) and \(\beta \) a general maximal monotone graph in \(\mathbb {R}\times \mathbb {R}\), is represented by [11]; the authors proved that locally bounded, weak solutions are locally continuous, and the modulus of continuity can be quantitatively estimated only in terms of the data, even though an explicit expression of such a modulus is not provided in the paper. The proof is given in full generality for \(N=2\), whereas for \(N\ge 3\) it relies on a proper comparison function, and therefore, it is limited to \({\textbf {A}}=Du\). The paper is quite technical, but a thorough and clear presentation of the methods employed is given in [10, Sect. 5]; the list of references therein gives a comprehensive state of the art at the moment of its publication.
To our knowledge, the first paper to deal with \(p>2\) is [20]: besides its intrinsic mathematical interest, the nonlinear diffusion operator with growth of order larger than 2 naturally takes into account non-Newtonian filtration phenomena.
For a few years there were basically no further improvements, as far as the continuity issue is concerned. Things changed with [1]: the authors consider (1.3) with \(p\ge 2\) and (1.2) with a single jump, and they derive an explicit modulus of continuity better than (1.6), namely
with \(\sigma \) precisely quantified just in terms of N and p, which they conjecture to be optimal. In [2] the result is extended up to the boundary: under the same conditions as before about the equation, and assuming a positive geometric density condition at the boundary \(\partial E\), solutions to the Dirichlet problem have a modulus of continuity as in (1.6), yet weaker than (1.7).
Further progress has been recently made in [16, 17]. Indeed, interior moduli sharper than (1.7) are provided in [17] for \(p=2\) and \(N=1,2\). On the other hand, under the same general conditions as in [2], the boundary modulus of continuity has been improved to (1.7) in [16]: for Dirichlet boundary conditions, any \(p\ge 2\) can do; whereas for Neumann boundary conditions only \(p=2\) could be dealt with, while the case \(p>2\) remains an open problem.
With respect to the existing literature described so far, the present work represents a step forward, at least under two different points of view.
First of all, we consider an arbitrary number of jumps of \(\beta \), and not just a single discontinuity; this case has already been dealt with in [12], but only for \(p=2\), whereas here we work with \(p\ge 2\). Moreover, even though some of the techniques employed in [12] and here are comparable, the general approach we follow is definitely different.
The other novelty is given by the explicit modulus of continuity in Theorem 1.1: to our knowledge, it is the first time that a modulus is explicitly stated for a \(\beta \) that is more general than the one considered in [1, 2, 16, 17]. Due to the wide generality assumed on \(\beta \), i.e. arbitrary number of jumps and arbitrary height for each single jump, the parameter \(\sigma \) depends on the data, that is, also on \(\Vert u\Vert _{\infty ,E_T}\). Providing an optimal modulus of continuity that carries global information of \(\beta \) is a difficult task, and we are well aware that the one shown in Theorem 1.1 seems far from being the best possible. Nevertheless, as we have pointed out in Corollary 1.1, the importance of a quantitative continuity statement lies in the fact that once we have it, the same result implies that the modulus can be automatically improved to the one for the two-phase problem (single-jump); indeed, by restricting the space-time distance, u can be confined, so that it experiences one jump of \(\beta \) at most, and we end up having the modulus given in (1.7). We refrained from going into details about the proof of Corollary 1.1, since we would basically have to reproduce what was done in [1].
Moreover, in a forthcoming paper we plan to address a multi-phase transition problem with a maximal monotone graph \(\beta \) as in (1.5), without assuming that \(\beta _{AC}'\) is bounded above: besides its intrinsic mathematical interest, this is what occurs, for example, in the so-called Buckley-Leverett model for the motion of two immiscible fluids in a porous medium (see [13, 15]). In such a case, \(\beta \) presents two singularities, say at \(u=0\) and \(u=1\), where \(\beta \) can become vertical with an exponential speed, or even faster, and might also exhibit a jump.
1.3 Definition of solution
A function
is a local, weak sub(super)-solution to (1.3) with the structure conditions (1.4), if for every compact set \(K\subset E\) and every sub-interval \([t_1,t_2]\subset (0,T]\), there is a selection \(v\subset \beta (u)\), i.e.
such that
for all non-negative test functions
Observe that \(v\in L^{\infty }_{{\text {loc}}} \big (0,T;L^2_{{\text {loc}}}(E)\big )\) and hence all the integrals are well-defined. A function that is both a local, weak sub-solution and a local, weak super-solution is termed a local, weak solution.
We will consider the regularized version of the Stefan problem (1.3). For a parameter \(\varepsilon \in (0,\tfrac{1}{2} d)\), we introduce the function
and define the mollification of \(\beta \) by
we now deal with
Sub(super)-solutions to (1.8) are defined like for the parabolic p-Laplacian as in [6, Chapter II]. Hence, the solutions to (1.8) are generally not smooth.
In this note we assume that local solutions to (1.3) can be approximated by a sequence of solutions to (1.8) locally uniformly. This approximating approach parallels the one in [11], yet with a more particular \(\beta _\varepsilon \) and the p-Laplacian here. The goal is to establish an estimate on the modulus of continuity for the approximating solutions uniform in \(\varepsilon \), which grants the same modulus to the limiting function.
There is yet another notion of solution, which requires a solution to possess time derivative in the Sobolev sense, cf. [4, 5, 16, 17]. Theorem 1.1 continues to hold for that kind of notion and the proof calls for minor modifications from the one given here. The advantage is that an approximating scheme is not needed. However, the preset requirement on the time derivative is usually too strong to guarantee the continuity of a constructed solution in the existence theory.
1.4 Structure of the proof
Since the paper is technically involved, we think it better to first discuss the main ideas in an informal way.
Roughly speaking, we follow an approach that is by now standard when dealing with the continuity of solutions to degenerate parabolic equations: starting from a properly built reference cylinder, we have two alternatives: either we can find a sub-cylinder, such that the set where u is close to its supremum is small, or such a sub-cylinder cannot be determined.
In the first case, we can show a reduction of oscillation near the supremum, and this is accomplished in Sect. 3.1; the second alternative is more difficult and will be taken on in Sects. 3.2–3.5, where we prove a reduction of oscillation near the infimum, assuming that such an infimum is actually close to one of the discontinuity points; finally, the case of the infimum being properly far from all the discontinuity points is dealt with in Sect. 3.6. Indeed, this last possibility is the easiest one, since the equation behaves as though it were the parabolic p-Laplacian with \(p\ge 2\).
All the alternatives are quantified, and the structural dependences of the various constants are carefully traced, and this eventually leads to an estimate of the modulus of continuity in Sect. 3.7. As pointed out in Corollary 1.1, once established, the modulus improves automatically to the one for the two-phase problem. Indeed, in general, \(\omega \) could be large, and d could be small: our argument shrinks \(\omega \) step by step to an oscillation less than d across all potential jumps, and this quantification is precisely what eventually gives the modulus of (1.7).
2 Preliminary tools
2.1 Energy estimates
Here and in the following, we denote by \(K_\varrho (x_o)\) the cube of side length \(2\varrho \) and center \(x_o\), with faces parallel to the coordinate planes of \(\mathbb {R}^N\), and for \(k\in \mathbb {R}\) we let the truncations \((u-k)_+\) and \((u-k)_-\) be defined by
We can repeat almost verbatim the calculations in [11, Sect. 2] modulo proper mollification in the time variable, and prove the following estimates.
Proposition 2.1
Let u be a local weak sub(super)-solution to (1.8) with (1.4) in \(E_T\). There exists a constant \(\gamma (C_o,C_1,p)>0\), such that for all cylinders \(Q_{R,S}=K_R(x_o)\times (t_o-S,t_o)\subset E_T\), every \(k\in \mathbb {R}\), and every non-negative, piecewise smooth cutoff function \(\zeta \) vanishing on \(\partial K_{R}(x_o)\times (t_o-S,t_o)\), there holds
The general formulation of (2.1) can be simplified, if we take into account the specific structure of \(H_\varepsilon \). In particular, since \(H'_\varepsilon \ge 0\), the second term on the left-hand side can be dropped. On the other hand, since \(H_\varepsilon \) is a linear combination of Heaviside functions (an increasing step function) modulo \(\varepsilon \), we have
provided \(\sum _{i=0}^{\ell }\nu _i\) is finite. Instead, if it is infinite, we let
and estimate
Hence, in this case the subsequent estimates will depend also on M, but are independent of \(\varepsilon \).
With all these remarks, (2.1) becomes,
where the constant \(\gamma \) depends only on the data but M, if \(\sum _{i=0}^{\ell }\nu _i\) is finite. If it is infinite, the constant \(\gamma \) also depends on M.
If we choose the cutoff function \(\zeta \) such that \(\zeta (\cdot ,t_o-S)=0\), then we obtain
which corresponds to estimate (2.5) of [11].
On the other hand, if we choose the cutoff function such that \(\zeta =\zeta (x)\), i.e. independent of t, we get
which corresponds to estimate (2.6) of [11].
2.2 Logarithmic estimates
The following logarithmic energy estimate will be useful; the case \(p=2\) has been derived in [12, (3.13)] (see also [11, (2.7)]). To this end, letting k, u and \(Q_{R,S}\) be as in Proposition 2.1, we set
take \(c\in (0,\mathcal {L})\), and introduce the following function in \(Q_{R,S}\):
This function enjoys the following estimate.
Proposition 2.2
Let the hypotheses in Proposition 2.1 hold. There exists \(\gamma >1\) depending only on the data and on M, such that for any \(\sigma \in (0,1)\),
Proof
To simplify the symbolism, we denote \(\Psi (s)\equiv \Psi \big (\mathcal {L},s, c\big )\) and its derivative \(\Psi '\). In the weak formulation we use the test function \(\pm \Psi \Psi '\zeta ^p\), with \(\zeta =\zeta (x)\) and
We work in the cylinder \(K_R(x_o)\times (t_o-S,t]\) with \(t\in (t_o-S,t_o]\). Observe that
By the arbitrariness of \(t\in (t_o-S,t_o]\), we easily obtain
Since \(H'_\varepsilon \ge 0\), the second term on the left-hand side can be discarded. As for the right-hand side, since \(\Psi \) is an increasing function of its argument \((u-k)_\pm \), we have
provided \(\sum _{i=0}^{\ell }\nu _i\) is finite. If instead it is infinite, we estimate
Hence, in this case the subsequent estimates will depend also on M.
By its definition, \(\Psi '\le 1/c\) in \(Q_{R,S}\) and therefore,
where \(\gamma \) depends only on the data if \(\sum _{i=0}^{\ell }\nu _i\) is finite, otherwise it depends also on M.
Moreover, since \(\Psi \ge 0\), an application of Young’s inequality yields that
Collecting all the terms, we conclude the proof. \(\square \)
2.3 De Giorgi type lemmas
For a cylinder \(\mathcal {Q}=K \times (T_1,T_2)\subset E_T\), we introduce the numbers \(\mu ^{\pm }\) and \(\omega \) satisfying
We now present the first De Giorgi type lemma that can be shown by using the energy estimates in (2.2); for the detailed proof we refer to [16, Lemma 2.1]. Here we denote the backward cylinder \((x_o,t_o)+Q_{\varrho }(\theta ):=K_{\varrho }(x_o)\times (t_o-\theta \varrho ^p,t_o)\). If no confusion arises, we omit the vertex \((x_o,t_o)\) for simplicity.
Lemma 2.1
Let u be a local weak sub(super)-solution to (1.8) with (1.4) in \(E_T\). For \(\xi \in (0,1)\), set \(\theta =(\xi \omega )^{2-p}\). There exists a constant \(c_o\in (0,1)\) depending only on the data, such that if
then
provided the cylinder \(Q_{\varrho }(\theta )\) is included in \(\mathcal {Q}\).
The next lemma is a variant of the previous one, involving quantitative initial data. For this purpose, we will use the forward cylinder at \((x_o,t_1)\):
We have the following.
Lemma 2.2
Let u be a local weak sub(super)-solution to (1.8) with (1.4) in \(E_T\). Assume that for some \(\xi \in (0,1)\) there holds
There exists a constant \(\gamma _o\in (0,1)\) depending only on the data, such that for any \(\theta >0\), if
then
provided the cylinder \((x_o, t_1)+Q^+_{\varrho }(\theta )\) is included in \(\mathcal {Q}\).
Proof
Let us deal with the case of super-solutions only, as the other case is similar. We use the energy estimate in (2.3) in the cylinder \(Q_{R,S}\equiv (x_o, t_1)+Q^+_{\varrho }(\theta )\), with the levels
Due to this choice of \(k_n\) and the assumed pointwise information at \(t_1\), the two space integrals at \(t_o-S\equiv t_1\) vanish and the energy estimates reduce to the ones for the parabolic p-Laplacian. As a result, the rest of the proof can be reproduced as in [9, Chapt. 3, Sect. 4]. \(\square \)
The next lemma quantifies measure conditions to ensure the degeneracy of the p-Laplacian prevails over the singularity of \(\beta (\cdot )\). Its proof can be attributed to the theory of parabolic p-Laplacian. Again we omit the vertex of \((x_o,t_o)\) from \(Q_{\varrho }(\theta )\) for simplicity.
Lemma 2.3
Let u be a local weak super-solution to (1.8) with (1.4) in \(E_T\). Assume that for some \(\alpha ,\,\eta \in (0,1)\) and \(A>1\), there holds
There exists \(\xi \in (0,\eta )\), such that if \(A\ge \xi ^{2-p}\) and
where \(k=\mu ^-+\xi \omega \) and \(\theta =(\xi \omega )^{2-p}\), then
provided the cylinder \(Q_{\varrho }(A\omega ^{2-p})\) is included in \(\mathcal {Q}\). Moreover, it can be traced that
Proof
Let us turn our attention to the energy estimate (2.1) written with \(Q_{R,S}=Q_r(\theta )\) for \(\frac{1}{2}\varrho \le r\le \varrho \), and with \(k=\mu ^-+\xi \omega \). The last integral on the right-hand side is estimated by using the given measure theoretical information:
As such it can be combined with an analogous term involving \(\partial _t\zeta ^p\) on the right-hand side of (2.1). Consequently, we end up with an energy estimate of \((u-k)_-\), departing from which the theory of parabolic p-Laplacian in [6] applies. Therefore, we may determine a constant \(\xi \) by the data and \(\alpha \), such that
The dependence of \(\xi \) can be traced as in [16, Appendix A]. \(\square \)
Remark 2.1
An analogous statement for sub-solutions holds near \(\mu ^+\). Since we do not use it in the argument, it is omitted.
2.4 Consequence of the logarithmic estimate
The setting is the same as in Sect. 2.3, i.e., we introduce the cylinder \(\mathcal {Q}\subset E_T\) and define the quantities \(\mu ^\pm \) and \(\omega \) connecting the supremum/infimum and the oscillation of u over \( \mathcal {Q}\). We will use also cylinders of the forward type (2.4), with vertex at \((x_o,t_1)\).
The following lemma indicates how the measure of sets where u is close to the supremum/infimum shrinks at each level of an arbitrarily long time interval, once initial pointwise information is given.
Lemma 2.4
Let u be a local weak sub(super)-solution to (1.3) with (1.4) in \(E_T\). For \(\xi \in (0,1)\), set \(\theta =(\xi \omega )^{2-p}\). Suppose that
Then for any \(\alpha \in (0,1)\) and \(A\ge 1\), there exists \(\bar{\xi }\in (0,\frac{1}{4}\xi )\), such that
provided the cylinder \(K_{\varrho }(x_o)\times (t_1,t_1+A\theta \varrho ^p)\) is included in \(\mathcal {Q}\). Moreover, the dependence of \(\bar{\xi }\) is given by
Proof
We will prove the estimate with \(\mu ^+\), since the one with \(\mu ^-\) is completely analogous. Moreover, for simplicity we omit the dependence on \(x_o\). Proposition 2.2 will be used in the cylinder \(K_{\varrho }(x_o)\times (t_1,t_1+A\theta \varrho ^p) \). To this end, let us put
with \(\bar{\xi }\in (0,\frac{1}{4}\xi )\) to be chosen. Due to (2.5) the integrals on the right-hand side at time \(t=t_1\) vanish. Therefore, we are left with
Let us relabel \(4\gamma \) as \(\gamma \). It is easy to see that
since
On the other hand,
Hence, we may estimate
bearing in mind that \(\bar{\xi }\in (0,\frac{1}{4}\xi )\) and \(\theta =(\xi \omega )^{2-p}\). If we consider
as integration set for the integral on the left-hand side, instead of the larger \(K_{{\frac{1}{2}\varrho }}\) and note that \(\Psi \) is decreasing in \(\mathcal {L}\), we may estimate over \(A_{\bar{\xi },{\frac{1}{2}\varrho }}(t)\):
Then we obtain
with \(\widetilde{\gamma }=2\gamma \), that is
If we choose \(\bar{\xi }\) such that
we conclude the proof. \(\square \)
3 Proof of Theorem 1.1
Assume \((x_o,t_o)=(0,0)\), introduce \(Q_o=K_\varrho \times (-\varrho ^{p-1},0)\) and set
Letting \(\theta =(\frac{1}{4} \omega )^{2-p}\), for some \(A(\omega )>1\) to be determined in terms of the data and \(\omega \), we may assume that
the case when the set inclusion does not hold will be incorporated later.
3.1 Reduction of oscillation near the supremum
In this section, we work with u as a sub-solution near its supremum. Recalling that \(\theta =(\frac{1}{4} \omega )^{2-p}\), suppose for some \(\bar{t}\in \big (-(A-1)\theta \varrho ^p,0\big )\), there holds
where \(c_o\) is the constant determined in Lemma 2.1 in terms of the data. According to Lemma 2.1 (with \(\xi =\frac{1}{4}\)), we have
This pointwise estimate (3.3) at \(t_1:=\bar{t}-\theta (\tfrac{1}{2}\varrho )^p\) serves as the initial datum for an application of Lemma 2.2 and Lemma 2.4. First of all, according to Lemma 2.2, there exists \(\gamma _o\in (0,1)\), such that if for some \(\eta \in (0,\frac{1}{8})\),
then
On the other hand, owing to (3.3), Lemma 2.4 implies that (3.4) is verified with the choice
and hence so is (3.5) due to Lemma 2.2. Note that all constants are stable as \(p\downarrow 2\). Consequently, the estimate (3.5) yields a reduction of oscillation
Keep in mind that \(A(\omega )\) is yet to be chosen.
3.2 Reduction of oscillation near the infimum: part I
Starting from this section, let us suppose (3.2) does not hold, that is, for any \(\bar{t}\in \big (-(A-1)\theta \varrho ^p,0\big ]\),
Since \(\mu ^+-\frac{1}{4}\omega \ge \mu ^-+\frac{1}{4}\omega \) may be assumed without loss of generality, we rephrase it as
Fixing such \(\bar{t}\) for the moment, we will analyze the local clustering phenomenon of u encoded in the measure information (3.7). The idea of the following argument is taken from [11, Sects. 5–8]. We will work with u as a super-solution near its infimum throughout Sects. 3.2–3.6.
Lemma 3.1
For every \(\lambda \in (0,1)\) and \(\eta \in (0,1)\), there exist a point \((x_*,t_*)\in (0,\bar{t})+Q_{\varrho }(\theta )\), a number \(\kappa \in (0,1)\) and a cylinder \((x_*,t_*)+Q_{\kappa \varrho }(\theta )\subset (0,\bar{t})+Q_{\varrho }(\theta )\), such that
The dependence of \(\kappa \) is traced by \(\kappa =\gamma (\text {data}) (1-\lambda )^{3+\frac{2}{p}}\eta ^{1+\frac{1}{p}}\alpha ^{2+\frac{1}{p}}\omega ^{1+\frac{1}{p}}\).
Proof
For simplicity of notation, we take \(\bar{t}=0\). Let \(\zeta \) be a standard cutoff function in \(Q_{2\varrho }(\theta )\) that vanishes on its parabolic boundary and equals the identity in \(Q_{\varrho }(\theta )\), satisfying \(|D\zeta |\le \gamma /\varrho \) and \(|\partial _t \zeta |\le \gamma /(\theta \varrho ^p) \). According to the energy estimate (2.2) written in \(Q_{2\varrho }(\theta )\) for \((u-k)_-\) with \(k=\mu ^-+\frac{1}{4}\omega \) and with the cutoff function \(\zeta \), a simple calculation yields
In terms of
this energy estimate may be written as
and meanwhile the measure information (3.7) reads
To proceed, let us define the set
and the set
Now we may estimate by using (3.9):
which implies \(|\mathcal {P}|\ge \tfrac{1}{2}\alpha \theta \varrho ^p\). This joint with (3.8) yields that
that is,
Therefore, there exists \(\widetilde{t}\in \mathcal {P}\), such that
Meanwhile, by the definition of \(\mathcal {P}\), there holds that
Based on (3.10) and (3.11), we are ready to apply [9, Chap. 2, Lemma 3.1]. Indeed, let \(\widetilde{\lambda }:=\tfrac{1}{2}(1+\lambda )\) and \(\widetilde{\eta }\in (0,1)\) to be determined: there exist \(\widetilde{x}\in K_\varrho \) and
such that
Reverting to u, we actually obtain
In order to propagate this measure information, we consider the forward cylinders
where \(\delta >0\) is to be determined. Let \(\zeta (x)\) be a cutoff function in \(K_{ \varepsilon \varrho }(\widetilde{x})\) that vanishes on \(\partial K_{ \varepsilon \varrho }(\widetilde{x})\) and equals the identity in \(K_{ \frac{1}{2}\varepsilon \varrho }(\widetilde{x})\), such that \(|D\zeta |\le \gamma / (\varepsilon \varrho )\). Employing (3.13), the energy estimate (2.3) for \((u-k)_-\) with \(k=\mu ^-+\tfrac{1}{4}\widetilde{\lambda }\omega \) in this setting gives that
for all \(t\in \big (\widetilde{t},\widetilde{t}+\delta \theta ( \varepsilon \varrho )^p\big )\).
We estimate the integral on the left-hand side from below by
Substituting this estimate back to the energy estimate yields that
Now we may choose \(\delta \) and \(\widetilde{\eta }\) to satisfy
Up to now, we have shown that
for all \(t\in \big (\widetilde{t},\widetilde{t}+\delta \theta (\varepsilon \varrho )^p\big )\). For simplicity let us denote \(r:=\frac{1}{2}\varepsilon \varrho \). The above slicewise measure information actually yields
Arranging \(L:=\tfrac{1}{2}\delta ^{-\frac{1}{p}}\) to be an integer, we partition Q along the space coordinate planes into \(L^N\) disjoint but adjacent sub-cylinders, each of which is congruent to
where \(\kappa :=\varepsilon \delta ^{\frac{1}{p}}\) can be traced by combining (3.12) and (3.14), i.e.
Due to (3.15), it is easy to see that at least one of them, say \((x_*,t_*)+Q_{\kappa \varrho }(\theta )\), will satisfy the desired property
The proof is concluded with such a choice of \((x_*,t_*)\) and \(\kappa \). \(\square \)
The location of the clustering within \((x_*,t_*)+Q_{\kappa \varrho }(\theta )\subset (0,\bar{t})+Q_{\varrho }(\theta )\) being only qualitative notwithstanding, the quantified measure concentration allows us to extract pointwise estimate with the aid of Lemma 2.1 and then use the logarithmic energy estimate to propagate the measure information up to the level \(\bar{t}\), cf. Fig. 2.
As a matter of fact, if in Lemma 3.1 we choose \(\lambda =\tfrac{1}{2}\) and \(\eta =c_o(\tfrac{1}{4}\omega )^{\frac{N+p}{p}}\) where \(c_o\) is determined in Lemma 2.1, then Lemma 3.1 and Lemma 2.1 would yield that
for some \((x_*,t_*)\in (0,\bar{t})+Q_{\varrho }(\theta )\) and the constant
In particular, we have
which serves as the initial datum to apply Lemma 2.4. Indeed, setting \(\alpha =\tfrac{1}{2}\) and \(\xi =\tfrac{1}{16}\) in Lemma 2.4 and choosing \(\widetilde{A}\) so large that
it yields a number \(\bar{\xi }\in (0,\tfrac{1}{4}\xi )\), such that
The dependence of \(\bar{\xi }\) is traced by
The measure information (3.16) permits us to claim that
Thanks to the arbitrariness of \(\bar{t}\), we have actually arrived at
The dependence of \(\bar{\alpha }\) is traced by
This measure information (3.18) lays the foundation for the analysis to be set out in the following sections. Since A is a large number, we will stipulate that (3.18) holds with \(A-1\) replaced by A for simplicity.
3.3 Reduction of oscillation near the infimum: part II
Let us first introduce the following intrinsic cylinders
for some \(\xi (\omega )\) and \(\delta (\omega )\) in (0, 1) to be determined later. We can always assume \(\xi \) and \(\delta \) to be sufficiently small, so that \( \widehat{\theta }\le \widetilde{\theta }\). On the other hand, we may assume that
for some \(A(\omega )\) yet to be determined.
Throughout Sects. 3.3–3.5, we always assume that
for the same \(\xi (\omega )\) and \(\delta (\omega )\) in (0, 1) introduced above, to be determined. When restriction (3.21) does not hold, the case will be examined in Sect. 3.6.
First of all, we turn our attention to Lemma 2.1 and Lemma 2.3. In view of the measure information (3.18), Lemma 2.3 is at our disposal, with \(\alpha \), \(\eta \) and A replaced by \(\bar{\alpha }\), \(\bar{\xi }\) and \(A/4^{2-p}\) respectively. Suppose \(\xi \) is determined in Lemma 2.3 in terms of the data and \(\bar{\alpha }\) fixed in (3.19), and recall that \( \widehat{\theta }=(\xi \omega )^{2-p}\). If there holds
then Lemma 2.1 yields that
Analogously, if for \(k=\mu ^-+ \xi \omega \), there holds
then Lemma 2.3 yields that, stipulating \(A\ge 4^{2-p}\xi ^{2-p}\),
Consequently, either (3.22) or (3.23) yields a reduction of oscillation
For later use, we record the dependence of \(\xi \) here, that is,
3.4 Reduction of oscillation near the infimum: part III
In this section, we continue to examine the situation when the measure condition in Lemma 2.1 is violated:
and when the condition in Lemma 2.3 is also violated: for \(k=\mu ^-+ \xi \omega \), there holds
Combining (3.26) and (3.27), we obtain that, for all \(r\in [2\varrho , 8\varrho ]\),
where \(\widetilde{\gamma }=c_o16^{-N-p}\) and \(b=1+\tfrac{N+p}{p}\).
Next, introduce a free parameter \(\bar{\delta }\in (\delta ,2\delta )\) and set \(\bar{\theta }:=(\bar{\delta }\xi \omega )^{1-p}\). Recall also that \( \theta =(\tfrac{1}{4}\omega )^{2-p}\), \(\widetilde{\theta }=(\delta \xi \omega )^{1-p}\), \(\widehat{\theta }=(\xi \omega )^{2-p}\), and that we have assumed \(\widetilde{\theta }(8\varrho )^p\le A \theta \varrho ^p\le \varrho ^{p-1}\) in (3.20). Therefore,
The estimate (3.28) implies that there exists \(t_*\in [-\widehat{\theta } r^p,0]\), such that
Observe also that for any \(t\in [-\bar{\theta } r^p, 0]\) and any \(\bar{\delta }\in (\delta ,2\delta )\), there holds
Denoting \(\bar{k}=\mu ^-+ \bar{\delta }\xi \omega \) and enforcing that for some \(i\in \{0,1,\cdots ,\ell \}\),
we use (3.29) and (3.30) to estimate
In the first inequality above, we have assumed \(\tfrac{9}{4}\xi \omega \le d\) by possibly further restricting the choice of \(\xi \) in (3.25), and hence \((\mu ^-,\bar{k})\subset (e_i-\tfrac{1}{4}\delta \xi \omega ,e_i+\tfrac{1}{4}\delta \xi \omega +2\delta \xi \omega )\subset (e_i-d,e_i+d)\). As such the constant \(\gamma \) in the definition of \(\xi \) in (3.25) depends on d and M. The above analysis together with (2.1) yields the following energy estimate.
Lemma 3.2
Let u be a weak super-solution to (1.8) with (1.4) in \(E_T\), under the measure information (3.18). Let (3.26) and (3.27) hold true. Denoting \(b:=1+\tfrac{N+p}{p}\) and setting \(k=\mu ^-+\bar{\delta }\xi \omega \) with \(\bar{\delta }\in (\delta ,2\delta )\), there exists a positive constant \(\gamma \) depending only on the data, such that for all \(\sigma \in (0,1)\) and all \(r\in [2\varrho , 8\varrho ]\) we have
provided that for some \(i\in \{0,1,\cdots ,\ell \}\),
Based on the energy estimate in Lemma 3.2, a De Giorgi type lemma can be derived. Notice that the time scaling used here is different from the one in Lemmas 2.1–2.3.
Lemma 3.3
Suppose the hypotheses in Lemma 3.2 hold. Let \(\delta \in (0,1)\). There exists a constant \(c_1\in (0,1)\) depending only on the data, such that if
then enforcing \(|\mu ^- - e_i |\le \tfrac{1}{4}\delta \xi \omega \) for some \(i\in \{0,1,\ldots ,\ell \}\) and \(\varepsilon \le \tfrac{1}{4}\delta \xi \omega \), we have
provided \(4^p\widetilde{\theta }\le A \theta \).
Proof
For \(n=0,1,\ldots ,\) we set
We will use the energy estimate in Lemma 3.2 with the pair of cylinders \(\widetilde{Q}_n\subset Q_n\). Note that the constant \(\bar{\delta }\) in Lemma 3.2 is replaced by \((1+2^{-n})\delta \), as indicated in the definition of \(k_n\). Enforcing \(|\mu ^- - e_i|\le \tfrac{1}{4}\delta \xi \omega \) and \(\varepsilon \le \tfrac{1}{4}\delta \xi \omega \), the energy estimate in Lemma 3.2 yields that
where \(A_n:=\big [u<k_n\big ]\cap Q_n\).
Let \(0\le \phi \le 1\) be a cutoff function that vanishes on the parabolic boundary of \(\widetilde{Q}_n\) and equals the identity in \(Q_{n+1}\). An application of the Hölder inequality, the Sobolev imbedding [6, Chap. I, Proposition 3.1] and the above energy estimate give that
In terms of \( Y_n=|A_n|/|Q_n|\), the recurrence is rephrased as
for a constant \(\gamma \) depending only on the data and with \(C=C(N,p)\). Hence, by [6, Chap. I, Lemma 4.1], there exists a positive constant \(c_1\) depending only on the data, such that \(Y_n\rightarrow 0\) if we require that \(Y_o\le c_1(\xi \omega )^b\). This concludes the proof. \(\square \)
The next lemma concerns the smallness of the measure density of the set \([u\approx \mu ^-]\). Its proof relies on (2.2) and the measure information (3.18) will be employed.
Lemma 3.4
Let u be a weak super-solution to (1.8) with (1.4) in \(E_T\), under the measure information (3.18). There exists a positive constant \(\gamma \) depending only on the data, such that for any \(j_*\in \mathbb {N}\) we have
with \(\bar{\alpha }\) as in (3.19), provided \(4^p\widetilde{\theta }\le A \theta \).
Proof
We employ the energy estimate (2.2) in \(Q_{8\varrho }(\widetilde{\theta })\) with a standard cutoff function \(\zeta \) that vanishes on the parabolic boundary of \(Q_{8\varrho }(\widetilde{\theta })\) and equals the identity in \(Q_{4\varrho }(\widetilde{\theta })\), satisfying \(|D\zeta |\le \gamma /\varrho \) and \(|\partial _t\zeta |\le \gamma /(\widetilde{\theta }\varrho ^p)\). The levels are chosen to be
Therefore, assuming \(j_*\) has been chosen, and taking into account the definition of \( \widetilde{\theta }(j_*)\), the energy estimate (2.2) yields that
where \( A_{j,8\varrho }:= \big [u<k_{j}\big ]\cap Q_{8\varrho }(\widetilde{\theta })\).
Observing \(\xi <\bar{\xi }\) from (3.17) and (3.25), we may derive the measure theoretical information from (3.18):
With this information at hand, we apply [6, Chap. I, Lemma 2.2] slicewise to \(u(\cdot ,t)\) for \(t\in \big (-\widetilde{\theta }(4\varrho )^p,0\big ]\), over the cube \(K_{4\varrho }\), for levels \(k_{j+1}<k_{j}\), followed by an application of Hölder’s inequality. Indeed, we estimate
where we have set \( A_{j,4\varrho }(t):= \big [u(\cdot ,t)<k_{j}\big ]\cap K_{4\varrho }\). We perform an integration in \(\textrm {d}t\) over the interval \(\big (-\widetilde{\theta }(4\varrho )^p,0\big ]\) on both sides and apply Hölder’s inequality. Setting \(A_{j,4\varrho }:=[u<k_j]\cap Q_{4\varrho }(\widetilde{\theta })\), we arrive at
Now take the power \(\frac{p}{p-1}\) on both sides of the above inequality to obtain
Add these inequalities from 0 to \(j_*-1\) to obtain
from which we easily obtain
The proof is completed. \(\square \)
3.5 Reduction of oscillation near the infimum: part IV
Under the conditions (3.26) and (3.27), we may reduce the oscillation in the following way. First of all, let \(\xi \) be determined in (3.25). Then we choose, according to Lemma 3.4, the integer \(j_*\) so large to satisfy that
where \(c_1\) is the constant appearing in Lemma 3.3. According to (3.19) and (3.25), the dependence of \(j_*\) can be traced by
for some positive \(\{q_o,q_1\}\) depending on the data.
Next, we can fix \(2\delta =2^{-j_*}\) in Lemma 3.3. Consequently, by the choice of \(j_*\) in (3.31), Lemma 3.3 can be applied, assuming that \(|\mu ^- - e_i|\le \tfrac{1}{4}\delta \xi \omega \) for some \(i\in \{0,1,\cdots ,\ell \}\) and \(\varepsilon \le \frac{1}{4}\delta \xi \omega \), and we arrive at
where we may trace, recalling (3.25),
for some generic \(\gamma \) and some positive \(\{q_o, q_1, q_2\}\) determined by the data. This would give us a reduction of oscillation
with the above-defined \(\delta \) and \(\xi \). The choice of A can be finally made from \(8^p\widetilde{\theta }\le A\theta \) as required in (3.20), i.e. \(A\ge 2^{p+4}\omega ^{-1}(\delta \xi )^{1-p}\). Thus we may choose
for some properly defined positive \(\gamma \) and q depending only on the data.
To summarize the achievements in Sects. 3.1–3.5, taking the reverse of (3.1), (3.6), (3.24) and (3.33) all into account, if \(|\mu ^- - e_i|\le \tfrac{1}{4}\delta \xi \omega \) for some \(i\in \{0,1,\cdots ,\ell \}\) and \(\varepsilon \le \frac{1}{4}\delta \xi \omega \) hold true, then for \(\theta =(\tfrac{1}{4}\omega )^{2-p}\) we have that
where
for some properly defined positive \(\gamma \) and q depending only on the data.
3.6 Reduction of oscillation near the infimum: part V
Let \(\xi (\omega )\) and \(\delta (\omega )\) be determined in (3.32). The analysis throughout Sects. 3.3–3.5 has been founded on the condition (3.21). We now examine the case when (3.21) does not hold, namely,
Notice that the analysis in Sect. 3.2 does not rely on the condition (3.21), and thus the measure information (3.18) derived there is still at our disposal. In view of the dependences of \(\delta \) and \(\xi \) in (3.32) and that of \(\bar{\xi }\) in (3.17), we may assume that \(\delta \xi <\bar{\xi }\) and that (3.18) holds true with \(\bar{\xi }\) replaced by \(\delta \xi \).
Next, for \(\widetilde{\xi }\in (0,\tfrac{1}{8})\) we introduce the levels \(k=\mu ^-+ \widetilde{\xi }\delta \xi \omega \). According to (3.36) and assuming that \(\varepsilon \le \frac{1}{4}\delta \xi \omega \), the energy estimate (2.1)\(_-\) written in \(Q_{\varrho }(\vartheta )\subset Q_\varrho (A\theta )\) for some \(0<\vartheta <A\theta \) yields that
Given this energy estimate and the measure information (3.18), the theory of parabolic p-Laplacian in [6] applies; see also [16, Appendix A] for tracing the constants.
Lemma 3.5
Let u be a weak super-solution to (1.8) with (1.4) in \(E_T\). Suppose (3.18) and (3.36) hold true, and \(\varepsilon \le \frac{1}{4}\delta \xi \omega \). There exists a positive constant \(\widetilde{\xi }\) depending on the data and \(\bar{\alpha }\) of (3.19), such that for \( \vartheta =(\widetilde{\xi }\delta \xi \omega )^{2-p}\) we have
provided \(\vartheta \le A\theta \). Moreover, the dependence of \(\widetilde{\xi }\) can be traced by
Remark 3.1
Note that the choice of A in (3.34) verifies \(\vartheta \le A\theta \).
3.7 Derivation of the modulus of continuity
This is the final part of the proof of Theorem 1.1. Let us summarize what has been achieved by the previous sections. To do so, we will first assume that \(\omega \le 1\). According to (3.35) and Lemma 3.5, we have
where \(\theta =(\tfrac{1}{4}\omega )^{2-p}\) and
for some properly defined positive \(\gamma \) and q depending only on the data.
In order to iterate the arguments, we set
and seek \(\varrho _1\) to verify the set inclusions, recalling A from (3.34):
where \(\theta _1:=(\tfrac{1}{4}\omega _1)^{2-p}\), \(A_1:=A(\omega _1)\). Note that we may assume \(\eta (\omega )\le \frac{1}{2}\), which yields \(\omega _1\ge \frac{1}{2}\omega \). Then we estimate
and hence choose
It is not hard to verify that the other set inclusion also holds with such a choice of \(\varrho _1\). Consequently, we arrive at
which takes the place of (3.1)\(_2\) in the next stage. Repeating the arguments of Sects. 3.1–3.6, we obtain that
Now we may construct for \(n\in \mathbb {N}\),
By induction, if up to some \(j\in \mathbb {N}\), we have
then for all \(n\in \{0,1,\cdots , j\}\), there holds
On the other hand, we denote by j the first index to satisfy
Observe that if there exist \(n_o\in \mathbb {N}\) and a sequence \(\{a_n\}\) satisfying
for all \(n\ge n_o\), and meanwhile \(a_{n_o}\ge \omega _{n_o}\), then \(a_n\ge \omega _n\) for all \(n\ge n_o\). We may choose
for some proper \(\sigma \in (0,\tfrac{1}{q})\) and an absolute constant a, such that \(a_o\ge 1\). Since we have assumed \(\omega \le 1\), we have \(a_o\ge \omega _o\) and hence, \(a_n\ge \omega _n\) for all \(n\ge 0\).
Let us take \(4r\in (0,\varrho )\). If for some \(n\in \{0,1,\cdots , j\}\), we have
then the right-hand side inequality yields
Next we examine the left-hand side inequality. For this purpose, we first note that it may be assumed that \(\eta (\omega _n)\le \frac{1}{2}\). Hence, we estimate \(\omega _n\ge (\frac{1}{2})^n\omega \),
and
By taking logarithm on both sides, we estimate
for some absolute constant \(c>0\). Substituting it back to (3.39), we obtain
for some \(C>0\) depending on the data and \(\omega \).
Finally, if \(4r<\varrho _{j+1}\) where j is the first index for (3.38) to hold, then we may use (3.38) and
to incorporate the \(\varepsilon \) term into the oscillation estimate:
Now according to our assumption in Sect. 1.3 we may let \(\varepsilon \rightarrow 0\) and obtain the desired modulus of continuity.
The assumption that \(\omega \le 1\) at the beginning of this section is not restrictive. For otherwise, the same arguments in the previous sections generate quantities
depending only on the data, but independent of \(\omega \). Consequently, instead of (3.37), we end up with
Given this, we may set up an iteration scheme as before and iterate \(n_*=n_*(\text {data})\) times, such that
for some \(\varrho _*\) and \(\omega _*\) depending on \(n_*\).
Without loss of generality, due to (3.34), we may take \(A=A(\omega _*)\). As such the above intrinsic relation plays the role of (3.1) and the previous arguments can be reproduced.
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Acknowledgements
U. Gianazza was supported by the grant 2017TEXA3H_002 “Gradient flows, Optimal Transport and Metric Measure Structures”. N. Liao was supported by the FWF-Project P31956-N32 “Doubly nonlinear evolution equations”. We thank the referees for their careful reading and comments.
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Gianazza, U., Liao, N. Continuity of the temperature in a multi-phase transition problem. Math. Ann. 384, 1–35 (2022). https://doi.org/10.1007/s00208-021-02255-x
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DOI: https://doi.org/10.1007/s00208-021-02255-x