We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the 𝑥-variable.
The paper deals with the regularity of minimizers of integral functionals of the Calculus of Variations of the form
where , , is a bounded open set, , , is a Sobolev map. The main feature of (1.1) is the possible degeneracy of the lagrangian with respect to the 𝑥-variable. We assume that the Carathéodory function is convex and of class with respect to , with , also Carathéodory functions and . We emphasize that the matrix of the second derivatives not necessarily is uniformly elliptic and it may degenerate at some .
In the vector-valued case , minimizers of functionals with general structure may lack regularity, see [19, 49, 43], and it is natural to assume a modulus-gradient dependence for the energy density, i.e. that there exists such that
Without loss of generality, we can assume ; indeed, the minimizers of 𝐹 are minimizers of
too. Moreover, by (1.2) and the convexity of 𝑓, is a nonnegative, convex and increasing function of .
As far as the growth and the ellipticity assumptions are concerned, we assume that there exist exponents , nonnegative measurable functions and a constant such that
for a.e. and for every . We allow the coefficient to be zero so that (1.3)1 is a not uniform ellipticity condition. As proved in Lemma 2.3, (1.3)1 implies the following possibly degenerate -growth conditions for 𝑓:
Our main result concerns the local Lipschitz regularity and the higher differentiability of the local minimizers of 𝐹.
If is a local minimizer of 𝐹, then for every ball , the estimates
It is well known that, to get regularity under -growth, the exponents 𝑞 and 𝑝 cannot be too far apart; usually, the gap between 𝑝 and 𝑞 is described by a condition relating and the dimension 𝑛. In our case, we take into account the possible degeneracy of and condition (1.3)2 on the mixed derivatives in terms of a possibly unbounded coefficient ; then we deduce that the gap depends on 𝑠, the summability exponent of that “measures” how much 𝑎 is degenerate, and the exponent 𝑟 that tells us how far is from being bounded. If , then (1.6) reduces to that is what one expects; see  and for instance . Moreover, if and , then (1.6) reduces to , and we recover the result of .
As it is well known, weak solutions to the elliptic equation in divergence form of the type
are locally Lipschitz continuous provided the vector field is differentiable with respect to 𝜉 and satisfies the uniform ellipticity conditions
Trudinger  started the study of the interior regularity of solutions to linear elliptic equations of the form
where the measurable coefficients satisfy the non-uniform condition
for a.e. and every . Here is the minimum eigenvalue of the symmetric matrix and . Trudinger proved that any weak solution of (1.9) is locally bounded in Ω, under the following integrability assumptions on 𝜆 and 𝜇:
Equation (1.9) is usually called degenerate when , whereas it is called singular when . These names in this case refer to the degenerate and the singular cases with respect to the 𝑥-variable, but in the mathematical literature, these names often refer to the gradient variable; this happens for instance with the 𝑝-Laplacian operator. We do not study in this paper the degenerate case with respect to the gradient variable, but we refer for instance to the analysis made by Duzaar and Mingione , who studied an -gradient bound for solutions to non-homogeneous 𝑝-Laplacian type systems and equations; see also Cianchi and Maz’ya  and the references therein for the rich literature on the subject.
The result by Trudinger was extended in many settings and directions: firstly, by Trudinger himself in  and later by Fabes, Kenig and Serapioni in ; Pingen in  dealt with systems. More recently, for the regularity of solutions and minimizers, we refer to [3, 4, 10, 15, 16, 32]. For the higher integrability of the gradient, we refer to  (see also ). Very recently, Calderon–Zygmund’s estimates for the 𝑝-Laplace operator with degenerate weights have been established in . The literature concerning non-uniformly elliptic problems is extensive, and we refer the interested reader to the references therein.
The study of the Lipschitz regularity in the -growth context started with the papers by Marcellini [35, 36], and since then, many and various contributions to the subject have been provided; see the references in [42, 40]. The vectorial homogeneous framework was considered in [37, 41] and by Esposito, Leonetti and Mingione [27, 28]. Condition (1.3)2 for general non-autonomous integrands has been first introduced in [23, 22, 24]. It is worth to highlight that, due to the 𝑥-dependence, the study of regularity is significantly harder and the techniques more complex. The research on this subject is intense, as confirmed by the many articles recently published; see e.g. [11, 13, 14, 20, 25, 30, 39, 38, 40, 45, 46, 47].
Let us briefly sketch the tools to get our regularity result. First, for Lipschitz and higher differentiable minimizers, we prove a weighted summability result for the second-order derivatives of minimizers of functionals with possibly degenerate energy densities; see Proposition 3.2. Next, in Theorem 3.3, we get an a priori estimate for the -norm of the gradient. To establish the a priori estimate, we use Moser’s iteration method  for the gradient and the ideas of Trudinger . An approximation procedure allows us to conclude. Actually, if 𝑢 is a local minimizer of (1.1), we construct a sequence of suitable variational problems in a ball with boundary value data 𝑢. In order to apply the a priori estimate to the minimizers of the approximating functionals, we prove a higher differentiability result, see Theorem 3.3, for minimizers of the class of functionals with -growth studied in , where only the Lipschitz continuity was proved. By applying the previous a priori estimate to the sequence of the solutions, we obtain a uniform control in of the gradient which allows to transfer the local Lipschitz continuity property to the original minimizer 𝑢.
Another difficulty due to the 𝑥-dependence of the energy density is that the Lavrentiev phenomenon may occur. A local minimizer of 𝐹 is a function such that and
for every ball . Therefore, in our context, a priori, the presence of the Lavrentiev phenomenon cannot be excluded. Indeed, due to the growth assumptions on the energy density, the integral in (1.1) is well defined if , but a priori, this is not the case if . However, as a consequence of Theorem 1.1, under the stated assumptions (1.2), (1.3), (1.5), (1.6), the Lavrentiev phenomenon for the integral functional 𝐹 in (1.1) cannot occur. For the gap in the Lavrentiev phenomenon, we refer to [52, 5, 28, 26].
We conclude this introduction by observing that, even in the one-dimensional case, the Lipschitz continuity of minimizers for non-uniformly elliptic integrals is not obvious. Indeed, if we consider a local minimizer 𝑢 to the one-dimensional integral
then Euler’s first variation takes the form
This implies that the quantity is constant in ; it is a nonzero constant, unless itself is constant in , a trivial case that we do not consider here. In particular, the sign of is constant, and we get
Therefore, if vanishes somewhere in , then is unbounded (and vice versa), independently of the exponent . Thus, for , the local Lipschitz regularity of the minimizers does not hold in general if the coefficient vanishes somewhere.
We can compare this one-dimensional fact with the general conditions considered in the Theorem 1.1. In the case for some , then, taking into account the assumptions in (1.5), for the integral in (1.11), we have and
These conditions are compatible if and only if . Therefore, also in the one-dimensional case, we have a counterexample to the -gradient bound in (1.7) if
which essentially is the complementary condition to (1.12) when .
The plan of the paper is the following. In Section 2, we list some definitions and preliminary results. In Section 3, we prove a priori estimates of the -norm of the gradient of local minimizers and a higher differentiability result; see Theorem 3.3. In the last section, we complete the proof of Theorem 1.1.
2 Preliminary results
We shall denote by 𝐶 or 𝑐 a general positive constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies will be suitably emphasized using parentheses or subscripts. In what follows, will denote the ball centered at 𝑥 of radius 𝑟. We shall omit the dependence on the center and on the radius when no confusion arises.
To prove our higher differentiability result (see Theorem 3.3 below), we use the finite difference operator. For a function , Ω open subset of , given , we define
where is the unit vector in the direction, and
We now list the main properties of this operator.
If , , then and
If 𝑓 or 𝑔 has support in , then
If , , and , then for every ℎ, ,
If , , and for , there exists such that, for every ℎ, ,
then letting ℎ go to 0, there exists and for every .
Let . There exists a constant such that
for any 𝜉, , .
Note that if is such that , then there exists a constant such that
Let be of class with respect to the 𝜉-variable. Let us assume that the quadratic form of the second derivatives of 𝑓 satisfies the conditions
for some exponents and nonnegative functions and for all . Then 𝑓 satisfies the following -growth condition:
where and .
As noticed by an anonymous referee, that we thank for her/his careful reading of the manuscript, in Lemma 2.3, it is sufficient to assume the validity of (2.1) only for every with 𝜆 proportional to 𝜉 (see formulas (2.5), (2.6) below in the proof); in this case, (2.1) simplifies into
For and , let us set , where we recall that 𝑔 is linked to 𝑓 by (1.2). The assumptions on 𝑓 imply that with . Since
the Taylor expansion formula in integral form yields
Recalling the definition of 𝜑, since and , we get
We recall [41, formula (3.3)],
which holds with equality when 𝜆 is proportional to 𝜉, when (2.5) simplifies to
Therefore, in the particular case with , assumption (2.1) translates into
Assume . For the integral in the left-hand side of (2.7), since and , we obtain
Similarly (and simpler) for the integral in the right-hand side of (2.7),
Combining the last two estimates and recalling in (2.7) that , we get
If , then we modify (2.8) into
then, if also , we modify (2.9) into
In this case, we obtain
The conclusion (2.2) follows by combining all these cases. ∎
We end this preliminary section with two well-known properties. The first lemma has important applications in the so-called hole-filling method. Its proof can be found for example in [31, Lemma 6.1].
Let be a nonnegative bounded function and , and . Assume that
for all . Then
We will also use the following (see e.g. ).
Let . For every and , there exist positive constants and such that
We end this section with a remark that we will use when considering the summability of 𝑘.
Let , . If and
then implies .
Since , we need to prove that that is equivalent to . This holds true because
and we conclude because . ∎
3 The a priori estimate
The main result in this section is an a priori estimate of the -norm of the gradient, and a higher differentiability result, of local minimizers of the functional 𝐹 in (1.1) satisfying assumptions (1.2), (1.3), (1.5) and (1.6), with exponents 𝑝 and 𝑞 satisfying and not the more restrictive condition , assumed in Theorem 1.1.
We use the following weighted Sobolev type inequality, whose proof relies on the Hölder’s inequality; see e.g. .
Let , and ( if ), with , and let be a measurable function such that . Then there exists a constant such that
where ( if ).
In establishing the a priori estimate, we need to deal with quantities that involve the -norm of the second derivatives of the minimizer weighted with the function . The next result tells that local Lipschitz continuous minimizers 𝑢 of 𝐹 possess weak second derivatives such that
More precisely, we have the following proposition.
with . If is a local minimizer of 𝐹, then
Since 𝑢 is a local minimizer of the functional 𝐹, then 𝑢 satisfies the Euler system
Fix , , . Consider a cut-off function , , in , and . Assume .
Fix , and consider the function
Thanks to the assumption , we can use 𝜑 as test function in equation (3.2), thus getting
where we used property (ii) of the finite difference operator. Now, we observe that
and so (3.3) can be written as follows:
By the use of Cauchy–Schwarz and Young’s inequalities and by virtue of the second inequality of (1.3), we can estimate the integral . If , we have
where we used in turn Lemma 2.6, property (iii) of the finite difference operator and the assumption .
If , a similar estimate of holds true, and it can be obtained in a simpler way. Indeed, in this case,
can be estimated as follows:
At the end, we obtain
By the last inequality in (1.3), Hölder’s inequality, again by property (iii) of the finite difference operator and the assumption , we obtain
and also, for every , there exists such that
Therefore, we get
where all the constants in the previous estimate depend also on . Reabsorbing the first integral in the right-hand side by the left-hand side, we obtain
Choosing , we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting
Using Lemma 2.1 to control the left-hand side of the previous estimate from below, we get
and so, by the assumption and Hölder’s inequality,
Dividing both sides of the previous inequality by and letting , and using (3.1), we obtain
with 𝐾 positive constant depending on and . Therefore, by property (iv) of the finite difference operator, we get
and for a sequence converging to 0, the sequence strongly converges to in and also a.e. up to a subsequence. Hence, by Fatou’s lemma, we also have
We are now ready to establish the main result of this section, that holds true with the condition and not only for .
Moreover, 𝑢 satisfies the Euler system
Fix . By considering in the equality above with , we get
Fix a cut-off function , and define for any the function
One can easily check that
is in , through a density argument, we can use 𝜑 as test function in equation (3.7), thus getting
Estimate of . By the use of Cauchy–Schwarz and Young’s inequalities and by virtue of the second inequality in (1.3), we can estimate the integral as follows:
where will be chosen later.
Estimate of . Since
Using the Cauchy–Schwarz inequality, i.e.
and observing that , we conclude
Estimate of . By using the last inequality in (1.3), we obtain
Estimate of . Using the last inequality in (1.3) and Young’s inequality, we have that
where will be chosen later and 𝑎 is the function appearing in (1.3).
where we used Young’s inequality again. Since equality (3.8) can be written as
by virtue of (3.11), we get
Choosing , we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting
By and Proposition 3.2, the first integral in the right-hand side of previous estimate is finite.
By choosing , we can reabsorb the first integral in the right-hand side by the left-hand side, thus getting
where we used Young’s inequality again. Now, we note that
and so, fixing with such that and choosing a cut-off function between and , by the assumption , we can use the Sobolev type inequality of Lemma 3.1 with , and , thus obtaining
with a constant 𝑐 depending on 𝑛 and .
Using (3.16) to estimate the last integral in the previous inequality, we obtain
where we used assumption (1.5) and Hölder’s inequality with exponents 𝑠, and .
Using the properties of 𝜂, we obtain
where we used that, by assumption (1.3), . Setting
and assuming without loss of generality that , we can write the previous estimate as follows:
and using the a priori assumption , we get
and noting that
we can write (3.19) as follows:
where, without loss of generality, we supposed .
Define the increasing sequence of exponents
and the decreasing sequence of radii by setting
We have that
where, recalling that , we used in the last equality that
Therefore, we can let in (3.23), thus getting
Since assumption (1.6) implies that
we can use Young’s inequality with exponents
to deduce that
with and .
We now estimate the last integral. By definition of 𝑚 and by the assumption on 𝑠, that implies , we get
Thus, by Hölder’s inequality,
If we denote
by (1.6) and , we get
Therefore, there exists such that
The precise value of 𝜃 is
By Hölder’s inequality with exponents and , we get
Hence, we can use the inequality above to estimate the last integral of (3.26) to deduce that
We can use Young’s inequality in the last term of (3.28) with exponents and to obtain that, for every ,
By applying Lemma 2.5, and noting that , we conclude that
i.e. (3.6), with 𝑐 depending on . ∎
4 Proof of Theorem 1.1
Using the previous results and an approximation procedure, we can prove of our main result.
Proof of Theorem 1.1
Note that satisfies the following set of conditions:
Now, fix a ball , and let be the unique solution to the problem
Since satisfies (4.1), by the minimality of , we get
Therefore, the sequence is bounded in , so there exists such that, up to subsequences,
with independent of ℎ and . Therefore, up to subsequences,
Our next aim is to show that . The lower semicontinuity of and the minimality of imply
The strict convexity of 𝑓 yields that . Therefore, passing to the limit as in (4.5), we get
where we used the minimality of and the definition of . By Remark 2.2, the above estimate implies that the sequence is bounded in the -norm. Using also (4.6) and that , we get that, up to subsequences, this sequence converges in the weak topology of -norm to . By the lower semicontinuity of the -norm with respect to the weak convergence, we conclude that
i.e. (1.8). ∎
Funding source: Università degli Studi di Napoli Federico II
Award Identifier / Grant number: FRA 000022–ALTRI_CDA_75_2021_FRA_PASSARELLI
Funding statement: A. Passarelli di Napoli has been supported by Università di Napoli “Federico II” through the project FRA 000022–ALTRI_CDA_75_2021_FRA_PASSARELLI.
We thank the referee for carefully reading the manuscript, for the valuable remarks and suggestions. The authors are members of GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), whose support is warmly acknowledged.
Communicated by: Frank Duzaar
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