Abstract
We consider autonomous integral functionals of the form
where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of \({\mathcal {F}}\) assuming \(\frac{q}{p}<1+\frac{2}{n-1}\), \(n\ge 3\). This improves earlier results valid under the more restrictive assumption \(\frac{q}{p}<1+\frac{2}{n}\).
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1 Introduction
In this note, we study regularity properties of local minimizers of integral functionals
where \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 3\), is a bounded domain, \(u:\varOmega \rightarrow {\mathbb {R}}^N\), \(N\ge 1\) and \(f:{\mathbb {R}}^{N\times n}\rightarrow {\mathbb {R}}\) is a sufficiently smooth integrand satisfying (p, q)-growth of the form
Assumption 1
There exist \(0<\nu \le L<\infty \) such that \(f\in C^2({\mathbb {R}}^{N\times n})\) satisfies for all \(z,\xi \in {\mathbb {R}}^{N\times n}\)
Regularity properties of local minimizers of (1) in the case \(p=q\) are classical, see, e.g., [24]. A systematic regularity theory in the case \(p<q\) was initiated by Marcellini in [27, 28], see [31] for an overview (for a more up-to-date overview see the introduction in [30]). In particular, Marcellini [29] proves (among other things):
-
(A)
If \(N=1\), \(2\le p<q\) and \(\frac{q}{p}<1+\frac{2}{n}\), then every local minimizer \(u\in W_\mathrm{loc}^{1,p}(\varOmega )\) of (1) satisfies \(u\in W_\mathrm{loc}^{1,\infty }(\varOmega )\).
Local boundedness of the gradient implies that the non-standard growth of f and \(\partial ^2f\) in (1) becomes irrelevant and higher regularity (depending on the smoothness of f) follows by standard arguments, see e.g. [27, Chapter 7].
Only very recently, Bella and the author improved in [6] the result (A) in the sense that ’n’ in the assumption on the ratio \(\frac{q}{p}\) can be replaced by ’\(n-1\)’ for \(n\ge 3\) (to be precise, [6] considers the non-degenerate version (4) of (2)). The argument in [6] relies on scalar techniques, e.g., Moser-iteration type arguments, and thus cannot be extended to the vectorial case \(N>1\).
For the vectorial case \(N>1\), Esposito, Leonetti and Mingione showed in [18] that
-
(B)
If \(2\le p<q\) and \(\frac{q}{p}<1+\frac{2}{n}\), then every local minimizer \(u\in W_\mathrm{loc}^{1,p}(\varOmega ,{\mathbb {R}}^N)\) of (1) satisfies \(u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\).
To the best of our knowledge, there is no improvement of (B) with respect to the relation between the exponents p, q and the dimension n available in the literature. Here we provide such an improvement for \(n\ge 3\).
Before we state the results, we recall a standard notion of local minimizer in the context of functionals with (p, q)-growth
Definition 1
We call \(u\in W_\mathrm{loc}^{1,1}(\varOmega )\) a local minimizer of \({\mathcal {F}}\) given in (1) iff
and
for any \(\varphi \in W^{1,1}(\varOmega ,{\mathbb {R}}^N)\) satisfying \(\mathrm{supp}\;\varphi \Subset \varOmega \).
The main result of the present paper is
Theorem 2
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 3\), and suppose Assumption 1 is satisfied with \(2\le p<q<\infty \) such that
Let \(u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Then, \(u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\).
Higher gradient integrability is a first step in the regularity theory for integral functionals with (p, q)-growth, see [7, 11, 19, 20] for further higher integrability results under (p, q)-conditions. Clearly, we cannot expect to improve from \(W_\mathrm{loc}^{1,q}\) to \(W_\mathrm{loc}^{1,\infty }\) for \(N>1\), since this even fails in the classic setting \(p=q\), see [34]. Direct consequences of Theorem 2 are higher differentiability and a further improvement in gradient integrability in the form:
-
(i)
(Higher differentiability). In the situation of Theorem 2 it holds \(|\nabla u|^\frac{p-2}{2} \nabla u\in W^{1,2}_\mathrm{loc}(\varOmega )\), see Theorem 5.
-
(ii)
(Higher integrability). Sobolev inequality and (i) imply \(\nabla u\in L^{\kappa p}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^{N\times n})\) with \(\kappa =\frac{n}{n-2}\). Note that \(\kappa p>q\) provided \(\frac{q}{p}<1+\frac{2}{n-2}\).
A further, on first glance less direct, consequence of Theorem 2 is partial regularity of minimizers of (1), see, e.g., [1, 7, 10, 32], for partial regularity results under (p, q)-conditons. For this, we slightly strengthen the assumptions on the integrand and suppose
Assumption 3
There exist \(0<\nu \le L<\infty \) such that \(f\in C^2({\mathbb {R}}^{N\times n})\) satisfies for all \(z,\xi \in {\mathbb {R}}^{N\times n}\)
In [7], Bildhauer and Fuchs prove partial regularity under Assumption 3 with \(\frac{q}{p}<1+\frac{2}{n}\) ( [7] contains also more general conditions including, e.g., the subquadratic case). Here we show
Theorem 4
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 3\), and suppose Assumption 3 is satisfied with \(2\le p<q<\infty \) such that (3). Let \(u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Then, there exists an open set \(\varOmega _0\subset \varOmega \) with \(|\varOmega \setminus \varOmega _0|=0\) such that \(\nabla u\in C^{0,\alpha }(\varOmega _0,{\mathbb {R}}^{N\times n})\) for each \(0<\alpha <1\).
We do not know if (3) in Theorems 2 and 4 is optimal. Classic counterexamples in the scalar case \(N=1\), see, e.g., [23, 28], show that local boundedness of minimizers can fail if \(\frac{q}{p}\) is to large depending on the dimension n. In fact, [28, Theorem 6.1] and the recent boundedness result [26] show that \(\frac{1}{p}-\frac{1}{q}\le \frac{1}{n-1}\) is the sharp condition ensuring local boundedness in the scalar case \(N=1\) (for sharp results under additional structure assumptions, see, e.g., [14, 22]).
For non-autonomous functionals, i.e., \(\int _\varOmega f(x,Du)\,dx\), rather precise sufficiently & necessary conditions are established in [20], where the conditions on p, q and n has to be balanced with the (Hölder)-regularity in space of the integrand. However, if the integrand is sufficiently smooth in space, the regularity theory in the non-autonomous case essentially coincides with the autonomous case, see [10]. Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g. p(x)-Laplacian or double phase functionals are a very active field of research, see, e.g., [2, 12, 13, 15,16,17, 25, 33].
Coming back to autonomous integral functionals: In [11] higher gradient integrability is proven assuming so-called ’natural’ growth conditions, i.e., no upper bound assumption on \(\partial ^2f\), under the relation \(\frac{q}{p}<1+\frac{1}{n-1}\). Moreover, in two dimensions we cannot improve the previous results on higher differentiability and partial regularity of, e.g., [7, 18], see [8] for a full regularity result under Assumption 3 with \(n=2\) and \(\frac{q}{p}<2\). Finally, we mention the recent paper [3] where optimal Lipschitz-estimates with respect to a right-hand side are proven for functionals with (p, q)-growth.
Let us briefly describe the main idea in the proof of Theorem 2 and from where our improvement compared to earlier results comes from. The main point is to obtain suitable a priori estimates for minimizers that may already be in \(W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\). The claim then follows by a known regularization and approximation procedure, see, e.g., [18]. For minimizers \(v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\) a Caccioppoli-type inequality
is valid for all sufficiently smooth cut-off functions \(\eta \), see Lemma 1. Very formally, the Caccioppoli inequality (5) can be combined with Sobolev inequality and a simple interpolation inequality to obtain
where \(\theta =\frac{\frac{1}{p}-\frac{1}{q}}{\frac{1}{p}-\frac{1}{\kappa p}}\in (0,1)\) and \(\kappa =\frac{n}{n-2}\). The \(\Vert Dv\Vert _{L^{\kappa p}}\)-factor on the right-hand side can be absorbed provided we have \(\frac{q\theta }{p}<1\), but this is precisely the ’old’ (p, q)-condition \(\frac{q}{p}<1+\frac{2}{n}\), this type of argument was previously rigorously implemented in, e.g., [7, 19]. Our improvement comes from choosing a cut-of function \(\eta \) in (5) that is optimized with respect to v, which enables us to use Sobolev inequality on \(n-1\)-dimensional spheres wich gives the desired improvement, see Sect. 3. This idea has its origin in joint works with Bella [4, 5] on linear non-uniformly elliptic equations.
With Theorem 2 at hand, we can follows the arguments of [7] almost verbatim to prove Theorem 4. In Sect. 4, we sketch (following [7]) a corresponding \(\varepsilon \)-regularity result from which Theorem 4 follows by standard methods.
2 Preliminary results
In this section, we gather some known facts. We begin with a well-known higher differentiability result for minimizers of (1) under the assumption that \(u\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\):
Lemma 1
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 2\), and suppose Assumption 1 is satisfied with \(2\le p<q<\infty \). Let \(v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Then, \(|Dv|^\frac{p-2}{2}Dv\in W_\mathrm{loc}^{1,2}(\varOmega ,{\mathbb {R}}^{N\times n})\) and there exists \(c=c(\frac{L}{\nu },n,N,p,q)\in [1,\infty )\) such that for every \(Q\in {\mathbb {R}}^{N\times n}\) and every \(\eta \in C_c^1(\varOmega )\)
The Lemma 1 is known, see e.g. [7, 18, 28]. Since we did not find a precise reference for estimate (6), we included a prove here following essentially the argument of [18].
Proof of Lemma 1
Without loss of generality, we suppose \(\nu =1\) the general case \(\nu >0\) follows by replacing f with \(f/\nu \) (and thus L with \(L/\nu \)). Throughout the proof, we write \(\lesssim \) if \(\le \) holds up to a multiplicative constant depending only on n, N, p and q.
Thanks to the assumption \(v\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\), the minimizer v satisfies the Euler-Largrange equation
(for this we use that the convexity and growth conditions of f imply \(|\partial f(z)|\le c(1+|z|^{q-1})\) for some \(c=c(L,n,N,q,)<\infty \)). Next, we use the difference quotient method, to differentiate the above equation: For \(s\in \{1,\dots ,n\}\), we consider the difference quotient operator
Fix \(\eta \in C_c^1(\varOmega )\). Testing (7) with \(\varphi :=\tau _{s,-h}(\eta ^2(\tau _{s,h}(v-\ell _Q)))\in W_0^{1,q}(\varOmega )\), where \(\ell _Q(x)=Qx\), we obtain
Writing \(\tau _{s,h}\partial f(D v)=\frac{1}{h} \partial f(D v+th\tau _{s,h}D v)\big |_{t=0}^{t=1}\), the fundamental theorem of calculus yields
where we use \(\tau _{h,s}\ell _Q=Qe_s\). Youngs inequality yields
where
Combining (8), (9) with the assumptions on \(\partial ^2f\), see (2), with the elementary estimate
for \(h>0\) sufficiently small (see e.g. [18, Lemma 3.4]), we obtain
Estimate (10), the fact \(v\in W_\mathrm{loc}^{1,q}(\varOmega )\) and the arbitrariness of \(\eta \in C_c^1(\varOmega )\) and \(s\in \{1,\dots ,n\}\) yield \(|Dv|^{\frac{p-2}{2}}Dv\in W_\mathrm{loc}^{1,2}(\varOmega )\). Sending h to zero in (10), we obtain
the desired estimate (6) follows by summing over s. \(\square \)
Next, we state a higher differentiability result under the more restrictive Assumption 3 which will be used in the proof of Theorem 4.
Lemma 2
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 2\), and suppose Assumption 3 is satisfied with \(2\le p<q<\infty \). Let \(v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Then, \(h:=(1+|Dv|^2)^\frac{p}{4}\in W_\mathrm{loc}^{1,2}(\varOmega )\) and there exists \(c=c(\frac{L}{\nu },n,N,p,q)\in [1,\infty )\) such that for every \(Q\in {\mathbb {R}}^{N\times n}\)
A variation of Lemma 2 can be found in [7] and we only sketch the proof.
Proof of Lemma 2
With the same argument as in the proof of Lemma 1 but using (4) instead of (2), we obtain \(v\in W^{2,2}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\) and the Caccioppoli inequality
for all \(\eta \in C_c^1(\varOmega )\), where \(c=c(\frac{L}{\nu },n,N,p,q)<\infty \). Formally, the chain-rule implies
where \(c=c(n,p)<\infty \), and the claimed estimate (11) follows from (12) and (13). In general, we are not allowed to use the chain rule, but the above reasoning can be made rigorous: Consider a truncated version \(h_m\) of h, where \(h_m:=\varTheta _m(|Dv|)\) with
For \(h_m\) we are allowed to use the chain-rule and (12) together with (13) with h replaced by \(h_m\) imply (11) with h replaced by \(h_m\). The claimed estimate follows by taking the limit \(m\rightarrow \infty \), see [7, Proposition 3.2] for details. \(\square \)
The following technical lemma is contained in [6] (see also [4, proof of Lemma 2.1, Step 1]) and plays a key role in the proof of Theorem 2
Lemma 3
( [6], Lemma 3) Fix \(n\ge 2\). For given \(0<\rho<\sigma <\infty \) and \(v\in L^1(B_{\sigma })\), consider
Then for every \(\delta \in (0,1]\)
For convenience of the reader we include a short proof of Lemma 3
Proof of Lemma 3
Estimate (14) follows directly by minimizing among radial symmetric cut-off functions. Indeed, we obviously have for every \(\varepsilon \ge 0\)
For \(\varepsilon >0\), the one-dimensional minimization problem \(J_{\mathrm{1d},\varepsilon }\) can be solved explicitly and we obtain
To see (15), we observe that using the assumption \(v\in L^1(B_\sigma )\) and a simple approximation argument we can replace \(\eta \in C^1(\rho ,\sigma )\) with \(\eta \in W^{1,\infty }(\rho ,\sigma )\) in the definition of \(J_{\mathrm{1d},\varepsilon }\). Let \({{\widetilde{\eta }}}:[\rho ,\sigma ]\rightarrow [0,\infty )\) be given by
Clearly, \({{\widetilde{\eta }}}\in W^{1,\infty }(\rho ,\sigma )\) (since \(b\ge \varepsilon >0\)), \({{\widetilde{\eta }}}(\rho )=1\), \({{\widetilde{\eta }}}(\sigma )=0\), and thus
The reverse inequality follows by Hölder’s inequality. Next, we deduce (14) from (15): For every \(s>1\), we obtain by Hölder inequality \(\sigma -\rho =\int _\rho ^\sigma (\frac{b}{b})^\frac{s-1}{s}\le \left( \int _\rho ^\sigma b^{s-1}\right) ^\frac{1}{s}\left( \int _\rho ^\sigma \frac{1}{b}\right) ^\frac{s-1}{s}\) with b as above, and by (15) that
Sending \(\varepsilon \) to zero, we obtain (14) with \(\delta =s-1>0\). \(\square \)
3 Higher integrability - Proof of Theorem 2
In this section, we prove the following higher integrability and differentiability result which clearly contains Theorem 2
Theorem 5
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 2\), and suppose Assumption 1 is satisfied with \(2\le p<q<\infty \) such that \(\frac{q}{p}<1+\min \{\frac{2}{n-1},1\}\). Let \(u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Then, \(u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)\) and \(|Du|^\frac{p-2}{2}Du\in W_\mathrm{loc}^{1,2}(\varOmega ,{\mathbb {R}}^{N\times n})\). Moreover, for
there exists \(c=c(\frac{L}{\nu },n,N,p,q,\chi )\in [1,\infty )\) such that for every \(B_{R}(x_0)\Subset \varOmega \)
where
Proof of Theorem 5
Without loss of generality, we suppose \(\nu =1\) the general case \(\nu >0\) follows by replacing f with \(f/\nu \). Throughout the proof, we write \(\lesssim \) if \(\le \) holds up to a multiplicative constant depending only on L, n, N, p and q.
Following, e.g., [7, 18, 19], we consider the perturbed integral functionals
We then derive suitable a priori higher differentiability and integrability estimates for local minimizers of \({\mathcal {F}}_\lambda \) that are independent of \(\lambda \in (0,1)\). The claim then follows with help of a by now standard double approximation procedure in spirit of [18].
Step 1. One-step improvement.
Let \(v\in W^{1,1}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}_\lambda \) defined in (19), \(B_1\Subset \varOmega \), and let \(\chi >1\) be defined in (16). We claim that there exists \(c=c(L,n,N,p,q,\chi )\in [1,\infty )\) such that for all \(\frac{1}{2}\le \rho <\sigma \le 1\) and every \(\lambda \in (0,1]\)
with the understanding \(\frac{\infty }{\infty -1}=1\) and
The growth conditions of \(f_\lambda \) and the minimality of v imply \(v\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\) and thus by Lemma 1
for all \(\eta \in C_c^1(\varOmega )\). Estimate (21) follows directly from (22) for \(\eta \in C_c^1(B_\sigma )\) with \(0\le \eta \le 1\), \(\eta \equiv 1\) on \(B_\rho \) and \(|\nabla \eta |\le \frac{2}{\sigma -\rho }\), combined with \(|z|^q\le \frac{1}{\lambda }f_\lambda (z)\) and \(\lambda \in (0,1]\).
Hence, it is left to show (20). For this, we use a technical estimate which follows from Lemma 3 and Hölders inequality: For given \(0<\rho<\sigma <\infty \) and \(w\in L^q(B_{\sigma })\) it holds
where J is defined as in Lemma 3. We postpone the derivation of (23) to the end of this step.
Combining (22) with \((1+|Dv|^2)^\frac{q-2}{2} |D v|^2\le (1+|Dv|)^q\) and estimate (23) with \(w=1+|Dv|\), we obtain
Next, we use the Sobolev inequality on spheres to estimate the second factor on the right-hand side in (24): For \(n\ge 2\) there exists \(c=c(n,N,\chi )\in [1,\infty )\) such that for all \(r>0\)
Combining (25) with elementary estimates and assumption \(\frac{1}{2}\le \rho <\sigma \le 1\), we obtain
Combining (24) and estimate (26), we obtain
The claimed estimate (20) now follows since \(|z|^p\le f(z)\le f_\lambda (z)\), \(\frac{\chi }{\chi -1}(1-\frac{q}{\chi p}+\frac{q}{p}-1)=\frac{q}{p}\ge 1\) and \(\int _{B_1}1+f_\lambda (Dv)\,dx\ge |B_1|\).
Finally, we present the computations regarding (23): Lemma 3 yields
Using two times the Hölder inequality, we estimate
Inequality (23) follows with the admissible choice
which ensures \(\theta q\delta \frac{s}{s-1}=(1-\theta ) q\delta s=p\).
Step 2. Iteration.
We claim that there exists \(c=c(L,n,N,p,q,\chi )\in [1,\infty )\) such that
where \(\alpha \) is defined in (18). For \(k\in {\mathbb {N}}\cup \{0\}\), we set
Estimate (21) and the choice of \(\rho _k\) imply for \(\lambda \in (0,1]\)
From (20) we deduce the existence of \(c=c(L,n,N,p,q,\chi )\in [1,\infty )\) such that for every \(k\in {\mathbb {N}}\)
Assumption \(\frac{q}{p}<1+\min \{1,\frac{2}{n-1}\}\) and the choice of \(\chi \) yield
where we use for \(n=3\) that \(\chi {\mathop {>}\limits ^{(16)}}\frac{1}{2-\frac{q}{p}}>0\) and
Hence, iterating (29) we obtain (using the uniform bound (28) on \(J_k\) and \({\frac{\chi }{\chi -1}\frac{q-p}{p}}<1\))
and the claimed estimate (27) follow from
Step 3. Conclusion.
We assume \(B_1\Subset \varOmega \) and show that there exists \(c=c(L,n,N,p,q,\chi )\in [1,\infty )\)
where \(\alpha \) is given as in (18) above. Clearly, standard scaling, translation and covering arguments yield
for all \(B_R(x_0)\Subset \varOmega \) and \(c=c(L,n,N,p,q,\chi )\in [1,\infty )\). The claimed estimate (17) then follows from Lemma 1.
Following [18], we introduce in addition to \(\lambda \in (0,1)\) a second small parameter \(\varepsilon >0\) which is related to a suitable regularization of u. For \(\varepsilon \in (0,\varepsilon _0)\), where \(0<\varepsilon _0\le 1\) is such that \(B_{1+\varepsilon _0}\Subset \varOmega \), we set \(u_\varepsilon :=u*\varphi _\varepsilon \) with \(\varphi _\varepsilon :=\varepsilon ^{-n}\varphi (\frac{\cdot }{\varepsilon })\) and \(\varphi \) being a non-negative, radially symmetric mollifier, i.e. it satisfies
Given \(\varepsilon ,\lambda \in (0,\varepsilon _0)\), we denote by \(v_{\varepsilon ,\lambda }\in u_\varepsilon +W_0^{1,q}(B_1)\) the unique function satisfying
Combining Sobolev inequality with the assumption \(\frac{q}{p}<1+\frac{2}{n-2}\) and estimate (27), we have
where we used Jensen’s inequality and the convexity of f in the last step. Similarly,
Fix \(\varepsilon \in (0,\varepsilon _0)\). In view of (33) and (34), we find \(w_\varepsilon \in u_\varepsilon +W_0^{1,p}(B_1)\) such that as \(\lambda \rightarrow 0\), up to subsequence,
Hence, a combination of (33), (34) with the weak lower-semicontinuity of convex functionals yield
Since \(w_\varepsilon \in u_\varepsilon +W_0^{1,q}(B_1)\) and \(u_\varepsilon \rightarrow u\) in \(W^{1,p}(B_1)\), we find by (36) a function \(w\in u+W_0^{1,p}(B_1)\) such that, up to subsequence,
Appealing to the bounds (35), (36) and lower semicontinuity, we obtain
Inequality (38), strict convexity of f and the fact \(w\in u+W_0^{1,p}(B_1)\) imply \(w=u\) and thus the claimed estimate (31) is a consequence of (37). \(\square \)
4 Partial regularity - Proof of Theorem 4
Theorem 4 follows from, the higher integrability statement Theorem 2, the \(\varepsilon \)-regularity statement of Lemma 4 below and a well-known iteration argument.
Lemma 4
Let \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 3\), and suppose Assumption 3 is satisfied with \(2\le p<q<\infty \) such that \(\frac{q}{p}<1+\frac{2}{n-1}\). Fix \(M>0\). There exists \(C^*=C^*(n,N,p,q,\frac{L}{\nu },M)\in [1,\infty )\) such that for every \(\tau \in (0,\frac{1}{4})\) there exists \(\varepsilon =\varepsilon (M,\tau )>0\) such that the following is true: Let \(u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)\) be a local minimizer of the functional \({\mathcal {F}}\) given in (1). Suppose for some ball \(B_r(x)\Subset \varOmega \)
where we use the shorthand , and
then
With the higher integrability of Theorem 5 and the Caccioppoli inequality of Lemma 2 at hand, we can prove Lemma 4 following almost verbatim the proof of the corresponding result [7, Lemma 4.1], which contain the statement of Lemma 4 under the assumption \(\frac{q}{p}<1+\frac{2}{n}\) (note that in [7] somewhat more general growth conditions including also the case \(1<p<q\) are considered). Thus, we only sketch the argument.
Proof of Lemma 4
Fix \(M>0\). Suppose that Lemma 4 is wrong. Then there exists \(\tau \in (0,\frac{1}{4})\), a local minimizer \(u\in W^{1,1}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\), which in view of Theorem 2 satisfies \(u\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)\), and a sequence of balls \(B_{r_m}(x_m)\Subset B_R\) satisfying
where \(C^*\) is chosen below. We consider the sequence of rescaled functions given by
where \(a_m:=(u)_{x_m,r_m}\) and \(A_m:=(Du)_{x_m,r_m}\). Assumption (39) implies \(\sup _m|A_m|\le M\) and thus, up to subsequence,
The definition of \(v_m\) yields
Assumptions (39) and (40) imply
The bound (42) together with (41) imply the existence of \(v\in W^{1,2}(B_1,{\mathbb {R}}^N)\) such that, up to extracting a further subsequence,
The function v satisfies the linear equation with constant coefficients
see, e.g., [21] or [7, Proposition 4.2]. Standard estimates for linear elliptic systems with constant coefficients imply \(v\in C_\mathrm{loc}^\infty (B_1,{\mathbb {R}}^N)\) and existence of \(C^{**}<\infty \) depending only on n, N and the ellipticity contrast of \(\partial ^2 f(A)\) (and thus on \(\frac{L}{\nu },p,q,\) and M) such that
Choosing \(C^*=2C^{**}\) we obtain a contradiction between (43) and (44) provided we have as \(m\rightarrow \infty \)
Exanctly as in [7, Proposition 4.3] (with \(\mu =2-p\), see also [9, Section 3.4.3.2] for a more detailed presentation of the proof), we have for all \(\rho \in (0,1)\),
where \(w:=v_m-v\), and thus the local \(L^2\)-convergence (45) follows. It is left to prove (46). For this, we introduce for \(\rho \in (0,1)\) and \(T>0\) the sequence of subsets
The local Lipschitz regularity of v, \(q>2\) and (45) imply for all \(\rho \in (0,1)\) and \(T>0\)
where here and for the rest of the proof \(\lesssim \) means \(\le \) up to a multiplicative constant depending only on L, n, N, p and q. Hence, it is left to show that there exists \(T>0\) such that
As in [7], we introduce a sequence of auxiliary functions
which satisfy
Indeed, by Theorem 2 and Lemma 2, we have for every \(\rho \in (0,1)\) and every \(Q\in {\mathbb {R}}^{N\times n}\)
and thus by rescaling and setting \(Q=A_m\)
The identity \(\psi _m=\lambda _m^{-1}\int _0^1\frac{d}{dt}\varTheta (A_m+t\lambda _mv_m)\,dt\) with \(\varTheta (F):=(1+|F|^2)^\frac{p}{4}\) implies
(see [7, p. 555] for details) and thus with help of (47), we obtain
For T sufficiently large (depending on M) there exists \(c>0\) such that for all \(z\in B_\rho \setminus U_m(\rho ,T)\)
Estimate (48) and Sobolev embedding imply \(\limsup _{m\rightarrow \infty }\Vert \psi _m\Vert _{L^{\frac{2n}{n-2}}(B_\rho )}\lesssim c(\rho )\in [1,\infty )\). Hence, using assumption \(\frac{q}{p}<1+\frac{2}{n-1}\) (and thus \(\frac{2q}{p}<\frac{2n}{n-2}\)), we obtain for every \(\rho \in (0,1)\)
which finishes the proof. \(\square \)
References
Acerbi, E., Fusco, N.: Partial regularity under anisotropic \((p, q)\) growth conditions. J. Differ. Equ. 107, 46–67 (1994)
Baroni, P., Colombo, M., Mingione, G.: Regularity for general functionals with double phase. Calc. Var. Partial Differ. Equ. 57(2), 62 (2018)
Beck, L., Mingione, G.: Lipschitz bounds and non-uniform ellipticity. Commun. Pure Appl. Math. 73, 944–1034 (2020)
Bella, P., Schäffner, M.: Local Boundedness and Harnack Inequality for Solutions of Linear Nonuniformly Elliptic Equations. Comm. Pure Appl. Math. 74, 453–477 (2021)
Bella, P., Schäffner, M.: Quenched invariance principle for random walks among random degenerate conductances. Ann. Probab. 48(1), 296–316 (2020)
Bella, P., Schäffner, M.: On the regularity of minimizers for scalar integral functionals with \((p, q)\)-growth. Anal. PDE 13(7), 2241–2257 (2020)
Bildhauer, M., Fuchs, M.: Partial regularity for variational integrals with \((s,\mu, q)\)-growth. Calc. Var. Partial Differ. Equ. 13(4), 537–560 (2001)
Bildhauer, M., Fuchs, M.: Twodimensional anisotropic variational problems. Calc. Var. Partial Differ. Equ. 16, 177–186 (2003)
Bildhauer, M.: Convex Variational Problems. volume 1818 of Lecture Notes in Mathematics. Springer, Berlin (2003)
Breit, D.: Dominic, New regularity theorems for non-autonomous variational integrals with (p, q)-growth. Calc. Var. Partial Differ. Equ. 44(1–2), 101–129 (2012)
Carozza, M., Kristensen, J., Passarelli di Napoli, A.: Regularity of minimizers of autonomous convex variational integrals. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13(4), 1065–1089 (2014)
Chlebicka, I., De Filippis, C., Koch, L.: Boundary regularity for manifold constrained \(p(x)\)-Harmonic maps. arXiv:2001.06243 [math.AP]
Colombo, M., Mingione, G.: Regularity for double phase variational problems. Arch. Ration. Mech. Anal. 215(2), 443–496 (2015)
Cupini, G., Marcellini, P., Mascolo, E.: Local boundedness of minimizers with limit growth conditions. J. Optim. Theory Appl. 166, 1–22 (2015)
De Filippis, C., Mingione, G.: On the regularity of minima of non-autonomous functionals. J. Geom. Anal. 30(2), 1584–1626 (2020)
De Filippis, C.: Partial regularity for manifold constrained \(p(x)\)-harmonic maps. Calc. Var. Partial Differ. Equ.58(2) (2019), Paper No. 47
Eleuteri, M., Marcellini, P., Mascolo, E.: Regularity for scalar integrals without structure conditions. Adv. Calc. Var. 13(3), 279–300 (2020)
Esposito, L., Leonetti, F., Mingione, G.: Higher integrability for minimizers of integral functionals with \((p, q)\) growth. J. Differ. Equ. 157(2), 414–438 (1999)
Esposito, L., Leonetti, F., Mingione, G.: Regularity results for minimizers of irregular integrals with \((p, q)\) growth. Forum Math. 14(2), 245–272 (2002)
Esposito, L., Leonetti, F., Mingione, G.: Sharp regularity for functionals with \((p, q)\) growth. J. Differ. Equ. 204(1), 5–55 (2004)
Evans, L.C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95(3), 227–252 (1986)
Fusco, N., Sbordone, C.: Some remarks on the regularity of minima of anisotropic integrals. Commun. Partial Differ. Equ. 18, 153–167 (1993)
Giaquinta, M.: Growth conditions and regularity, a counterexample. Manuscr. Math. 59(2), 245–248 (1987)
Giusti, E.: Direct Methods in the Calculus of Variations, p. viii+403. World Scientific Publishing Co., Inc., River Edge, NJ (2003)
Harjulehto, P., Hästö, P., Toivanen, O.: Hölder regularity of quasiminimizers under generalized growth conditions. Calc. Var. Partial Differential Equations 56(2) (2017), Paper No. 22, pp 26
Hirsch, J., Schäffner, M.: Growth conditions and regularity, an optimal local boundedness result. Commun. Contemp. Math. (2020). https://doi.org/10.1142/S0219199720500297
Marcellini, P.: Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Rational Mech. Anal. 105(3), 267–284 (1989)
Marcellini, P.: Regularity and existence of solutions of elliptic equations with \(p, q\)-growth conditions. J. Differ. Equ. 90(1), 1–30 (1991)
Marcellini, P.: Regularity for some scalar variational problems under general growth conditions. J. Optim. Theory Appl. 90(1), 161–181 (1996)
Marcellini, P.: Growth conditions and regularity for weak solutions to nonlinear elliptic pdes. J. Math. Anal. Appl. (2020), (to appear)
Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)
Passarelli Di Napoli, A., Siepe, F.: A regularity result for a class of anisotropic systems. Rend. Ist. Mat. Univ. Trieste 28(1–2), 13–31 (1996)
Rǎdulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents. Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)
Sverák, V., Yan, X.: Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA 99, 15269–15276 (2002)
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