Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
-
Antonio Giuseppe Grimaldi
Abstract
We establish the higher fractional differentiability of bounded minimizers to a class of obstacle problems with non-standard growth conditions of the form
where Ω is a bounded open set of
is the class of admissible functions. If the obstacle ψ is locally bounded, we prove that the gradient of solution inherits some fractional differentiability property, assuming that both the gradient of the obstacle and the mapping
1 Introduction
In this paper, we study the higher fractional differentiability properties of the gradient of bounded solutions
Here Ω is a bounded open set of
is the class of admissible functions, where
In what follows, we assume that the energy density
for a.e.
Moreover, we also assume that there exist positive constants
for a.e.
In the vectorial case, assumption (F1) is quite common to prove regularity results for local minimizers (see [49]), but, in the scalar case, it plays a crucial role, since it allows to approximate F with a sequence of integrands
Very recently, in [15] it has been proved, under the structure assumption (F1), that (F3) and (F4) imply (F2), i.e. if
We say that the function F satisfies assumption (F5) if there exists a non-negative function
with
for a.e.
On the other hand, we say that assumption (F6) is satisfied if there exists a sequence of measurable non-negative functions
with
for some
for a.e.
It is well known that the regularity of the minima often comes from the fact that they are also solutions to the corresponding Euler–Lagrange system, in the unconstrained setting, or, in the case of obstacle problems, to the corresponding variational inequality.
In the case of standard growth conditions, that is,
for all
On the other hand, in the case of non-standard growth conditions, even in the unconstrained case, the relation between extremals and minima is an issue that requires a careful investigation (see for example [7, 8, 22]).
From conditions (F2)–(F4), we deduce the existence of positive constants
for a.e.
Furthermore, if one of the conditions (F5) or (F6) holds, then
for a.e.
for a.e.
The study of obstacle problems started with the works by Stampacchia [47] and Fichera [23] and has since then attracted much attention. It is usually observed that the regularity of the solutions to the obstacle problems is influenced by the one of the obstacle; for example, for linear obstacle problems, obstacle and solutions have the same regularity [3, 5, 38]. This does not apply in the nonlinear setting, and hence there have been intense research activities in this direction (see [11, 12, 19, 42], just to mention a few).
When referring to functionals with non-standard growth conditions, it is well known that in order to get regularity of minimizers, even the boundedness, a restriction between p and q need to be imposed, usually expressed in the form
In recent years, there has been a considerable interest in analyzing how an extra differentiability of integer or fractional order of the obstacle transfers to the gradient of solutions: for instance we quote [9, 20, 21, 26, 27, 34] in the setting of standard growth conditions, and [10, 16, 24, 25, 28, 35, 36, 50] in the setting of non-standard growth conditions.
The analysis comes from the fact that the regularity of the solutions to obstacle problem (1.2) is strictly connected to the analysis of the regularity of the solutions to partial differential equations of the form
whose higher differentiability properties have been widely investigated (see for instance [1, 2, 13, 31, 30, 32, 44, 45].
It is well known that no extra differentiability properties for the solutions can be expected even if the obstacle ψ is smooth, unless some assumption is given on the coefficients of the operator
In particular, in [35] the author and Ipocoana obtained higher fractional differentiability properties for the solution to (1.1) in the sense of
provided the obstacle function satisfies
for
For precise definition and properties of Besov spaces, see Sections 2.2 and 3.
When one deals with bounded minimizers, both for unconstrained and constrained problems with
Recently, in [18], it has been proved that, assuming the local boundedness of the obstacle ψ, the solution to the obstacle problem (1.1) is locally bounded under a sharp relation between p and q.
Here, we study higher fractional differentiability properties of bounded solutions to obstacle problems satisfying
More precisely, we shall prove the following theorems.
Theorem 1.1.
Let
and
Let
provided
On the other hand, a Besov regularity of the type
Theorem 1.2.
Let
Let
for a constant
Note that Theorems 1.1 and 1.2 improve the results contained in [35].
Here, the local boundedness allows us to use an interpolation inequality (see Lemma 3.2) that gives the higher local integrability
and, moreover, under our assumption on the gap, i.e.
The structure of this paper is the following. After recalling some notation and preliminary results in Section 2, a Gagliardo–Nirenberg type inequality in Besov spaces is established in Section 3 for a priori bounded minimizers. Then we concentrate on proving our main result. The strategy is to prove uniform a priori estimates for solutions to a family of approximating problems. Therefore, in Section 4, we present the approximation lemma that allows to construct a sequence of functions satisfying p-growth conditions that converges to the initial problem. In Section 5, we prove Theorem 1.1. In particular, we derive the a priori estimates in Section 5.1, and, in Section 5.2, we pass to the limit in the approximating problems. Eventually, in Section 6, we give the proof of Theorem 1.2, focusing only on the a priori estimate since the approximation procedure works exactly in the same way as in the proof of Theorem 1.1.
2 Notations and preliminary results
For the rest of the paper, we denote by C, c, π general positive constants. Different occurrences from line to line will still be denoted using the same letters. Relevant dependencies on parameters will be emphasized using parentheses or subscripts.
We denote by
will denote the integral mean of the function u over the set B.
It is convenient to introduce an auxiliary function
defined for all
For the auxiliary function
Lemma 2.1.
Let
for any
Now we state a well-known iteration lemma (see [33, Lemma 6.1] for the proof).
Lemma 2.2.
Assume
where
The following regularity result, whose proof can be found in [18] in a more general setting, allows us to obtain the local boundedness of solutions to the obstacle problem (1.1).
Theorem 2.3.
Let
holds for every ball
2.1 Difference quotient
We recall some properties of the finite difference quotient operator that will be needed in the sequel. Let us recall that, for a function
We start with the description of some elementary properties that can be found, for example, in [33].
Proposition 2.4.
Let
Then the following assertions hold:
If at least one of the functions F or G has support contained in
We have
The next result about the finite difference operator is a kind of integral version of the Lagrange theorem.
Lemma 2.5.
If
Moreover,
2.2 Besov spaces
We give the definition of Besov spaces as done in [37, Section 2.5.12].
Definition 2.6.
Let
Let
Equivalently, we could simply say that
Let
Accordingly, for
Observe that, if
In the case
Lemma 2.7.
Suppose that
If
If
For further needs, we recall the following inclusions [37, Proposition 7.10 and (7.35)].
Lemma 2.8.
Suppose that
If
If
Given a domain
We have the following lemma.
Lemma 2.9.
A function
for any ball
Proof.
If
Now, let
holds.
It is clear that
So, since
Moreover, it holds
and therefore we have
As a consequence, we have the equivalence
However, it is clear that
if and only if the same happens for every
It is known that Besov spaces of fractional order
Definition 2.10.
Suppose
holds whenever
The following result was proved in [39].
Theorem 2.11.
Let
where the infimum runs over all possible fractional α-Hajlasz gradients for v.
3 Gagliardo–Nirenberg inequality
In this section, we collect some results on Besov spaces that will be useful later.
Lemma 3.1.
Let
holds for every ball
Proof.
We give the proof of Lemma 3.1 only for
Fix
Now, using
which is finite by the assumption on Dv. This completes the proof. ∎
The following interpolation inequality can be found in [46].
Lemma 3.2.
Let
holds for every
For the next result, see, e.g., [43].
Lemma 3.3.
Let
Now, we are able to prove the following higher integrability result.
Proposition 3.4.
Let
holds for every ball
Proof.
Thanks to Lemma 3.1, we obtain
Then Lemma 3.2 yields
for every
Choosing
Let us consider
From Lemma 3.2, we get
Using identity (2.2) and the properties of η, we infer
Now, exploiting Lemma 2.5 and using the fact that
Combining inequalities (3.1), (3.2) and (3.3) and Lemma 3.1, we derive
i.e. the desired estimate. ∎
The following proposition is an immediate consequence of the previous result and will be a fundamental tool for the proof of Theorem 1.1.
Proposition 3.5.
Let
holds for every ball
Proof.
By Lemma 2.1, we get
where in the last inequality we used the fact that
Estimate (3.4) implies
Now, taking the supremum over h in (3.5) and by virtue of the assumption on
Thanks to Lemma 3.1 and (3.6), we obtain
By virtue of Lemma 3.4 and (3.5), it follows
for every
Moreover, we have the following embeddings between Sobolev and Besov spaces (see [48]).
Lemma 3.6.
Let
Arguing similarly, but assuming that
Proposition 3.7.
Let
Moreover, the inequality
holds for every ball
4 Approximation lemma
The main tool to prove Theorem 1.1 is the following approximation lemma (see [35] for the proof).
Lemma 4.1.
Let
For a.e.
For a.e.
For a.e.
with
For a.e.
There exists a constant
for a.e.
Assuming (F6) instead of (F5), statement (v) would change as follows:
There exists a constant
for a.e.
5 Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. In particular, in Section 5.1, we derive the a priori estimates for regular minimizers of obstacle problems (1.1), while in Section 5.2 we conclude through an approximation argument.
5.1 A priori estimate
We have the following theorem.
Theorem 5.1.
Let
for
then the estimates
and
hold for all balls
Proof.
By virtue of assumption (1.4) and Theorem 2.3,
Notice that
Hence, by minimality of u, we get
which leads to
From Lagrange’s theorem, for
Since
Now, from assumption (A1), we obtain
where we also used that
On the other hand, by virtue of assumption (1.3) and (5.1), we have
Therefore, by applying the dominated convergence theorem, we can pass to the limit for
Fix
Now, for
and
Then
are admissible test functions for all
Inserting (5.3) and (5.4) in (1.2), we obtain
By means of a simple change of variable, we can write the second integral on the left-hand side of the previous inequality as follows:
and so inequality (5.5) becomes
We can write the previous inequality as follows:
which yields
The ellipticity assumption (A2) and Lemma 2.1 imply
From the growth condition (A3), Young’s and Hölder’s inequalities, Lemma 2.1
and the assumption on
Arguing analogously, we get
Using Lemma 2.5, we obtain
In order to estimate the integral
We now take care of
Now, we observe that
Hence,
From assumption (A4), the hypothesis
Using once again Hölder’s inequality, inequality (5.12) and Lemma 2.5, we have
Inserting estimates (5.8)–(5.11), (5.13) and (5.14) in (5.7), we infer
Using Lemma 2.1,
Choosing
where we set
From Young’s inequality, we infer
where
Using Young’s inequality, we estimate the third and the second last integral appearing in the right-hand side of estimate (5.15) as follows:
and analogously
where
Inserting (5.16) and (5.17) in (5.15), we get
We can rewrite the previous estimate as
for a constant
Dividing both sides of the previous estimate by
for every
Since
for a constant
Taking the supremum over h in the left-hand side of (5.18) and using estimate (5.19), we obtain
for every
Now, choosing s such that
Setting
we can write inequality (5.20) as
By virtue of Lemma 2.2, choosing θ such that
we obtain
with
Now, recalling the definition of Φ, we obtain
Thus, inserting (5.21) in (5.18), we deduce the a priori estimate
for constants
Now, we are able to establish the following higher differentiability result for obstacle problems with p-growth.
Theorem 5.2.
Assume that
for a.e.
holds, provided
5.2 Passage to the limit
Proof of Theorem 1.1.
Let
Setting
one can easily check that
for a.e.
Let
and
an easy computation shows that
and
for a.e.
Step 1.
For fixed
By the ellipticity assumption (A2), we have
Since
Combining (5.28) and (5.29), we get
where in the last inequality we used Hölder’s inequality.
Since
Since
and
Therefore, passing to the limit for
Moreover, since
we have
By virtue of Theorem 5.2,
for constants
Finally, by weak lower semicontinuity, (5.32) and (5.33), we get
Step 2.
From Lemma 4.1 (iv), there exists
The previous estimate and the minimality of
where in the last inequality we used Lemma 4.1 (ii). Thus, up to subsequences,
Now, fix
From weak lower semicontinuity of
Combining these last inequalities, we get
where we also applied the monotone convergence theorem, according to Lemma 4.1 (ii).
Moreover, by the weak convergence (5.35), the limit function
Now, from estimates (5.34) and (5.2), it follows
for constants
Hence, from (5.35) and weak lower semicontinuity, it follows
Eventually, proceeding as in the proof of Theorem 5.1,
we derive that
6 Proof of Theorem 1.2
Proof of Theorem 1.2.
We derive only the a priori estimates, since the approximation procedure is achieved using the same arguments presented in Section 5.2.
We a priori assume that
Arguing analogously as in the proof of Theorem 5.1,
we define the integrals
Consider the integral
we get
Performing the same computations which led us to (5.10) with
Now, we take care of the integral
Exploiting assumption (A5) and Hölder’s inequality, we infer
Now, we observe that
Hence,
where
Similarly to above, from assumption (A5), hypothesis
Using once again Hölder’s inequality, inequality (6.4) and Lemma 2.5, we have
Inserting estimates (5.8), (6.1)–(6.3), (6.5) and (6.6) in (5.7) and reabsorbing the integral
in the right-hand side by the left-hand side, we infer
for a positive constant
We now need to take the
We obtain the following estimate:
Note that, since
is controlled by the norm in the Besov space
on
can be calculated in polar coordinates as follows:
since
Now, we take care of the integral
We write the right-hand side of the previous estimate in polar coordinates, so
where we set
where in the last inequality we used that
which is finite by assumption (F6).
Using Young’s inequality with exponent 2, we deduce the following estimate:
where the two integrals in the right-hand side can be estimated as the integrals
Similarly, we obtain
The first term and the latter one can be estimated as the integral
Estimate (6.7) can be written in the following way:
for a constant
From Young’s inequality, we infer
for
We estimate the second and the penultimate integral appearing in the right-hand side of the estimate as follows:
Similarly,
where we set
Inserting estimates (6.9) and (6.10) in (6.8), we obtain
Now, by virtue of Proposition 3.7, we infer the following inequality:
Combining inequalities (6.11) and (6.12) and arguing as in the proof of Theorem 5.1, we obtain
which yields
for every
for constants
Acknowledgements
The author wishes to thank the anonymous referee for his/her comments which improved the presentation.
References
[1] P. Ambrosio, Besov regularity for a class of singular or degenerate elliptic equations, J. Math. Anal. Appl. 505 (2022), no. 2, Paper No. 125636. 10.1016/j.jmaa.2021.125636Search in Google Scholar
[2] A. L. Baisón, A. Clop, R. Giova, J. Orobitg and A. Passarelli di Napoli, Fractional differentiability for solutions of nonlinear elliptic equations, Potential Anal. 46 (2017), no. 3, 403–430. 10.1007/s11118-016-9585-7Search in Google Scholar
[3] H. Brézis and D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J. 23 (1973/74), 831–844. 10.1512/iumj.1974.23.23069Search in Google Scholar
[4] S.-S. Byun, Y. Cho and J. Ok, Global gradient estimates for nonlinear obstacle problems with nonstandard growth, Forum Math. 28 (2016), no. 4, 729–747. 10.1515/forum-2014-0153Search in Google Scholar
[5] L. A. Caffarelli and D. Kinderlehrer, Potential methods in variational inequalities, J. Anal. Math. 37 (1980), 285–295. 10.1007/BF02797689Search in Google Scholar
[6] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Higher differentiability of minimizers of convex variational integrals, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 3, 395–411. 10.1016/j.anihpc.2011.02.005Search in Google Scholar
[7] M. Carozza, J. Kristensen and A. Passarelli di Napoli, Regularity of minimizers of autonomous convex variational integrals, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), no. 4, 1065–1089. 10.2422/2036-2145.201208_005Search in Google Scholar
[8] M. Carozza, J. Kristensen and A. Passarelli di Napoli, On the validity of the Euler-Lagrange system, Commun. Pure Appl. Anal. 14 (2015), no. 1, 51–62. 10.3934/cpaa.2015.14.51Search in Google Scholar
[9] M. Caselli, A. Gentile and R. Giova, Regularity results for solutions to obstacle problems with Sobolev coefficients, J. Differential Equations 269 (2020), no. 10, 8308–8330. 10.1016/j.jde.2020.06.015Search in Google Scholar
[10] I. Chlebicka and C. De Filippis, Removable sets in non-uniformly elliptic problems, Ann. Mat. Pura Appl. (4) 199 (2020), no. 2, 619–649. 10.1007/s10231-019-00894-1Search in Google Scholar
[11] H. J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems, Arch. Ration. Mech. Anal. 114 (1991), no. 4, 383–394. 10.1007/BF00376141Search in Google Scholar
[12] H. J. Choe and J. L. Lewis, On the obstacle problem for quasilinear elliptic equations of p Laplacian type, SIAM J. Math. Anal. 22 (1991), no. 3, 623–638. 10.1137/0522039Search in Google Scholar
[13] A. Clop, R. Giova and A. Passarelli di Napoli, Besov regularity for solutions of p-harmonic equations, Adv. Nonlinear Anal. 8 (2019), no. 1, 762–778. 10.1515/anona-2017-0030Search in Google Scholar
[14] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218 (2015), no. 1, 219–273. 10.1007/s00205-015-0859-9Search in Google Scholar
[15]
G. Cupini, P. Marcellini, E. Mascolo and A. Passarelli di Napoli,
Lipschitz regularity for degenerate elliptic integrals with
[16]
C. De Filippis,
Regularity results for a class of non-autonomous obstacle problems with
[17] C. De Filippis and G. Mingione, On the regularity of minima of non-autonomous functionals, J. Geom. Anal. 30 (2020), no. 2, 1584–1626. 10.1007/s12220-019-00225-zSearch in Google Scholar
[18] M. De Rosa and A. G. Grimaldi, A local boundedness result for a class of obstacle problems with non-standard growth conditions, J. Optim. Theory Appl. 195 (2022), no. 1, 282–296. 10.1007/s10957-022-02084-1Search in Google Scholar
[19] M. Eleuteri, P. Harjulehto and T. Lukkari, Global regularity and stability of solutions to obstacle problems with nonstandard growth, Rev. Mat. Complut. 26 (2013), no. 1, 147–181. 10.1007/s13163-011-0088-1Search in Google Scholar
[20] M. Eleuteri and A. Passarelli di Napoli, Higher differentiability for solutions to a class of obstacle problems, Calc. Var. Partial Differential Equations 57 (2018), no. 5, Paper No. 115. 10.1007/s00526-018-1387-xSearch in Google Scholar
[21] M. Eleuteri and A. Passarelli di Napoli, Regularity results for a class of non-differentiable obstacle problems, Nonlinear Anal. 194 (2020), Article ID 111434. 10.1016/j.na.2019.01.024Search in Google Scholar
[22] M. Eleuteri and A. Passarelli di Napoli, On the validity of variational inequalities for obstacle problems with non-standard growth, Ann. Fenn. Math. 47 (2022), no. 1, 395–416. 10.54330/afm.114655Search in Google Scholar
[23] G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 7 (1963/64), 91–140. Search in Google Scholar
[24]
C. Gavioli,
A priori estimates for solutions to a class of obstacle problems under
[25] C. Gavioli, Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions, Forum Math. 31 (2019), no. 6, 1501–1516. 10.1515/forum-2019-0148Search in Google Scholar
[26] A. Gentile, Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions, Forum Math. 33 (2021), no. 3, 669–695. 10.1515/forum-2020-0299Search in Google Scholar
[27] A. Gentile and R. Giova, Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions, Nonlinear Anal. Real World Appl. 68 (2022), Paper No. 103681. 10.1016/j.nonrwa.2022.103681Search in Google Scholar
[28] A. Gentile, R. Giova and A. Torricelli, Regularity results for bounded solutions to obstacle problems with non-standard growth conditions, Mediterr. J. Math. 19 (2022), no. 6, 270. 10.1007/s00009-022-02162-8Search in Google Scholar
[29] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), no. 2, 245–248. 10.1007/BF01158049Search in Google Scholar
[30] R. Giova, Higher differentiability for n-harmonic systems with Sobolev coefficients, J. Differential Equations 259 (2015), no. 11, 5667–5687. 10.1016/j.jde.2015.07.004Search in Google Scholar
[31] R. Giova, Besov regularity for solutions of elliptic equations with variable exponents, Math. Nachr. 293 (2020), no. 8, 1459–1480. 10.1002/mana.201900185Search in Google Scholar
[32] R. Giova and A. Passarelli di Napoli, Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients, Adv. Calc. Var. 12 (2019), no. 1, 85–110. 10.1515/acv-2016-0059Search in Google Scholar
[33] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, River Edge, 2003. 10.1142/5002Search in Google Scholar
[34] A. G. Grimaldi, Regularity results for solutions to a class of obstacle problems, Nonlinear Anal. Real World Appl. 62 (2021), Paper No. 103377. 10.1016/j.nonrwa.2021.103377Search in Google Scholar
[35] A. G. Grimaldi and E. Ipocoana, Higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions, Adv. Calc. Var. (2021), 10.1515/acv-2021-0074. 10.1515/acv-2021-0074Search in Google Scholar
[36] A. G. Grimaldi and E. Ipocoana, Higher differentiability results in the scale of Besov spaces to a class of double-phase obstacle problems, ESAIM Control Optim. Calc. Var. 28 (2022), Paper No. 51. 10.1051/cocv/2022050Search in Google Scholar
[37] D. D. Haroske, Envelopes and Sharp Embeddings of Function Spaces, Chapman & Hall/CRC Res. Notes in Math. 437, Chapman & Hall/CRC, Boca Raton, 2007. 10.1201/9781584887515Search in Google Scholar
[38] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure Appl. Math. 88, Academic Press, New York, 1980. Search in Google Scholar
[39] P. Koskela, D. Yang and Y. Zhou, Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), no. 4, 3579–3621. 10.1016/j.aim.2010.10.020Search in Google Scholar
[40] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 5, 391–409. 10.1016/s0294-1449(16)30379-1Search in Google Scholar
[41]
P. Marcellini,
Regularity and existence of solutions of elliptic equations with
[42] J. H. Michael and W. P. Ziemer, Interior regularity for solutions to obstacle problems, Nonlinear Anal. 10 (1986), no. 12, 1427–1448. 10.1016/0362-546X(86)90113-6Search in Google Scholar
[43] P. Mironescu, E. Russ and Y. Sire, Lifting in Besov spaces, Nonlinear Anal. 193 (2020), Article ID 111489. 10.1016/j.na.2019.03.012Search in Google Scholar
[44] A. Passarelli di Napoli, Higher differentiability of minimizers of variational integrals with Sobolev coefficients, Adv. Calc. Var. 7 (2014), no. 1, 59–89. 10.1515/acv-2012-0006Search in Google Scholar
[45]
A. Passarelli di Napoli,
Higher differentiability of solutions of elliptic systems with Sobolev coefficients: the case
[46] T. Runst, Mapping properties of nonlinear operators in spaces of Triebel–Lizorkin and Besov type, Anal. Math. 12 (1986), no. 4, 313–346. 10.1007/BF01909369Search in Google Scholar
[47] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Math. Acad. Sci. Paris 258 (1964), 4413–4416. Search in Google Scholar
[48] W. Trebels, Inequalities for moduli of smoothness versus embeddings of function spaces, Arch. Math. (Basel) 94 (2010), no. 2, 155–164. 10.1007/s00013-009-0078-4Search in Google Scholar
[49] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math. 138 (1977), no. 3–4, 219–240. 10.1007/BF02392316Search in Google Scholar
[50] X. Zhang and S. Zheng, Besov regularity for the gradients of solutions to non-uniformly elliptic obstacle problems, J. Math. Anal. Appl. 504 (2021), no. 2, Paper No. 125402. 10.1016/j.jmaa.2021.125402Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- On the asymptotics of the shifted sums of Hecke eigenvalue squares
- On the splitting conjecture in the hybrid model for the Riemann zeta function
- Almost Dedekind domains without radical factorization
- Groups in which the centralizer of any non-central primary element is maximal
- GKM actions on cohomogeneity one manifolds
- Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
- Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
- Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
- Measures of noncompactness of interpolated polynomials
- The homotopy category of acyclic complexes of pure-projective modules
- The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
- Sharp Li–Yau inequalities for Dunkl harmonic oscillators
- A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
- Free polynilpotent groups and the Magnus property
Articles in the same Issue
- Frontmatter
- On the asymptotics of the shifted sums of Hecke eigenvalue squares
- On the splitting conjecture in the hybrid model for the Riemann zeta function
- Almost Dedekind domains without radical factorization
- Groups in which the centralizer of any non-central primary element is maximal
- GKM actions on cohomogeneity one manifolds
- Gruenberg–Kegel graphs: Cut groups, rational groups and the prime graph question
- Sobolev regularity for nonlinear Poisson equations with Neumann boundary conditions on Riemannian manifolds
- Higher differentiability for bounded solutions to a class of obstacle problems with (p,q)-growth
- Measures of noncompactness of interpolated polynomials
- The homotopy category of acyclic complexes of pure-projective modules
- The Gelfand–Kirillov dimension of Hecke–Kiselman algebras
- Sharp Li–Yau inequalities for Dunkl harmonic oscillators
- A Shimura–Shintani correspondence for rigid analytic cocycles of higher weight
- Free polynilpotent groups and the Magnus property
