Higher regularity in congested traffic dynamics
In this paper, we consider minimizers of integral functionals of the type F (u) : = ∫ Ω [ 1 p (| D u | - 1) + p + f · u ] d x for p > 1 in the vectorial case of mappings u : R n ⊃ Ω → R N with N ≥ 1 . Assuming that f belongs to L n + σ for some σ > 0 , we prove that H (D u) is continuous in Ω for any continuous function H : R Nn → R Nn vanishing on { ξ ∈ R Nn : | ξ | ≤ 1 } . This extends previous results of Santambrogio and Vespri (Nonlinear Anal 73:3832–3841, 2010) when n = 2 , and Colombo and Figalli (J Math Pures Appl (9) 101(1):94–117, 2014) for n ≥ 2 , to the vectorial case N ≥ 1 .
Keywords: 49N60; 35D10; 35J70
Introduction and main result
In this paper, we study -regularity of minimizers of integral functionals of the Calculus of Variations with widely degenerate convex integrands of the form
1.1
Graph
where , , is a bounded domain and , , a possibly vector valued function. We concentrate ourself on the study of the prototype integrand
1.2
Graph
for some . The datum f is required to belong to for some . The functional with the specific integrand F from (1.2) is the prototype for a class of more general functionals where F is a convex function vanishing inside some convex set, and satisfying specific growth and ellipticity assumptions. For sake of clarity, the results in this paper are stated and proved for the functionals as in (1.1)–(1.2). However, we expect our techniques to apply to a general class of integrands with a widely degenerate structure as well. The functional and its associated Euler–Lagrange system
1.3
Graph
naturally arise in problems of optimal transport with congestion effects. In fact, minimizing (1.1) with and the integrand from (1.2) is equivalent to the dual minimization problem
1.4
Graph
where the integrand
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is the convex conjugate of F, or equivalently , and represents the traffic flow. The function models the congestion effect. Note that is increasing and , so that moving in an empty street has nonzero cost. As shown in [[4]] the unique minimizer of (1.4) is given by . We refer to [[2]–[4], [6], [36]] and the references therein for detailed motivations and for the physical meaning of the regularity of minimizers. It would be interesting to investigate if there are applications of the vectorial problem. However, our main motivation to consider the very degenerate system (1.3) was from a mathematical point of view.
In connection with congested traffic dynamic problems the regularity of minimizers, as well as the regularity of weak solutions of the associated autonomous Euler–Lagrange system has been an active field of research in recent years. For instance, in [[2], [4], [9], [24]] Lipschitz regularity of minimizers has been established under suitable assumptions on the datum f.
At this point it is worthwhile to observe that, in general, no more than Lipschitz regularity can be expected for solutions of equations or systems as in (1.3). Indeed when , every 1-Lipschitz continuous function solves (1.3). On the other hand, in the scalar case assuming for some , it was shown by Santambrogio and Vespri [[36]] for and Figalli and Colombo [[10]] for , that the composition of an arbitrary continuous function vanishing on the set with is continuous.
Our aim in this paper is to investigate -regularity of minimizers in the vectorial case . In general, regularity in the vectorial case is much more delicate and minimizers may be irregular although the integrand is smooth, cf. [[15], [38]]. In this respect, regularity can be expected only for integrands with special structure. The first result in this direction has been obtained by Uhlenbeck [[40]] for the p-Laplace system when . She proved that weak solutions are of class . The scalar case had previously been established by Ural'ceva [[41]], while the case was obtained by Tolksdorf [[39]]. As already mentioned, we cannot hope for a -regularity result for the elliptic system (1.3), since Lipschitz continuity is optimal. However, we are able to establish in the vectorial setting that the composition is continuous for any continuous function vanishing on . This phenomenon is somewhat reminiscent of comparable results for the Stefan problem, in which the continuity of the energy cannot been shown, but the temperature shows a logarithmic type continuity, cf. [[16], [31]].
Statement of the main result
Before formulating the main regularity result, we need to introduce a few notations. The natural energy space to deal with (local) minimizers of the integral functional is the Sobolev space . Then, (local) minimizers in of the functional are weak solutions of the Euler–Lagrange system
1.5
Graph
and vice versa, where
Graph
for some . A function is a weak solution of the Euler–Lagrange system (1.5) if and only if
Graph
holds true for any testing function . Our main result proves the continuity of the composition explained above.
Theorem 1.1
Let , for some and be a weak solution of (1.5) in . Then,
Graph
for any continuous function vanishing on .
By carefully tracing the dependence of constants on the parameter in the proof of Theorem 3.6, one could determine an explicit modulus of continuity of , where is defined in (2.2). However, it is not clear if is Hölder continuous in general. For a different very degenerate elliptic equation a counterexample to Hölder continuity is provided in [[11]].
Theorem 1.1 can be regarded as the vectorial analog of the regularity results of Santambrogio and Vespri [[36], Theorem 11] and of Figalli and Colombo [[11], Theorem 1.1] as far as the model type integral functional is considered. The vectorial case cannot be treated with the methods from [[11], [36]], since these are tailored to the scalar case. Nevertheless, some steps in our proof are similar, for example, the approximation procedure by some sequence of uniformly elliptic problems. The main difference in our proof is that we are establishing a variant of DiBenedetto's and Manfredi's proofs of -regularity of minimizers to p-energy type functionals [[17], [32]]. It is also inspired by the arguments from DiBenedetto and Friedman's pioneering results on -regularity for parabolic p-Laplacian systems [[18], [20]]. Roughly speaking, our strategy is the adaptation of De Giorgi's approach to the level of gradients in combination with Campanato type comparison arguments. The past has shown that De Giorgi's approach is extremely flexible. Therefore, we expect that our approach can be transferred to larger classes of widely degenerate functionals in the vectorial case. However, and in order to keep the individual steps as simple as possible, we limit ourselves to treating the model case.
Strategy of the proof
Concerning the overall strategy of proof a few words are in order. First, we observe that weak solutions of (1.5) are Lipschitz continuous. This has been proved in [[2], [4], [9]]. Moreover, functionals as in (1.1) fit into the broader context of asymptotically convex functionals, i.e. functionals having a p-Laplacian type structure only at infinity. This class of functionals has been widely studied, since the local Lipschitz regularity result by Chipot and Evans [[8]]. In particular we mention generalizations allowing super- and sub-quadratic growth [[28], [30], [35]], lower order terms [[34]]. Extensions to various other settings can be found in the non-complete list [[12]–[14], [23]–[27], [37]].
The proof of Theorem 1.1 is divided into several steps and starts by an approximation procedure. Indeed, by replacing h(t) by for and considering instead of (1.5) the Dirichlet-problem on a ball compactly contained in associated to the regularized coefficients and with Dirchlet boundary datum u we obtain a sequence of approximating more regular mappings . In particular, has second weak derivatives in . In Sect. 3.1 we summarize the most important properties, i.e. uniform energy bounds, uniform quantitative interior -gradient bounds, uniform quantitative higher differentiability -estimates, and finally strong -convergence of in the limit . The nonlinear mapping with is defined by
Graph
Observe that vanishes on the larger set . The reason for considering is that on the complement of the system (1.5) behaves non-degenerate in the sense that the vector field admits a uniform ellipticity bound from below, of course, with constants depending on . This point of view has already been exploited in [[11], [36]]. As a first main result we prove that is Hölder continuous uniformly with respect to . However, the constants in the quantitative estimate, i.e. the Hölder exponent and the Hölder norm, may blow up when . We distinguish between two different regimes: the degenerate and non-degenerate regime. The degenerate regime is characterized by the fact that the measure of those points in a ball in which is far from its supremum is large, while the non-degenerate regime is characterized by the opposite. In the non-degenerate regime we compare with a solution of a linearized system. This allows us to derive a quantitative -excess-improvement for on some smaller ball (see Proposition 3.4). This step utilizes a suitable comparison estimate and the higher integrability of . On the smaller ball we are again in the non-degenerate regime, so that the argument can be iterated yielding a Campanato-type estimate for the -excess of . In the degenerate regime we establish that is a subsolution to a linear uniformly elliptic equation with measurable coefficients; of course the ellipticity constants depend on and blow up as . At this stage a De Giorgi type argument allows a reduction of the modulus of on some smaller ball (see Proposition 3.5). However, on this smaller scale it is not clear whether or not we are in the degenerate or non-degenerate regime. Therefore one needs to distinguish between these two regimes again. In the non-degenerate regime we can conclude as above, while in the degenerate regime the reduction of the modulus of applies again. This argument can be iterated as long as we stay in the degenerate regime. However, if at a certain scale the switching from degenerate to non-degenerate occurs, the above Campanato type decay applies. If no switching occurs, we have at any scale of the iteration process a reduction of the modulus of . This, however, shows that the supremum of —and hence also the one of —on shrinking concentric balls converges to 0. Altogether this leads to a quantitative Hölder estimate for which remains stable under the already established convergence as . The final step consits in passing to the limit and conclude that is continuous. This can be achieved by an application of Ascoli–Arzela's theorem. It is here where we loose control on the quantitative Hölder exponent. At this point the continuity of for any continuous function vanishing on is an immediate consequence.
Notation and preliminary results
Notation
For the open ball of radius and center we write . The mean value of a function is defined by
Graph
If the center is clear from the context we omit the reference to the center and write respectively for short. For the standard scalar product on Euclidean spaces as well as the space of matrices, we use the notation . Finally, we use the notion for the gradient of a scalar function u, while we use Du for a vector field u.
Throughout this paper we abbreviate
2.1
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and
2.2
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Observe that
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Moreover, for we define
2.3
Graph
and note that .
Generic constants are denoted by c. They may vary from line to line. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts.
Algebraic inequalities
In this section, we summarize the relevant algebraic inequalities that will be needed later on. The first lemma follows from an elementary computation.
Lemma 2.1
For , we have
Graph
The next lemma can be deduced as in [[29], Lemma 8.3].
Lemma 2.2
For any , there exists a constant such that, for all , , we have
Graph
Lemma 2.3
Let and , . Then, for as defined in (2.3) we have
Graph
Moreover, if and there holds
Graph
Proof
We distinguish between different cases. If the inequality holds trivially since . If , we apply Lemma 2.1 and obtain
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If and , we have
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The case when and is similar, we just have to interchange the role of and . Joining the three cases gives the first assertion of the Lemma.
Now, we come to the proof of the second assertion. First, we consider the case in which . In this case we have
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Next, we consider the case . Recall that by assumption . We start with the observation that is a one to one mapping whose inverse mapping is given by . We let and and estimate with Lemma 2.1
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In the second to last estimate we used . Using in the previous inequality the definition of and the claim immediately follows.
Lemma 2.4
There exists a constant such that for any and we have
Graph
Proof
We apply Lemma 2.2 with to obtain
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This proves the claim.
Lemma 2.5
For we have
Graph
Moreover, for there holds
Graph
Proof
By direct computation we have for that
2.4
Graph
from which the first claim immediately follows. We now turn our attention to the second claim. We may assume that ; otherwise we interchange the role of a and b. In view of (2.4) we find
Graph
For the first term we have
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For the second term we estimate
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This completes the proof of the lemma.
Lemma 2.6
For any we have
Graph
and
Graph
Proof
The first assertion can be achieved, since for we have
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For the second claim, we first observe that both sides are zero for . Therefore, it remains to consider . Recalling (2.4) we compute
2.5
Graph
Moreover, we have
Graph
so that
Graph
Dividing the previous inequality by we infer that
Graph
holds, proving the second claimed inequality.
Bilinear forms
For we define
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We note that for , while for , we have . For we additionally have . In any case, on the interval [0, 1]. Moreover, we let
2.6
Graph
For with if we define the bilinear forms
Graph
and
2.7
Graph
and
2.8
Graph
Observe that all forms are symmetric in the arguments and . Due to the special structure of and the compositions , and are well definined for as integrable functions. Therefore integral calculations involving these quantities make sense.
The next Lemma provides the relevant ellipticity and boundedness properties of the bilinear forms , and . The following abbreviations
Graph
and
Graph
and for prove to be useful in the formulation of the Lemma.
Lemma 2.7
Let and . The bilinear form defined above satisfies
Graph
The analogous estimates hold for the bilinear form and any , as well as for and any .
Proof
If the inequality holds trivially. Therefore, it remains to consider the case . We first establish the lower bound. If we omit the second term in the definition of and obtain
Graph
while for we use Cauchy-Schwarz inequality and (2.5) to conclude
Graph
Now, we turn our attention to the upper bound. If we have
Graph
while for we obtain
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This proves the claim for the bilinear form . The corresponding estimates for and follow in the same way.
It should also be mentioned that the coercive symmetric bilinear forms fulfill Cauchy-Schwarz inequality. In particular, we have
Graph
In the next Lemma we put together the monotonicity and growth properties of the vector field .
Lemma 2.8
Let and with . Then, we have
Graph
and
Graph
Proof
The first inequality results from the following chain of inequalities
Graph
From the third last line to the second last we used Lemma 2.4. For the proof of the second inequality we abbreviate
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Keeping this in mind we compute
Graph
In turn we used Lemma 2.7 and the elementary inequality . Now, we distinguish whether or not . If , then
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For this implies a bound from below in the form
Graph
So it follows that
Graph
holds true. In the case that , we estimate from below by
Graph
Therefore, for we obtain
Graph
This yields
Graph
Inserting this above, we obtain the second claim of the Lemma.
Lemma 2.9
Let and . Then, we have
Graph
for a constant .
Proof
If , we have . Therefore, the desired inequality follows from the second inequality in Lemma 2.8 after omitting the positive second term in the bracket on the right-hand side.
Therefore, it remains to consider the case where either or . We again distinguish two cases and start with . Note that this implies . If we use Lemma 2.3 to conclude
Graph
while in the case we use Young's inequality and Lemma 2.3 to obtain
Graph
for a constant . Combining both cases, taking into account the elementary inequalities and , and finally applying Lemma 2.8, we obtain
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where . This proves the claimed inequality in this case. In the remaning case, i.e. , we obtain a similar estimate. We only have to replace on the right-hand side by . Then, we use the estimate and absorb by Lemma 2.8 into the second term on the right-hand side. In this way, we obtain
Graph
proving the claim also in this case.
In the following Lemma we quantify the remainder term in the linearization of . In the application, it can be assumed that the linearization only takes place in points with in a quantifiable way. The precise statement is
Lemma 2.10
Let , and with and for some . Then, we have
Graph
Proof
We distinguish two cases. We start with the case . For we write . Note that
2.9
Graph
Similarly to the computations in the proof of Lemma 2.8 we have
Graph
This allows us to re-write
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We decompose and estimate the integrand appearing on the right-hand side and obtain in this way
Graph
where
Graph
In view of Lemma 2.4 we have
Graph
which immediately implies
Graph
For the second term we use Lemma 2.1, Lemma 2.5, the assumptions on and (2.9) to obtain
Graph
Inserting the preceding estimates above, we find that
Graph
At this stage it remains to consider the case . Note that
Graph
Therefore, by Lemma 2.7 and the assumption we obtain
Graph
Joining both cases yields the claim.
Lemma 2.11
For any , any ball and any we have
Graph
where g is defined in (2.1).
Proof
For we compute
Graph
Summing with respect to , we obtain
Graph
In the preceding inequality we replace the term by , which is possible by an application of the first inequality in Lemma 2.6. Moreover, we would also like to replace by . To this aim we have to distinguish two cases. If , we use the second inequality in Lemma 2.6 to replace by . Thereby, we may omit the positive term on the left-hand side. Inserting this above and using also Kato's inequality in the form , we obtain
Graph
Otherwise, if we use Kato's inequality twice and again the second inequality in Lemma 2.6 to obtain
Graph
This proves the asserted inequality also in the second case.
Proof of Theorem 1.1
In this section we will prove Theorem 1.1 under the hypothesis that Propositions 3.4 and 3.5 below are true. The remainder of the paper is then devoted to the proof of those two propositions.
Here and in the following we denote by a weak solution of (1.5). We first observe that ; cf [[2], [4]] for the scalar case and [[9]] for the vectorial case. Therefore, we may always assume that Du is locally bounded in .
Regularization
The first step in the proof consists in the construction of more regular approximating solutions. To this aim we consider a fixed ball . We let and and recall the definition of the regularized coefficients from (2.6). By we denote the unique weak solution of the regularized elliptic system
3.1
Graph
The weak formulation of (3.1) is
3.2
Graph
Note that . This can be retrieved similarly as in [[33]].
Lemma 3.1
For any we have . Moreover, for any ball the (uniform with respect to ) quantitative -gradient bound
Graph
with and the quantitative -estimate
Graph
with hold true.
Our first observation is a uniform energy bound for .
Lemma 3.2
There exists a constant such that for any we have
Graph
Proof
The desired estimate can be deduced with a standard argument by testing the weak formulation (3.2) with the test-function . Indeed, we have
Graph
By Young's inequality we obtain for the first integral on the right-hand side
Graph
The second integral on the right-hand side is estimated with Hölder's and Sobolev's inequality, so that
Graph
where . We insert these inequalities above and reabsorb the terms containing from the right-hand side into the left. In this way, we get
Graph
with a constant . The desired uniform energy bound can easily be deduced from the preceding inequality.
The next lemma ensures strong convergence of the approximating solutions, in the sense that strongly converges to in .
Lemma 3.3
Let and with be the unique solution of the Dirichlet problem (3.1). Then, we have
Graph
Proof
Testing (3.2) and the weak formulation of (1.5) with we have
Graph
Using Lemma 2.9 and Young's inequality we find
Graph
Re-absorbing the first integral from the right into the left-hand side, we obtain
Graph
The integral on the right-hand side is finite, since . Therefore, the preceding inequality implies strong convergence in as .
Hölder-continuity of Gδ(Duε)
We recall the definition of from (2.3). In this subsection we will prove that is locally Hölder continuous in for any . This will be achieved in Theorem 3.6. Thereby, it is essential that all constants are independent of .
The proof of Theorem 3.6 relies on the distinction between two different regimes—the degenerate and non-degenerate regime. In the non-degenerate regime we will prove an excess-decay estimate for (see Proposition 3.4), while in the degenerate regime we establish a reduction of the modulus of (see Proposition 3.5). The precise setup is as follows. We consider a ball and denote by the unique weak solution of the Cauchy–Dirichlet problem (3.1). For we let . Then, by Lemma 3.1 and Lemma 3.2 we have (uniform with respect to ) boundedness of on . More precisely, there exists a constant
3.3
Graph
independent of , such that . We may assume that . Now, we consider a center and a radius such that . On this ball we have
3.4
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for some such that
3.5
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Next, for we define the super-level set of by
Graph
The definition of the super-level set allows us to distinguish between the degenerate regime which is characterized by the measure condition and the non-degenerate regime which is characterized by the reversed inequality. Roughly speaking, in the degenerate regime the set of points with small has large measure, while in the non-degenerate regime the set of points with small is small in measure. We start with the latter one. In the following we abbreviate .
Proposition 3.4
Let and
3.6
Graph
Then, there exist and such that there holds: Whenever is a ball with radius and center , and is the unique weak solution of the Dirichlet problem (3.1) and hypothesis (3.4) and (3.5) and the measure condition
3.7
Graph
are satisfied, then the limit
3.8
Graph
exists, and the excess decay estimate
3.9
Graph
holds true. Moreover, we have
Graph
The statement for the degenerate regime is as follows.
Proposition 3.5
Let , and . Then, there exist constants and such that the there holds: Whenever is a ball with center , and is the unique weak solution of the Dirichlet problem (3.1) and hypothesis (3.4) and (3.5) and the measure condition
3.10
Graph
are satisfied, then, either
Graph
or
Graph
hold true.
We postpone the proofs of Proposition 3.4 and 3.5 to Chapters 4–6 and continue with formulating the main result of this subsection.
Theorem 3.6
Let and be the unique weak solution of the Dirichlet problem (3.1) in . Then, is Hölder continuous in for any with Hölder-exponent and a Hölder constant both depending on and .
Proof
By
Graph
and
Graph
we denote the constants from Proposition 3.4 and by
Graph
we denote the ones from Proposition 3.5. The dependence of on the structural parameters implies that depends on and . Finally, we let and
Graph
We consider a ball with center and as described above. On this ball we have (3.3)–(3.5) satisfied. Our aim now is to prove that is Hölder continuous in with Hölder exponent
Graph
In turn, substituting by , this proves the claim of the proposition. We proceed in two steps.
Step 1: We prove that the limit
Graph
exists and that
3.11
Graph
holds true for a constant . To this aim, we define for radii
Graph
and observe that
3.12
Graph
Now, suppose that assumption (3.10) holds on . Then, Proposition 3.5 yields that either or
Graph
Note that the first alternative cannot happen, since it would imply
Graph
Hence, we conclude that (3.4) holds on with . If the measure condition (3.10) is satisfied with and , then a second application of Proposition 3.5 yields that either or
Graph
As before, the first alternative cannot happen, since it would imply
Graph
Assume now that (3.10) is satisfied for up to some , i.e. that (3.10) holds true on the balls with . Then, we iteratively conclude that
3.13
Graph
Now assume that (3.10) fails to hold for some . If , the hypothesis of Proposition 3.4 are satisfied on and we conclude that the limit
Graph
exists and that
Graph
Moreover, we have
3.14
Graph
Therefore, we obtain from the preceding inequality and (3.12) that
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holds true for any . For a radius there exists such that . Using (3.13), (3.14) and (3.12) we obtain
Graph
Combining the preceding two inequalities, we have shown (3.11) provided .
In the case , we have on . Combining this with (3.13) and keeping in mind that , we obtain
3.15
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In the final case when (3.10) holds for any , then (3.13) is satisfied for any and hence we obtain (3.15) also in this case. (3.15), however, implies
Graph
For we find such that . Then, (3.15) and (3.12) imply
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This establishes (3.11) in the remaining cases.
Step 2: Here, we prove that the Lebesgue representative of is Hölder continuous in . The proof is standard once the excess decay (3.11) is established. For convenience of the reader we give the details. We consider . If we define and obtain from Step 1 that
Graph
This can be re-written in the form
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Otherwise, if , we trivially have
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Together, this establishes that the Lebesgue representative of is Hölder continuous in with Hölder exponent . Note that admits the same dependencies as , i.e. . This finishes the proof of the theorem.
Continuity of G(Du)
In this subsection it is important that all estimates are independent of . More precisely, constants might depend on , but are independent of .
Proof of Theorem 1.1
We let and consider a fixed ball . By we denote the weak solution to (3.1) constructed in Sect. 3.1. Next, we fix and . From Theorem 3.6 we know that is Hölder continuous in with Hölder-exponent and constant , both depending at most on , and . From Lemma 3.3 we know that in as . This implies that there exists a subsequence as such that a.e. in . On the other hand, by Ascoli-Arzelà's Theorem we conclude that converges uniformly on compact subsets of . Therefore the limit is Hölder continuous in with Hölder-exponent and constant . In particular, is continuous in for any . Moreover, we have uniformly in as . Indeed
Graph
in . As the uniform limit of a sequence of continuous functions, itself is continuous on . Observe that is also uniformly continuous on .
Now, let be any continuous function vanishing on . Since , we find such that on . By we denote the modulus of continuity of on , i.e. for any with we have . Next, given we choose such that
3.16
Graph
We now distinguish two cases. First, we assume . If we use on to conclude
Graph
The preceding estimate trivially holds if . Moreover, by (3.16) we have
Graph
This implies . Similarly as above we conclude
Graph
Combining the estimates from above, we end up with
Graph
Now, we consider the case . Here, Lemma 2.3 and (3.16) imply
Graph
proving
Graph
Hence, is continuous on . Since were arbitrary, we have shown that is continuous in . This completes the proof of the theorem.
As mentioned above, we have now finished the proof of the main theorem Theorem 1.1 under the condition that Propositions 3.4 and 3.5 are true. The rest of the paper is now devoted to the proof of those two propositions.
Conclusions from the differentiated system
The main integral inequality for second derivatives
Throughout this subsection we assume as a general requirement that is a weak solution to the regularized system (3.1). Instead of , we write u for the sake of simplicity. In contrast, we will continue to use the subscript in the notation for the coefficients and its associated bilinear forms, such as . We recall that the bilinear forms have been defined in Sect. 2.3.
For some index we differentiate the regularized system (3.1) with respect to and obtain
4.1
Graph
for any . We have
Graph
In (4.1) we choose the testing function , where is non-negative and is non-decreasing. Note that
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The resulting equations are then summed with respect to from 1 to n. This leads to
Graph
Now, we compute the right-hand side.
Graph
with the obvious meaning of . For the first term, we have by Young's inequality
Graph
for any . Similarly, we get
Graph
and
Graph
Inserting this above and re-absorbing the terms containing second derivatives from the right-hand side into the left, we obtain
Graph
for any non-negative function . In the preceding inequality the parameter is at our disposal. We choose . For the first factor, i.e. the term , in the integrand of the first integral on the left-hand side we have
Graph
In fact, if the inequality is obvious. If otherwise , which can only happen if and , the result follows by an application of Kato's inequality and (2.4). Indeed
Graph
For the term in brackets of the second integral on the left-hand side a similar computation applies. The result of the calculation is
Graph
Using the last and second last inequality above, we obtain an inequality which can be interpreted in two ways. On the one hand it can be seen as an energy inequality for the second derivatives of u. On the other hand—by discarding on the left-hand side the two non-negative terms containing second derivatives—the inequality implies that |Du| is a subsolution of an elliptic equation with measurable coefficients.
Lemma 4.1
Let and a weak solution to the regularized system (3.1) on . Then, for any non-decreasing function and any non-negative testing function we have
Graph
where .
Subsolution to an elliptic equation
We start by showing that is a sub-solution of a certain elliptic equation. More precisely
Lemma 4.2
Let and be a weak solution of the regularized system (3.1) satisfying (3.4) and (3.5) on . Then, the function
4.2
Graph
is a sub-solution of a linear elliptic equation on , in the sense that
4.3
Graph
holds true for any non-negative test function and with a universal constant . The coefficients are given by
Graph
where is the bilinear form defined in (2.8).
Proof
We apply Lemma 4.1 with . Due to Lemma 2.7 the first two integrals on the left-hand side are non-negative and therefore can be discarded. In this way, we obtain with that
4.4
Graph
holds true for any non-negative . To proceed further we compute
Graph
and
Graph
The above inequalities together with the general assumptions (3.4), (3.5) allow to estimate
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Moreover, we have . In this way, we obtain for the right-hand side in (4.4) the estimate
Graph
where . Now, we consider the left-hand side in (4.4). Observe that , so that by the linearity of with respect to the first variable we have
Graph
For the left-hand side this has the consequence that
Graph
holds true. Here, we have taken into account the definition of the coefficients . Altogether we have shown the claim (4.3).
The coefficients in (4.3) can be explicitly written as
Graph
They are only degenerate elliptic due to the factor which, for , vanishes on the set . On the other hand has its support in the set . This allows us to modify the coefficients on . This idea will lead us to an energy estimate for in the next lemma.
Lemma 4.3
Let and be a weak solution of the regularized system (3.1) satisfying (3.4) and (3.5) on and denote by the function defined in (4.2). Then, for any and any we have
Graph
where and .
Proof
We let be the coefficients defined in Lemma 4.2 and extend them from to the complement by letting . The new coefficients are thus defined by
Graph
From this definition and Lemma 4.2 we observe that is a weak sub-solution also with the modified coefficients. More precisely, we have that
4.5
Graph
for any non-negative .
We now investigate the upper bound and ellipticity of the coefficients . We will show that there exist both depending at most on p, M and such that
Graph
for any and . We start with the former one. On the set where the upper bound holds with , while on the set where we have from Lemma 2.7 that
Graph
which proves the claim with . Similarly, on the set where the ellipticity holds with , while on the set where we have from Lemma 2.7 that
Graph
which proves the claim with .
Now, the claimed energy estimate follows in a standard way by choosing in (4.5) a test-function of the form with a cut-off function with on and , cf. [[19], Chapter 10.1].
Energy estimates
Here, we assume that the hypothesis of Proposition 3.4 are in force. Our starting point is again Lemma 4.1. This time we keep the two non-negative terms containing the quadratic forms and on the left-hand side. For any non-decreasing function and any non-negative function we have
Graph
with the obvious meaning of – . For we have
Graph
Moreover, we estimate the integral by Cauchy-Schwarz inequality and obtain
Graph
From the second to last inequality we used the upper bound from Lemma 2.7 to estimate the second integral. Inserting the results above and re-absorbing the first term from the right into the left, we find that
4.6
Graph
holds true with a constant . We now choose , where is non-decreasing. Note that
Graph
For we compute
Graph
In turn we used from (3.5) and (3.6), which implies on the one hand , and on the other hand . Next, we compute
Graph
Due to assumptions (3.4) and (3.6) we know that on . This allows us to use the preceding estimates in (4.6) to bound the right-hand side from above. Moreover, by Lemma 2.11 the left-hand side in (4.6) can be estimated from below. Proceeding in this way we obain
4.7
Graph
for any . The constant c depends only on p, M, and .
Different concrete choices of in (4.7) result in two important energy inequalities. The first one is
Lemma 4.4
Let and be a weak solution of the regularized system (3.1) such that hypotheses (3.4), (3.5) and (3.6) are in force on . Then, for any there holds
Graph
for some universal constant .
Proof
We apply inequality (4.7) with the choice . The cut-off function is chosen such that in , , and . This leads us to
Graph
with a constant , which is the claimed energy estimate.
The second energy estimate is
Lemma 4.5
Let , and be a weak solution of the regularized system (3.1) such that hypotheses (3.4), (3.5), (3.6) and (3.7) are in force on . Then, for any we have
Graph
for a constant .
Proof
This time we choose
Graph
in inequality (4.7), and obtain
Graph
On we have
Graph
and
Graph
since . Again, we choose to be a non-negative cut-off function with in , , and . This, together with the fact that on , allows us to estimate the right-hand side in the above inequality. Indeed, we have
Graph
Therefore it remains to estimate the left-hand side from below. The integral has to be taken only on the set of points with . We shrink this set to those points satisfying the stronger condition , i.e. to . On this set we have
Graph
Inserting this above we conclude that
Graph
holds true. This proves the claim.
The non-degenerate regime
The aim of this section is to prove Proposition 3.4. Throughout this section we presume the following general assumptions. For given we denote by the unique weak solution of the regularized system (3.1). Moreover, we assume that for some and and a ball with assumptions (3.4)–(3.6) are in force. We denote by
5.1
Graph
the -excess of on , i.e. the -mean square deviation of from its mean value .
Higher integrability
An ingredient in the proof of Proposition 3.4 is the following higher integrability result.
Lemma 5.1
Under the general assumptions of Sect. 5 there exist and such for any satisfying
Graph
we have
Graph
Proof
We consider a ball . We test the weak form (3.2) of the elliptic system by the testing function
Graph
and is a standard cut-off function with in , , and . We obtain
Graph
We use the monotonicity of from Lemma 2.8 in order to estimate the first term from below. Due to our assumption on and (3.4) we have and therefore obtain
Graph
where . To bound the second integral from above we use the structural upper bound from Lemma 2.8. Moreover, we observe that due to our assumption on . This allows us to estimate
Graph
for a constant . Re-absorbing terms on the left-hand side, we find that
Graph
where . Due to the assumption on , the assumption (3.5), and the particular choice of , we conclude with an application of Hölder's and Sobolev–Poincaré's inequality a reverse Hölder inequality of the form
Graph
with a constant . The dependence of c upon M only occurs in the sub-quadratic case of . The claim, i.e. the higher integrability, now follows with Gehring's lemma, since , cf. [[1], Theorem 3.22], [[22], Theorem 2.4] and [[42], Theorem 3.3]. Note that can always be diminished if necessary.
Comparison with a linear system
In this section we will consider the weak solution of the linear elliptic system
5.2
Graph
for any as comparison function to our solution of the regularized elliptic system (3.1). Recall that has been defined in (2.7).
Lemma 5.2
Let the general assumptions of Sect. 5 be in force and assume that
5.3
Graph
Then, there exists and , such that
Graph
Here, v is the unique weak solution of the Dirichlet problem (5.2) and .
Proof
Throughout the proof we omit the reference to the center and write instead of . Moreover, we abbreviate . Using the weak form (3.2) of the elliptic system we obtain
Graph
for any . Using also the fact that v is a weak solution of the linear elliptic system (5.2), we find that
Graph
Here we used from the second to last line Lemma 2.10. This is possible since (3.4) and (5.3) are in force. Since , we may choose the testing function . Together with the bound from below from Lemma 2.7 and Hölder's and Poincaré's inequality this leads us to
Graph
for a constant . We divide both sides by
Graph
square the result and finally take means. This implies
Graph
with a constant . Here we have also used . At this stage arrived, we want to reduce the integrability exponent on the right-hand side from 4 to , where is the integrability exponent from the higher integrability Lemma 5.1. This is possible since and are bounded by on account of (3.4) and (3.6). Then, the application of the higher integrability lemma yields
Graph
where . Inserting this inequality above and noting that finishes the proof of the lemma.
The following a priori estimate for solutions to linear elliptic systems can be inferred from [[5]] once the ellipticity conditions for the quadratic form are established; see also [[21], Theorem 2.3].
Lemma 5.3
Let the general assumptions of Sect. 5 be in force and assume that (5.3) holds true. Then, the weak solution of the linear elliptic system (5.2) satisfies and there exists a constant such that for any we have
Graph
Proof
As mentioned before, the a priori estimate is standard. The constant depends on the dimensions n, N and the ellipticity constant and the upper bound of the quadratic form . Due to assumption (5.3) and Lemma 2.7 these quantities only depend on p and .
Exploiting the measure theoretic information
The aim of this subsection is to convert the measure theoretic information (3.7) into a lower bound for the mean value of and smallness of the excess.
Lemma 5.4
Let the general assumptions of Sect. 5 be in force. Furthermore, assume that (3.7) holds for some . Then there exists a constant such that for any there holds
Graph
Proof
Throughout the proof we omit the reference to the center and write instead of . We define by
Graph
and let
Graph
Note that by (3.4) and . Due to the minimality of the integral average with respect to the mapping , we have
Graph
We recall that and hence by (3.6) we have . Due to assumption (3.7) we therefore obtain for the second integral
Graph
For the estimate of we first note that on since and . Therefore, the application of Lemma 2.3 yields
Graph
Next, we note that
Graph
Using this information, Lemma 2.2, the choice of , and Poincaré's inequality we obtain
Graph
We once again decompose the domain of integration into and . Subsequently applying Hölder's inequality and taking into account assumption (3.7) leads us to
Graph
We note that and hence . For the first integral we use Lemma 4.5, while for the second one we use Lemma 4.4 and the assumption . In this way we obtain
Graph
for a constant . Inserting this above yields the desired estimate.
Lemma 5.5
Let the general assumptions of Sect. 5 be in force. Then, for any there exist and such that the smallness assumption and the measure theoretic hypothesis (3.7) imply
5.4
Graph
Proof
We let , and . Consider with . For convenience in notation we omit the reference to the center . Using the minimality of with respect to the mapping and decomposing the domain of integration into and , we obtain
Graph
For the first integral we use Lemma 5.4 and obtain
Graph
where . For the second integral we use and get
Graph
so that
Graph
for a constant . Now, we first choose in dependence on and in such a way that . Subsequently, we choose in dependence on and such that
Graph
Finally, we choose such that
Graph
In this way we obtain (5.4) .
To prove (5.4) , we first observe that the measure theoretic assumption (3.7) implies
Graph
Hence, due to the definition of the set , we obtain
Graph
On the other hand, due to (5.4) , we have
Graph
so that
Graph
Due to the choice of and the fact that we have and . Together with the assumptions and we obtain
Graph
Inserting this above yields the claim (5.4) and finishes the proof of the lemma.
Proof of Proposition 3.4
Our aim in this subsection is to prove Proposition 3.4. We start with an excess-decay estimate for the excess of .
Lemma 5.6
Assume that the general hypotheses of Sect. 5 are in force. Let and be the exponent from Lemma 5.2. If
5.5
Graph
hold true, then we have the quantitative excess decay estimate
Graph
with a constant .
Proof
Throughout the proof we omit the reference to the center and write instead of . By we denote the unique weak solution of the linear elliptic system (5.2). For , we have
Graph
In view of Lemma 5.3 we deduce
Graph
where . Inserting this above and applying Lemma 5.2 and assumption (5.5) , we end up with
Graph
Note that the constant depends on and .
Proof of Proposition 3.4
By we denote the constant from Lemma 5.2 and by the one from Lemma 5.6. For we define by
Graph
For the particular choice we let be the radius from Lemma 5.5. Finally, we define
Graph
so that depends on . In the following we consider a ball with . As before, we omit the reference to the center and write instead of . By we denote the constant from Lemma 5.5 and assume that (3.7) is satisfied for this particular choice of . Note that by our choice of the parameter depends on and . From Lemma 5.5 applied with we infer that
5.6
Graph
By induction we shall prove that for any we have
Graph
and
Graph
For we can apply Lemma 5.6, since (5.6) ensures that the assumptions of the lemma are satisfied. Then, (I) follows from Lemma 5.6, (5.6) and our choices of and , since
Graph
For the proof of (II) we use (5.6) and to obtain
Graph
so that
Graph
Together with (5.6) this implies (II) .
Now, we consider and prove (I) and (II) assuming that (I) and (II) hold. From (I) and (II) we observe that the assumptions of Lemma 5.6 as formulated in (5.5) are satisfied on . Therefore, applying the lemma with instead of , recalling the choices of and and joining the result with (I) yields
Graph
This proves (I) . Moreover, from (I) and we obtain
Graph
so that
Graph
by our choice of . Together with (II) , this proves (II) .
We now come to the proof of (3.8) and (3.9). For we obtain from the minimizing property of the mean value, Lemma 2.3, (I) , (5.6) and our choice of that
5.7
Graph
This allows us to compute
Graph
Given , we use the preceding inequality to conclude that
5.8
Graph
This shows that is a Cauchy sequence and therefore the limit
Graph
exists. Passing to the limit in (5.8) yields
Graph
Joining this with (5.7), we find
Graph
For there exists such that . Then, we obtain from the last inequality
Graph
This implies
Graph
so that also
Graph
Finally, due to assumption (3.4) we have for any , which implies . This finishes the proof of Proposition 3.4.
The degenerate regime
Our aim in this section is to prove Proposition 3.5, which treats the degenerate regime. The proof relies on a De Giorgi type reduction argument reducing the supremum of under the measure theoretic assumption (3.10). The starting point is the energy estimate for from Lemma 4.3.
As in Section 5, we first formulate the general assumptions. For we denote by the unique weak solution to the Dirichlet problem (3.1) associated to the regularized system. We assume that (3.4) is in force for some on some ball . Let denote the function defined in (4.2). Note that (3.4) implies
Graph
Moreover, we set .
We start by a De Giorgi type lemma for , which can for instance be deduced as in [[19], Chap. 10, Proposition 4.1] by the use of the energy estimate from Lemma 4.3. For the readers convenience we provide the proof in the appendix Sect. 7.
Lemma 6.1
(Reducing the supremum) Assume that the general assumptions of Sect. 6 are in force and let . Then, there exists such that the measure theoretic assumption
Graph
implies that either
Graph
or
Graph
hold true.
The proof of the next Lemma can be deduced as in [[19], Chap. 10, Proposition 5.1] utilizing the energy estimate from Lemma 4.3; see also Sect. 7.
Lemma 6.2
Assume that the general assumptions of Sect. 6 are in force and assume that (3.10) is satisfied for some . Then, for any we either have
Graph
or
Graph
for a constant .
Now, we have all the prerequisites at hand to provide the
Proof of Proposition 3.5
Let and be the constants from Lemmas 6.1 and 6.2. Note that both depend on and . Choose such that
Graph
Then depends and . Lemma 6.2 implies that either , or
Graph
In the first case the proposition is proved with , while in the second case we may apply Lemma 6.1 with . Therefore either or
Graph
The first alternative coincides with the first alternative above, while the second one implies the sup-bound for for any since . Therefore we my choose as required.
Acknowledgements
R. Giova and A. Passarelli di Napoli have been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Passarelli di Napoli has been partially supported by Università degli Studi di Napoli Federico II through the Project FRA (000022-75-2021-FRA-PASSARELLI)
We would like to thank N. Liao for discussions concerning the modulus of continuity.
Funding Information
Open Access funding enabled and organized by Projekt DEAL.
Appendix
Here, we provide the proofs of Lemmas 6.1 and 6.2. We first state a well known iteration lemma which can be found for instance in [[19], Chap. 9.15.1, Lemma 15.1].
Lemma 7.1
Let be a sequence of non-negative numbers satisfying
Graph
with some positive constants and . If
Graph
then as .
The next lemma can be seen as a discrete version of the isoperimetric inequality, cf. [[19], Chap. 10.5.1, inequality (5.4)].
Lemma 7.2
Let and with . Then, there exists a constant c depending on n such that
Graph
Proof of Lemma 6.1
For we let
Graph
and consider for the levels
Graph
the normalized measure of the associated super-level sets
Graph
In view of Hölder's, Sobolev's inequality and Lemma 4.3 we now estimate[1]
Graph
where . On the other hand, by the definition of we have
Graph
Here we used that on . Joining the preceding two inequalities and recalling the definition of shows
Graph
If , the lemma is proved. Otherwise, the term in brackets on the right-hand side is bounded by 2, so that
Graph
where . Therefore, Lemma 7.1 ensures that in the limit , provided that and hence on .
Proof of Lemma 6.2
For convenience in notation we omit the reference to the center . For we consider the super-level sets
Graph
Then, due to assumption (3.10) we have for any . Applying Lemma 7.2 to on with and , we obtain with a constant that
Graph
In view of the energy estimate from Lemma 4.3 and the definition of we have
Graph
with a constant . If , the lemma is proved. Otherwise, the term in brackets on the right-hand side is bounded by 2. Inserting this above yields
Graph
for . Now, we add these inequalities for and obtain
Graph
Therefore, we have
Graph
This proves the assertion of the lemma.
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[
Footnotes
If we may choose any exponent instead of .
]
By Verena Bögelein; Frank Duzaar; Raffaella Giova and Antonia Passarelli di Napoli
Reported by Author; Author; Author; Author