1 Introduction and results
In this article, we are interested in the self-improving property of higher integrability of weak solutions to porous medium-type systems, whose prototype is
This problem has been open for some time. For non-negative solutions to porous medium-type equations it has recently been solved by Gianazza and Schwarzacher . Here, we are able to treat signed solutions and the vectorial case. More precisely, we consider equations (the case ) or systems (the case ) of the form
with , in a space-time cylinder , where is a bounded open domain, , , and we abbreviated . The assumptions on the vector field are as follows. We assume that is measurable with respect to for all , continuous with respect to for a.e. , and moreover that satisfies for some structural constants the following growth and ellipticity conditions:
for a.e. and any . Note that these assumptions are compatible with the ones in  and [11, Chapter 3.5] in the scalar case. For the inhomogeneity we assume that . As usual, we suppose that the solutions to (1.1) lie in a parabolic Sobolev space; the precise definition will be given below in Definition 1.1.
In the stationary elliptic case it is by now well known that weak solutions to elliptic systems of the type
locally belong to a slightly higher Sobolev space than a priori assumed. The so-called self-improving property of higher integrability was first detected by Elcrat and Meyers . Their proof is based, among other things, on a reverse Hölder-type inequality – a direct consequence of a Caccioppoli-type inequality (also called reverse Poincaré inequality) – and some adaptation of the famous Gehring Lemma ; the nowadays standard interior version can be retrieved from [17, Chapter 11, Theorem 1.2], for the boundary version we refer to  and [13, Theorem 2.4]. Originally, Gehring’s lemma was developed to establish the higher integrability of the Jacobian of quasi-conformal mappings. Over time, the self-improving property of higher integrability was first established for solutions of stationary elliptic systems  and later for minima of variational integrals  by Giaquinta and Modica. A unified treatment in the language of quasi-minima is given in [21, Theorem 6.7]. Corresponding global results for stationary elliptic problems with a Dirichlet boundary condition were established in [21, Section 6.5], [13, Section 3].
The first higher integrability result for vectorial evolutionary problems goes back to Giaquinta and Struwe [20, Theorem 2.1]. More precisely, quasilinear parabolic systems of the type
whose coefficients continuously depend on have been investigated. The technique of Giaquinta and Struwe does not carry over to the parabolic p-Laplacian system
or general parabolic systems with p-growth (the growth and coercivity condition from (1.2) have to be replaced by and ). The obstruction relies in the fact that the parabolic p-Laplacian equation has a different homogeneity in the time and the diffusion term. In particular, multiples of a solution do not anymore solve the differential equation. This problem has finally been solved by Kinnunen and Lewis  who proved the higher integrability result for general parabolic systems with p-growth. More precisely, they have shown that weak solutions from the natural energy space have a more integrable spatial gradient, namely
This shows that also in the case of parabolic systems with coefficients of p-growth and coercivity energy solutions enjoy the self-improving property of higher integrability for the gradient. The key to the result was the use of intrinsic cylinders in the sense of DiBenedetto and Friedman [8, 10, 9, 7], i.e. cylinders of the form whose space-time scaling depends on the spatial gradient of the solution via
This important result has been generalized over time in various directions. The global result with a Dirichlet boundary condition at the lateral boundary was established by Parviainen . Interior higher integrability for weak solutions of higher order degenerate parabolic systems has been shown by Bögelein , while the corresponding global result was established in . The case of parabolic equations with non-standard -growth was treated by Antonsev and Zhikov , while systems were treated by Zhikov and Pastukhova  and independently by Bögelein and Duzaar .
For the porous medium equation, the question of higher integrability of the gradient, even for non-negative solutions in the scalar case, remained an open problem for a while. The reason was that when proving regularity of the gradient the degeneracy with respect to u is much more difficult to handle. This difficulty has recently been overcome by Gianazza and Schwarzacher  who proved that non-negative weak solutions to porous medium equations of the type (1.1) enjoy the self-improving property of higher integrability. More precisely, this means that the integrability of weak solutions was improved to
The main novelty with respect to the proof for the parabolic p-Laplacian in  is that Gianazza and Schwarzacher work with cylinders which are intrinsically scaled with respect to u rather than the spatial gradient Du. This means that they consider cylinders of the type whose space-time scaling is adapted to the solution u via the coupling
This is exactly the intrinsic scaling which is typically used in the proof of regularity of u, as for instance Hölder continuity of u, cf. . At first glance it is quite surprising that this approach also yields regularity of the spatial gradient. However, these cylinders are better adapted to the equation and this is crucial for the proof. Nevertheless, the argument becomes much more involved than the one for the parabolic p-Laplacian. The overall strategy can be outlined as follows. First, one has to prove a reverse Hölder-type inequality on certain intrinsic cylinders. To achieve this, Gianazza and Schwarzacher distinguish whether a cylinder Q belongs to the non-degenerate regime in which the inequality
holds true for some particular , or Q belongs to the degenerate regime in which the opposite inequality is valid. In the non-degenerate regime they rely on the expansion of positivity in order to guarantee that the solution does not become too small on the cylinder. In a second step, one usually constructs a covering of super-level sets of the spatial gradient with intrinsic cylinders. However, this is not possible for the cylinders which are intrinsically scaled with respect to u. Gianazza and Schwarzacher overcame this problem by a very elegant idea. They weakened this property to the so-called sub-intrinsic cylinders for which they succeeded to prove the covering property. Thereby, they call a cylinder sub-intrinsic if (1.3) holds as an inequality, i.e. the mean value integral is bounded from above by the right-hand side.
The methods of proof of this important result are only applicable in the scalar case for non-negative solutions, because tools as the expansion of positivity are neither available in the vectorial case, nor for signed solutions.
The present paper has its origin in the effort to extend the purely scalar result to the vector-valued case. As a by-product of the vectorial case, we are able to deal also with signed solutions in the scalar case. Moreover, contrary to Gianazza and Schwarzacher, we start from the definition of weak (energy) solutions introduced in [28, Theorem 5.5], i.e. we start with solutions satisfying , see (1.6). As main result, we prove that
We note that starting from a vectorial version of the energy estimate used in , a modification of our method also applies to the definition of weak solution as considered there. The key to the higher integrability result in the vectorial case is to prove the reverse Hölder-type inequality just by the use of an energy estimate and a gluing lemma as stated in Lemmas 3.1 and 3.2. In particular, it is important to omit the use of the expansion of positivity. In fact, for the proof of the Sobolev–Poincaré-type inequality in Lemma 4.3 we only use the Gluing Lemma 3.2, the standard Sobolev inequality and some algebraic lemmas. Here, we note that contrary to (1.3) we work with differently scaled cylinders which reflect more clearly the behavior of the porous medium equation and which are adapted to the energy space (1.6) (for the heuristics see also [16, Remark 5.6]). These cylinders are given by with an intrinsic scaling of the form
so that in case that the mean value of on the cylinder is zero, the scaling parameter θ is comparable to . A cylinder is called sub-intrinsic if (1.4) holds as an inequality, where the mean value integral is bounded from above by the right-hand side. Contrary to  we present a unified proof of the Sobolev–Poincaré-type inequalities on sub-intrinsic cylinders that works likewise in the non-degenerate and degenerate regime. These inequalities are subsequently used to derive reverse Hölder-type inequalities on intrinsic cylinders and sub-intrinsic cylinders additionally satisfying
For the final proof of the higher integrability we cover the super-level-sets of by sub-intrinsic cylinders. Here, we rely on the construction by Gianazza and Schwarzacher. The idea is to choose with the help of the intermediate value theorem for a given center and radius the scaling parameter in such a way that
is satisfied, where
Unfortunately, the mapping is not monotone. Therefore, we modify the parameter by a rising sun-type construction, i.e. we define
Then the mapping is monotonically decreasing and furthermore one can show that the cylinders are still sub-intrinsic. A crucial observation at this point is that by construction either the cylinders are intrinsic or satisfy (1.5). This allows to apply our reverse Hölder inequality. As in  our cylinders satisfy a Vitali covering property which allows to cover the super-level-sets of by countably many of these cylinders. In this way, we obtain a reverse Hölder inequality on the super-level-sets of . In a standard way, this implies the higher integrability by a Fubini-type argument.
1.1 General setting and results
In this subsection we fix the notations, describe the general setup and present our main result. First, we define what we mean by a weak energy solution to the porous medium-type system.
Assume that the vector field satisfies the conditions in (1.2) and that . We identify a measurable map in the class
as a weak solution to the porous medium-type system (1.1) if and only if the identity
holds true, for any testing function .
Throughout the paper we work with parabolic cylinders of the type
whose associated parabolic dimension is
Our main result reads now as follows.
Let and . There exist constants and such that the following holds true: Whenever and
where . Moreover, for every and every cylinder , we have the quantitative local higher integrability estimate
The quantitative local estimate (1.8) can be converted easily into an estimate on the standard parabolic cylinders . The precise statement is as follows.
Under the assumptions of Theorem 1.2, the estimate
holds true on any parabolic cylinder and for every and with a constant .
In order not to overburden the notation, we abbreviate in the following the power of a vector (or possibly negative number) by
where we interpret in the case and . Throughout the paper we write and use the space-time cylinders
with some scaling parameter . One of the most important notions for this paper is the notion of sub-intrinsic cylinders. We call a cylinder sub-intrinsic if and only if
holds true. If the preceding inequality actually is an equality, we call the cylinder intrinsic. In the case , we simply omit the parameter in our notation and write
instead of , and, analogously, instead of . If is the origin, we write , and for , and . Moreover, if the center is clear from the context, we omit it in our notation.
For a map and given measurable sets and with positive Lebesgue measure the slicewise mean of u on A is defined by
whereas the mean value of u on E is defined by
Note that if the slicewise means are defined for any . If the set A is a ball , then we abbreviate and if E is a cylinder of the form , we use the shorthand notation . Finally, we define the boundary term
that will appear in the energy estimate from Lemma 3.1.
2.2 Auxiliary material
In order to “re-absorb” certain terms, we will use the following iteration lemma, which can be retrieved by a change of variable from [21, Lemma 6.1].
Let , and . Then there exists a constant such that there holds: For any and any non-negative bounded function satisfying
For any , there exists a constant such that, for all the following assertions hold true:
The proof of (i) and (ii) can be found in [19, Lemma 2.2]. Inequality (iii) can be derived by combining the proof of [7, Chapter I, Lemma 4.4] with (i). The next lemma provides useful estimates for the boundary term introduced in (2.3).
There exists a constant such that for any the following assertions hold true:
Using the auxiliary function , , we can re-write the boundary term to
The Hessian of ϕ is given by the matrix
whose eigenvalues are and . Therefore, the integral formula for the remainder in Taylor’s expansion yields
Now, we distinguish between the cases and . In the first case, for any we have
from which we infer
where . In the second case , we restrict ourselves to values . Interchanging the roles of u, a and t, we end up with the same estimate for . In view of (2.4), this implies also in the remaining case for estimate (2.5). Combining this with Lemma 2.2 (i), we arrive at the first claimed estimate, since
For the second asserted estimate, we apply Lagrange’s formula for the remainder in Taylor’s expansion, which yields
In view of Lemma 2.2 (i), this yields the second estimate from (i), since
The inequalities in (ii) are a consequence of (i) and Lemma 2.2 (i) applied with , and , since
The following estimate, which is known as the quasi-minimality of the mean value, can be established by Young’s and Hölder’s inequality.
Let . Then for any bounded domain , , any , and any there holds
The following statement shows that mean values over subsets are still quasi-minimizing. This is well known for . Here, we state the version for powers. As expected, the quasi-minimality constant depends on the ratio of the measures of the set and the subset.
Let . Then there exists a universal constant such that whenever , , are two bounded domains and , there holds
We start by estimating the difference . Using Lemma 2.2 (i)–(ii), we obtain for a constant that
From this estimate we conclude
which proves the claim. ∎
The following lemma is from [12, Lemma 6.2]. For convenience of the reader, we nevertheless include the proof.
Let . Then there exists a universal constant such that for any bounded domain , any non-negative , and any there holds
Using Lemma 2.2 (iii), we obtain for a constant that
In order to estimate the integrand from above, we distinguish between two cases. In the case , we have
and hence . In turn, this allows us to estimate
which by Lemma 2.2 (ii) implies
In the remaining case , Lemma 2.2 (i) shows
An application of Lemma 2.2 (i) therefore yields
holds true for a constant . We insert this into (2.2) and apply Young’s inequality twice. This leads to
Here we re-absorb the term into the left-hand side and obtain the asserted inequality. ∎
Finally, we ensure that the mean value is also a quasi-minimizer of .
There exists a universal constant such that for any bounded domain , any non-negative , and any there holds
3 Energy bounds
In this section we derive an energy inequality and a gluing lemma which follow from the weak formulation (1.7) of the differential equation by testing with suitable testing functions. Later on, they will be used in order to prove Sobolev–Poincaré and reverse Hölder-type inequalities.
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the growth and ellipticity assumptions (1.2). Then there exists a constant such that on any cylinder with and , and for any and any the following energy estimate:
holds true, where has been defined in (2.3).
For , we define the following mollification in time:
From the weak form (1.7) of the differential equation we deduce the mollified version (without loss of generality we may assume that )
for any . Let be the standard cut off function with in and and defined by
Furthermore, for given and we define the cut-off function by
as testing function in the mollified version (3.1) of the differential equation. For the integral containing the time derivative we compute
where we also used the identity , cf. [24, Chapter 2]. Since in , we may pass to the limit in the integral on the right-hand side and therefore find that
At this point, we pass to the limit and obtain for the first term
for any , whereas the term can be estimated in the following way (observe that the boundary term is non-negative):
for a constant . Finally, we consider the right-hand side integrals in (3.1). The second integral disappears in the limit , since . In the integral containing the inhomogeneity F we pass to the limit and subsequently apply Hölder’s inequality. In this way, we obtain
We combine these estimates and then pass to the limit . This leads to
for any , with a constant . In the preceding inequality we take in the first term on the left-hand side the supremum over , and then pass to the limit . Finally, we take means on both sides. This procedure leads to the claimed inequality. ∎
The following lemma serves to compare the slice-wise mean values at different times. This is necessary since Poincaré’s and Sobolev’s inequality can only be applied slice-wise. Such a result, which connects means on different time slices, is termed Gluing Lemma.
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the growth and ellipticity assumptions (1.2). Then for any cylinder with and there exists such that for all there holds
for a constant .
Let with and assume that . For and , we define by
and a radial function by , where
for . For fixed we choose as testing function in the weak formulation (1.7), where denotes the i-th canonical basis vector in . In the limit we obtain
We multiply the preceding inequality by and sum over . This yields
Here, we use the growth condition (1.2) and immediately get for any and any that there holds
there exists a radius with
Therefore, we choose in the above inequality and then take means on both sides of the resulting inequality. This implies
for any and with a constant . ∎
4 Parabolic Sobolev–Poincaré-type inequalities
Throughout this section we consider so-called sub-intrinsic cylinders. These cylinders are characterized as follows: On the scaled cylinder with and the following coupling between the mean of on and θ holds true:
The following lemma is the first step towards a Poincaré-type inequality for weak solutions to the porous medium system. This is necessary because the standard Poincaré inequality in cannot be applied directly, since weak solutions u a priori do not possess the necessary regularity with respect to time; note that we only assume for the spatial derivative , while no regularity assumption with respect to time is incorporated in the definition of weak solutions. Nevertheless, we are able to prove some sort of Poincaré inequality. This is achieved by considering the space and time direction separately. In x-direction we can apply the Poincaré inequality on , while in t-direction the needed regularity is gained from the gluing lemma.
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the growth and ellipticity assumptions (1.2). Then on any cylinder satisfying the sub-intrinsic coupling (4.1) for some and some , the inequality
holds true with a universal constant .
In the following we shall again omit for simplification the reference point in our notation. Moreover, we let be the radius from Lemma 3.2. By adding and subtracting the slice-wise means as defined in (2.2), we obtain the inequality
with the obvious meaning of . In the following, we treat the terms of the right side in order. We start with the term I. Using the fact that , we can first replace the slice-wise means by with the help of Lemma 2.5, and afterwards apply Lemma 2.6, to obtain
where . Taking squares on both sides, integrating with respect to t and τ over and applying Hölder’s inequality and the sub-intrinsic coupling (4.1), we infer
for a constant c depending only on , and L. At this point, we use the estimates for I – III in (4) and obtain the claimed inequality. ∎
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the growth and ellipticity assumptions (1.2). Then on any cylinder satisfying the sub-intrinsic coupling (4.1) for some and some , the Poincaré-type inequality
holds true with a universal constant .
In the following we shall again omit for simplification the reference point in our notation. We will take estimate (4.2) from Lemma 4.1 as starting point for our considerations. To the first integral on the right-hand side, we apply Poincaré’s inequality slice wise for a.e. . In this way, we obtain
where . Applying Hölder’s inequality to the second integral on the right-hand side of (4.2) yields the claimed Poincaré-type inequality on sub-intrinsic cylinders. ∎
The next statement can be interpreted as some sort of Sobolev–Poincaré inequality for the -deviation of from its mean value on the sub-intrinsic cylinder . Later on, we shall use this inequality to estimate the right-hand side in the energy inequality from Lemma 3.1. As usual, this leads to a reduction in the integration exponent of the energy term of the right-hand side, i.e. the integral containing . Similar to Lemma 4.2, we take Lemma 4.1 as starting point in the proof. Then the idea is to extract a part of the integration exponent from the -oscillation integral by the sup-term (occurring in the left-hand side of the energy estimate) and then to apply Sobolev’s inequality to the remainder.
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the growth and ellipticity assumptions (1.2). Then on any sub-cylinder as in (4.1) for some and some , and for any given the following Sobolev-type inequality holds:
for a universal constant and .
In the following, we shall again omit the reference point in our notation. As in the proof of Lemma 4.2 we take inequality (4.2) from Lemma 4.1 as starting point. Moreover, we abbreviate by . From the context, it is clear that is to be interpreted as a function of t. To the first integral on the right-hand side, we apply the lower bound for the boundary term from Lemma 2.3 (ii) and Hölder’s inequality with exponents , . In this way, we obtain
Now, we use the sub-intrinsic coupling (4.1), Hölder’s inequality with exponents , and for a.e. Sobolev’s inequality slicewise (note that , since ). This yields
with a universal constant . In the last line we have used Lemma 2.7 in order to replace in the boundary term the slice wise mean by the mean . Inserting this inequality into (4.2) and applying Young’s and Hölder’s inequality, this results for any in
This completes the proof of the Sobolev–Poincaré-type inequality. ∎
5 Reverse Hölder inequality
As it is well known, the core of each higher-integrability result is a so-called reverse Hölder inequality for the quantity in question, which in our case is the gradient . These reverse Hölder inequalities result in a certain way from the previously established Caccioppoli-type estimate and Sobolev–Poincaré-type inequalities. In principle, the right-hand side integrals of the Caccioppoli inequality are estimated by applying the Sobolev–Poincaré inequalities. However, the proof turns out to be more subtle than originally expected. The assumption that a sub-intrinsic coupling assumption must be imposed for the cylinder is obvious, since this was presupposed in Lemma 4.3. However, this is not sufficient because the factor in the energy estimate has to be converted into an -oscillation integral of u. This is done by a super-intrinsic coupling on the cylinder ; see the assumption (5.1). Both assumptions together, i.e. (5.1) and (5.1), mean that the cylinder is intrinsic in some sense. On such an intrinsic cylinder the oscillations of u are small compared to the mean value of u. This case could be called the non-degenerate case.
Let and let u be a weak solution to (1.1) in in the sense of Definition 1.1, where the vector-field fulfills the structural assumptions (1.2). Then on any cylinder with an intrinsic coupling of the form
for some and , the following reverse Hölder-type inequality holds true:
for some universal constant and where .
Once again, we omit the reference to the center in the notation. We consider radii with . From the energy estimate in Lemma 3.1, we obtain
with the obvious meaning of I, II, III. We abbreviate
and observe that
This together with Lemma 2.5 yields for the first term
Inserting the estimates for I and II above and applying Lemma 4.3, we find for any that
With the choice , this yields
for a constant . To re-absorb the term from the right-hand side into the left-hand side, we apply the Iteration Lemma 2.1. This leads to the claimed reverse Hölder-type inequality, i.e. to
This finishes the proof of Proposition 5.1. ∎
The next lemma deals with the degenerate case which is characterized by the fact that u is small compared to the oscillations of u. In terms of integral quantities this means that on the one hand is sub-intrinsic, and on the other hand the scaling parameter is smaller than the mean of on . As in the non-degenerate case, we need the assumption (5.5), i.e. that is sub-intrinsic, as a prerequisite for the application of Lemma 4.3, which serves to deal with some of the right-hand side integrals of the Caccioppoli-type estimate. However, during this procedure, a term of the order of magnitude appears, and it is precisely there where we need assumption (5.5), which converts this term into the oscillation term that can be re-absorbed into the left-hand side of Caccioppoli’s inequality.
for some scaling parameter and some constant , the following reverse Hölder-type inequality holds true:
with a constant and .
We omit in our notation the reference to the center . Furthermore, we consider radii with . As in the proof of Proposition 5.1 we start from inequality (5.2) which follows from the energy estimate in Lemma 3.1 and we recall the abbreviation (5.3). Estimate (5.4) for I is the same as in the proof of Proposition 5.1. This is clear, since we did not use hypothesis (5.1) for their proof. Therefore, it remains to consider the term II. Applying Young’s inequality, Lemma 2.3 (iii), and Lemma 2.5, we infer for any that
Moreover, from the coupling (5.5) we infer that
We insert the estimates for I and II into (5.2) and choose . This allows us to re-absorb the integral of into the left-hand side. Proceeding in this way, we obtain
At this stage the choice
Now, we apply the Iteration Lemma 2.1 to re-absorb the sup-term from the right-hand side into the left. This leads us to
where the constant c is of the form . This finishes the proof of the proposition. ∎
6 Proof of the higher integrability
As we have seen in the last section, one can establish reverse Hölder inequalities in both the degenerate and the non-degenerate regime. It should be recalled, however, that the cylinders on which these reverse Hölder inequalities are valid, are essentially scaled by the solution u. More precisely, the relationship between , the scaling parameter θ and plays the decisive role. Therefore, the main objective in the proof of the higher integrability theorem is to find parabolic cylinders covering the super-level set of the spatial gradient of in the sense of a Vitali-type covering, such that on each cylinder either a coupling in the form of (5.1) or in the form of (5.5) holds true. These cylinders will be constructed by some sort of stopping time argument, combined with a rising sun-type construction. This very nice idea, which has already been explained in the introduction, goes back to . Once the covering has been constructed by means of such cylinders, the application of the reverse Hölder inequalities leads to a quantitative estimate of on the super-level sets in terms of for . The decay in terms of the super-level sets can then be converted into the higher integrability of .
Before we start the construction of the system of non-uniform cylinders reflecting the character of the porous medium system as explained above, we fix the setup. We consider a fixed cylinder
with . In the following, we abbreviate for and define
At this point, we recall the notation for space-time cylinders from (2.1), which will be used in the following construction. Moreover, we observe that
whenever , and .
6.1 Construction of a non-uniform system of cylinders
Note that is well defined, since the set of those for which the integral condition is satisfied, is non-empty. In fact, in the limit the integral on the left-hand side converges to zero, while the right-hand side blows up with speed . Note also that the condition in the infimum above can be rewritten as
Therefore, we either have that
holds true. In any case we have . On the other hand, if , then (again by definition and the fact that ), we have
Therefore, we end up with the bound
Next, we establish that the mapping is continuous. To this end, consider and , and define . Then there exists such that
for all radii with . Indeed, the preceding strict inequality holds by the very definition of with , since the integral on the left-hand side decreases with the replacement of by (note that the domain of integration shrinks), while the right-hand side strictly increases. The claim now follows, since both, i.e. the integral on the right- and the left-hand side, are continuous with respect to the radius. With other words, we have shown that for r sufficiently close to ϱ. Therefore, it remains to prove for r close to ϱ. This is clear from the construction if , since for any r. In the other case, after diminishing if necessary, we get
for all with . For , this is a direct consequence of the definition of , since otherwise, we would have , which is a contradiction. For r with the claim follows from the continuity of both sides as a function of r. By definition of , the preceding inequality implies , as claimed. This completes the proof of the continuity of .
Unfortunately, the mapping might not be monotone. For this reason we modify in a way, such that the modification – denoted by – becomes monotone. The precise construction is as follows: We define
This construction can be viewed as a rising sun construction, because on those intervals on which , for , one replaces by . Then by construction the mapping is continuous and monotonically decreasing; see Figure 1 for an illustration of the construction.
Moreover, the cylinders are sub-intrinsic whenever . More specifically, we have
In fact, the definition of and its monotonicity imply , so that . Therefore, we have
We now define
In particular, we have for any ; see again Figure 1. Next, we claim that
In the case that we know that also , so that (6.5) trivially holds. Therefore, it remains to consider the case . If , then , and the claim (6.5) follows again. Finally, if , then the monotonicity of , (6.1) and (6.3) imply
In the following, we consider the system of concentric cylinders with radii and . Note that the cylinders are nested in the sense that
The inclusion holds true due to the monotonicity of the mapping . The disadvantage of this system of nested cylinders is, that in general the cylinders only fulfill a sub-intrinsic coupling condition.
6.2 Covering property
Here, we will prove a Vitali-type covering property for the cylinders constructed in the last subsection. The precise result is the following:
There exists a constant such that the following holds true: Let be any collection of cylinders , where is a cylinder of the form constructed in Section 6.1 with radius . Then there exists a countable subfamily of disjoint cylinders in such that
where denotes the -times enlarged cylinder Q, i.e. if , then .
For we consider the sub-collection
and choose as follows: We let be any maximal disjoint collection of cylinders in . Note that is finite, since by (6.6) and the definition of the -measure of each cylinder is bounded from below. Now, assume that have already been selected for some integer . Then we choose to be any maximal disjoint subcollection of
Note again that also is finite. Finally, we define
Then is a countable collection of disjoint cylinders and . At this point it remains to prove that for each there exists a cylinder with , and that this implies .
To this end, fix . Then there exists such that . By the maximality of , there exists a cylinder
with . We know that and , so that . This ensures that . In the following, we shall prove
By we denote the radius from (6.4) associated to the cylinder . Recall that either is intrinsic or and . In the latter case we have due to the definition of that
Therefore, we may assume that is intrinsic, which means
In the following, we distinguish between the cases and , where . In the latter case we exploit (6.9) and the definition of and to obtain
This shows that
Therefore, it suffices to consider the case . Since and , we know that . In addition, we have
Without restriction one can now assume , because otherwise (6.8) trivially holds. Now, the monotonicity of and yield
But this means
This implies that
for a constant . This yields the inclusion . After possibly enlarging , so that , this implies . This establishes (6.7) and completes the proof of the Vitali covering-type lemma. ∎
6.3 Stopping time argument
For and , we define the super-level set of the function by
The Lebesgue points are to be understood with regard to the cylinders constructed in Section 6.1. Note that a.e. point is a Lebesgue point with respect to these cylinders; cf. [14, Section 2.9.1] and the Vitali-type covering Lemma 6.1. For fixed radii , we consider the concentric parabolic cylinders
Note that the inclusion
holds true whenever , and . We fix and abbreviate for throughout this section. By Lebesgue’s differentiation theorem, cf. [14, Section 2.9.1] we have that
In the following, we consider values of λ satisfying
where denotes the constant from the Vitali-type covering Lemma 6.1. For radii s with
we have, by the definition of , for any s as in (6.13) that
In the last chain of inequalities we used (6.6), (6.13) and . On the other hand, on behalf of (6.11) we find a sufficiently small radius such that the above integral with as domain of integration, possesses a value larger than . Consequently, by the absolute continuity of the integral there exists a maximal radius such that
The maximality of the radius implies in particular that
Finally, we know from the construction that is contained in , which in turn is contained in .
6.4 A reverse Hölder inequality
As before, we consider with λ as in (6.12) and abbreviate . As in (6.4) we construct the radius . Exactly at this point, we pass from the possibly sub-intrinsic cylinder to the intrinsic cylinder . Observe that for any , and, in particular, . Our aim now is to prove the following reverse Hölder inequality:
where . In the other case , we want to apply Proposition 5.2 on the cylinder . However, this is only permitted if the hypothesis (5.5) is satisfied. First, we notice that (5.5) is an immediate consequence of (6.3), and therefore we only need to verify (5.5). To this end, we consider two cases. If , we obtain (5.5) by the following computation:
Here we used (6.14) for the last identity. If , then by construction is intrinsic. Moreover, since , we can apply (6.3) with replaced by . This together with Lemma 4.2 and (6.15) (applied with ) ensures that
for a constant . Re-absorbing into the left-hand side and using (6.14), we find that
In conclusion, we have shown that in any case the claimed reverse Hölder inequality (6.16) holds true.
6.5 Estimate on super-level sets
So far we have shown that if λ satisfies (6.12), then for every there exists a cylinder with such that (6.14), (6.15) and (6.16) hold true on this specific cylinder. As before, we abbreviate . We define the super-level set of the inhomogeneity F by
As for the super-level set the Lebesgue points have to be understood with regard to the cylinders constructed in Section 6.1. Using (6.14) and (6.16), we obtain for (to be specified later in a universal way) that
for a constant . In the preceding inequality we choose the η in the form . This choice allows the re-absorption of into the left-hand side. Furthermore, we use Hölder’s inequality and (6.15) to estimate
We insert this above, and multiply the result, i.e. the inequality where we already fixed η and re-absorbed , by . This leads to the inequality
Inserting this above and keeping in mind that depends only on n and m, we deduce
So far, we showed that for any value the super-level set can be covered by a family of parabolic cylinders with center , which are contained in , and such that on each cylinder estimate (6.5) holds true. At this point, we use the Vitali-type Covering Lemma 6.1 and gain a countable subfamily
consisting of pairwise disjoint cylinders, such that the -times enlarged cylinders are contained in and cover the super-level set , i.e.
Since the cylinders are pairwise disjoint, we obtain from (6.5) that
where the constant c depends only on n, m, ν, and L. On we have the pointwise bound and therefore
We combine the last two inequalities and get the following reverse Hölder inequality on super-level sets:
Here, we replace by λ and recall that depends only on , and L. With this replacement we obtain for any that
holds true with a constant . This is the desired estimate on super-level sets.
6.6 Proof of the gradient estimate
For we define the truncation of by
and for the corresponding super-level set
Note that a.e., as well as for and for . Therefore, it follows from (6.18) that
whenever . Since for , the last inequality also holds in this case. Now, we multiply the preceding inequality by , where will be chosen later in a universal way, and integrate the result with respect to λ over the interval . This gives
Here we exchange the order of integration with the help of Fubini’s theorem. For the integral on the left-hand side Fubini’s theorem implies
while for the first integral on the right-hand side we find that
Finally, for the last integral in (6.19) we obtain
We insert these estimates into (6.19) and multiply by . This leads to
The last inequality is now combined with the corresponding inequality on the complement , i.e. with the inequality
We also take into account that . All together this gives the inequality
where . Now, we choose
Note that depends only on , and L. Moreover, observe that , since , and . Therefore, from the previous inequality we conclude that for any pair of radii , with there holds
We can now apply the Iteration Lemma 2.1 to the last inequality, which yields
On the left side we apply Fatou’s lemma and pass to the limit . In the result, we go over to means on both sides. This gives
At this point, we estimate with the help of the energy estimate from Lemma 3.1 applied with and and Hölder’s inequality. This leads to the bound
where . Inserting this above, we deduce
6.7 Proof of Corollary 1.3
It remains to deduce a corresponding estimate on a standard parabolic cylinder
To this end, we rescale the solution u, the vector-field , and the right-hand side F via
whenever and . Then v is a weak solution of the differential equation
for every , with a constant . Scaling back and recalling that , we arrive at the estimate
Dividing both sides by yields the assertion of Corollary 1.3. ∎
We would like to thank Juha Kinnunen for many constructive discussions on the subject and his persuasive work to pursue this topic further. Without his motivational efforts this work would probably not have come about.
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About the article
Published Online: 2018-06-06
Funding Source: Suomen Akatemia
Award identifier / Grant number: Project 308063
The third author was supported by the Academy of Finland, project 308063.
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1004–1034, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0270.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0