Abstract
We prove a full Harnack inequality for local minimizers, as well as weak solutions to nonlocal problems with non-standard growth. The main auxiliary results are local boundedness and a weak Harnack inequality for functions in a corresponding De Giorgi class. This paper builds upon a recent work on regularity estimates for such nonlocal problems by the same authors.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The goal of the present work is to prove a full Harnack inequality for local minimizers and weak solutions to a class of nonlocal problems which exhibit non-standard growth. This article builds upon the recent paper [13], in which we study regularity properties for local minimizers of nonlocal energy functionals, as well as weak solutions to nonlocal equations with non-standard growth. We prove that these objects satisfy a suitable fractional Caccioppoli inequality and therefore belong to corresponding De Giorgi classes. In this work, we show that any function in such De Giorgi class satisfies a full Harnack inequality. As a consequence, we obtain the full Harnack inequality for local minimizers and weak solutions.
Before we state the main result of this paper, let us formulate the main assumptions and briefly present the energy functionals respectively the nonlocal operators considered in this work. We point out that the setup of this article is in align with [13].
Let \(\Omega \subset \mathbb {R}^d\) be open, \(s \in (0,1)\) and \(1 \le p \le q\). Throughout the paper, let \(f: [0, \infty ) \rightarrow [0, \infty )\) be convex, strictly increasing and differentiable with \(f(0)=0\) and \(f(1)=1\). We say that f satisfies \((f_{p}^{q})\) if for all \(t \ge 0\):
The growth function f is naturally associated with nonlocal energy functionals and nonlocal operators. On the one hand, consider
where \(k: \mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}\) is a measurable function satisfying
for some \(\Lambda \ge 1\). In [13, Theorem 6.2], we prove that local minimizers of \({\mathcal {I}}_f\) belong to the De Giorgi class \(DG(\Omega ; q, c, s, f)\) for some constant \(c = c(d,q,\Lambda ) > 0\) if f satisfies (\(f^{q}\)) for some \(q > 1\). For the precise definition of the De Giorgi class, see Definition 2.8.
On the other hand, we consider weak solutions to
where \({\mathcal {L}}_{h}\) is a nonlocal operator of the form
Here, \(h : \mathbb {R}^d\times \mathbb {R}^d\times \mathbb {R}\rightarrow \mathbb {R}\) is a measurable function satisfying the structure condition
for a.e. \(x, y \in \mathbb {R}^d\) and for all \(t \in \mathbb {R}\). We show in [13, Theorem 7.3] that weak solutions to (1.2) are in \(DG(\Omega ; q, c, s, f)\) for some constant \(c = c(d,q,\Lambda ) > 0\) if (\(f^{q}\)) holds true for \(q > 1\).
The goal of this article is to prove a full Harnack inequality of the following form.
Theorem 1.1
Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(0< s_0 \le s < 1,\) \(1 < p \le q,\) \(c > 0\) and assume that f satisfies \((f_{p}^{q}).\) There exists a constant \(C > 0\) such that if \(u \in DG(\Omega ; q, c, s, f)\) is nonnegative in \(B_{R}(x_0) \subset \Omega ,\) then
The constant C depends only on d, \(s_0\), p, q and c.
The Harnack inequality was originally proved for harmonic functions and later obtained for several elliptic and parabolic local operators. It is known to have important consequences such as a priori estimates in Hölder spaces or convergence theorems. Therefore, it plays an important role in several mathematical fields such as geometric analysis, probability or analysis of partial differential equations. For an introduction to Harnack inequalities, their history and consequences, we refer the reader to the article by Kassmann [26]. The appearance of the tail term on the right-hand side of (1.3) is a purely nonlocal phenomenon. It is shown in [25] that the classical Harnack inequality fails for s-harmonic functions if nonnegativity of the function is assumed in the solution domain only. In [27], a new formulation of the Harnack inequality is introduced. It involves a nonlocal tail term as in (1.3) which captures the negative values of the s-harmonic function outside the solution domain. In our setup the nonlocal tail is of the following form
see Sect. 2.2 for details.
Further important contributions to investigation of Harnack inequalities for nonlocal operators are, among others, the articles [2, 3, 5, 7, 10, 12, 14,15,16,17, 31,32,33] and the references therein.
Since local minimizers of (1.1) belong to the De Giorgi class, we have the following corollary of Theorem 1.1, that is the full Harnack inequality for local minimizers.
Corollary 1.2
Let \(s_0 \in (0,1),\) \(1 < p \le q,\) \(\Lambda \ge 1\) and assume \(s \in [s_0, 1).\) Assume that f satisfies \((f_{p}^{q})\) and let \(k: \mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}\) be a measurable function satisfying (K). There exists a constant \(C > 0,\) depending only on d, \(s_0,\) p, q and \(\Lambda ,\) such that if \(u \in V^{s,f}(\Omega | \mathbb {R}^d)\) is a local minimizer of (1.1) that is nonnegative in \(B_{R}(x_0) \subset \Omega ,\) then the full Harnack inequality (1.3) holds true for u.
Another direct consequence of Theorem 1.1, together with the observation that weak solutions belong to the De Giorgi class, is the full Harnack inequality for weak solutions.
Corollary 1.3
Let \(s_0 \in (0,1),\) \(1 < p \le q,\) \(\Lambda \ge 1\) and assume \(s \in [s_0, 1).\) Assume that f satisfies \((f_{p}^{q})\) and let \(h: \mathbb {R}^d\times \mathbb {R}^d\times \mathbb {R}\rightarrow \mathbb {R}\) be a measurable function satisfying (h). There exists a constant \(C > 0,\) depending only on d, \(s_0,\) p, q and \(\Lambda ,\) such that if \(u \in V^{s,f}(\Omega | \mathbb {R}^d)\) is a weak solution to (1.2) that is nonnegative in \(B_{R}(x_0) \subset \Omega ,\) then the full Harnack inequality (1.3) holds true for u.
The proof of the main result Theorem 1.1 follows from a weak Harnack inequality together with the local boundedness of functions in the De Giorgi class. Our approach roughly follows the ideas of Mascolo and Papi [29], where they establish a Harnack inequality for minimizers of functionals with non-standard growth in the local case.
We would like to point out that all results in the present paper are robust in the sense that the constants stay uniform as \(s \rightarrow 1^{-}\), since they depend on \(s_0\) and are independent of the actual order of differentiability s. Since the tail contribution vanishes as \(s \rightarrow 1^{-}\), we recover purely local estimates in the limit case. Note that the nonlocal energy functional is known to converge to a local energy form as \(s \rightarrow 1^{-}\), see for instance [1, 8].
The family of operators studied in this paper exhibits non-standard growth behavior. The regularity theory has been intensively studied in recent years. In [13] and [4], local boundedness and local Hölder regularity are established for such operators using two different approaches and under slightly different conditions on the growth function. Both papers prove Hölder estimate under the condition \((f_{p}^{q})\) using similar approaches. While [13] establishes the local boundedness under the additional condition \(q < p^{*}\), in [4] such an estimate is proved without this restriction. Byun, Kim and Ok derive a Poincaré Sobolev-type inequality, which takes into account the specific growth of the functionals under consideration. However, their method is not robust as \(s \rightarrow 1^-\) in the aforementioned sense. In this paper, (see Theorem 3.1) we prove a robust estimate without any restriction on the exponents p and q, improving the corresponding results from [4, 13].
Recently, Fang and Zhang have investigated Harnack inequalities for nonlocal operators with general growth [20]. In comparison to our setup, they impose more restrictive structural assumptions on the growth function f. Similar to the approach in this article, they derive local boundedness and a tail estimate as in Theorem 3.1, Lemma 5.1, as well as a weak Harnack inequality. By combining these results, [20] derive an upper estimate for \(\sup u\) in terms of \(\inf u\) and a nonlocal tail term. However, for \(p < q\), this result is not optimal due to the appearance of an additional power \(\iota = q/p\) in the Harnack inequality. In this article, we prove a different version of a weak Harnack inequality taking into account the growth function f, see Theorem 4.1. This allows us to deduce a full Harnack inequality in the classical form (1.3).
For a deeper discussion on the literature about nonlocal operators with different types of non-standard growth behavior and their regularity theory, we refer the reader to the references given in those two articles. See also [6, 9, 11, 18, 19, 22,23,24, 30] and the references therein.
1.1 Notation
Throughout the paper, we will denote by \(C > 0\) a universal constant, which may be different from line to line.
1.2 Outline
This article is structured as follows. In Sect. 2 we collect several auxiliary results for the growth functions under consideration and provide definitions of related function spaces and De Giorgi classes. Sections 3 and 4 are devoted to the proof of local boundedness and a weak Harnack inequality for functions u in appropriate De Giorgi classes. Finally, the proof of the main result Theorem 1.1 is provided in Sect. 5.
2 Preliminaries
This section contains several auxiliary results on the growth function f and introduces the function spaces related to our setup.
2.1 Properties of growth functions
We collect several properties of growth functions \(f: [0, \infty ) \rightarrow [0, \infty )\) which were proved in [13] and will be used in the course of this article. Recall that we will assume throughout this paper that f is convex, strictly increasing and differentiable with \(f(0) = 0\) and \(f(1) = 1\).
Lemma 2.1
[13, Lemma 2.1] Let \(q \ge 1\). Then the following are equivalent :
-
(i)
\((f^{q}),\)
-
(ii)
\(t \mapsto t^{-q}f(t)\) is decreasing,
-
(iii)
\(f(\lambda t) \le \lambda ^q f(t)\) for all \(\lambda \ge 1,\)
-
(iv)
\(\lambda ^q f(t) \le f(\lambda t)\) for all \(\lambda \le 1.\)
Lemma 2.2
[13, Lemma 2.2] Let \(p \ge 1\). Then the following are equivalent :
-
(i)
\((f_{p}),\)
-
(ii)
\(t \mapsto t^{-p}f(t)\) is increasing,
-
(iii)
\(\lambda ^p f(t) \le f(\lambda t)\) for all \(\lambda \ge 1,\)
-
(iv)
\(f(\lambda t) \le \lambda ^p f(t)\) for all \(\lambda \le 1.\)
Corollary 2.3
[13, Corollary 2.4] Let \(1 \le p \le q.\) Assume that f satisfies \((f_{p}^{q}).\) Then,
Lemma 2.4
[13, Lemma 2.5] Let \(c > 1\) and assume that for some \(t,s > 0\) it holds that \(f(t) \le c f(s)\). Then \(t \le cs\).
Note that under the assumptions on f, it does not necessarily follow that \(f'\) is invertible. Throughout this article, we will work with the following generalized inverse of \(f'\):
We collect a few properties of \((f')^{-1}\). First, we recall a proposition from [13].
Proposition 2.5
[13, Proposition 3.1] It holds that
The following are simple consequences of the previous results.
Lemma 2.6
For every \(t,s \ge 0{:}\)
Proof
By (2.5), (2.6) and monotonicity of \((f')^{-1} \):
Lemma 2.7
Let \(1 < p \le q\). Assume that f satisfies \((f_{p}^{q})\). Then
where \(c_{\lambda } = (q\lambda /p)^{1/(p-1)}\) if \(\lambda \ge p/q\) and \(c_{\lambda } = (q\lambda /p)^{1/(q-1)}\) if \(\lambda \le p/q.\)
Proof
First, we observe that by (2.1) and (2.2), \(\lambda f'(t) \le f'(c_{\lambda } t)\). Therefore, using (2.5), (2.6) and monotonicity of \((f')^{-1} \):
\(\square \)
2.2 Function spaces and De Giorgi classes
Let \(s \in (0, 1)\) and \(\Omega \subset \mathbb {R}^d\) be open. We define the Orlicz and Orlicz–Sobolev spaces by
where \(\Phi _{L^f(\Omega )}\), \(\Phi _{W^{s, f}(\Omega )}\) and \(\Phi _{V^{s,f}(\Omega |\mathbb {R}^d)}\) are modulars defined by
Next, we define nonlocal tails, which capture the behavior of functions \(u\in V^{s, f}(\Omega |\mathbb {R}^d)\) at large scales. We define the nonlocal \(f'\)-Tail by
Recall that the function \(f'\) does not have to be invertible. Here \((f')^{-1} \) denotes the generalized inverse, see (2.4). In our previous work [13], we prove that this expression is finite if \(u\in V^{s, f}(\Omega |\mathbb {R}^d)\) for \(B_R(x_0)\subset \Omega \).
We are now ready to provide the definition of De Giorgi classes.
Definition 2.8
(De Giorgi classes) Let \(\Omega \) be an open subset in \(\mathbb {R}^d\). Let \(s \in (0,1)\), \(q > 1\) and \(c > 0\). We say that \(u \in DG_+(\Omega ; q, c, s, f)\) if \(u \in V^{s,f}(\Omega |\mathbb {R}^d)\) and if for every \(x_0 \in \Omega \), \(0< r < R \le d(x_0, \partial \Omega )\) and \(k \in \mathbb {R}\), it holds that
where \(w_{\pm }(x) = (u(x)- k)_{\pm }\) and \(A_{k}^{-} = \lbrace y \in \mathbb {R}^d: u(y) < k \rbrace \). We say that \(u \in DG_-(\Omega ; q, c, s, f)\) if (2.8) holds with \(w_+\), \(w_-\) and \(A_{k}^{-}\) replaced by \(w_-\), \(w_+\) and \(A_{k}^{+} = \lbrace y \in \mathbb {R}^d: u(y) > k \rbrace \), respectively. Moreover, we denote by \(DG(\Omega ; q, c, s, f) = DG_+(\Omega ; q, c, s, f) \cap DG_-(\Omega ; q, c, s, f)\).
3 Local boundedness
The goal of this section is to prove local boundedness of functions \(u \in DG_+(\Omega ; q, c, s,\! f)\) under (\(f^{q}\)), see Theorem 3.1. This result significantly improves [13, Theorem 5.1]. Let us mention that a similar estimate has been obtained in [4] using a different technique based on a Poincaré–Sobolev-type inequality for nonlocal Orlicz–Sobolev spaces. Our proof solely relies on the classical fractional Sobolev embedding and our estimate is robust for \(s \rightarrow 1^{-}\).
Theorem 3.1
Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(0< s_0 \le s < 1,\) \(q > 1,\) \(c > 0\) and assume that f satisfies \((f^{q}).\) There exists a constant \(C > 0\) such that if \(u \in DG_+(\Omega ; q, c, s, f),\) then for any \(B_R(x_0) \subset \Omega ,\) \(1/2 \le \rho < \tau \le 1\) and \(\delta \in (0,1),\) it holds that
where \(\gamma = 2d(d+q)/s_0.\) The constant C depends only on d, \(s_0,\) q and c.
Remark 3.2
In particular, Theorem 3.1 implies that functions \(u \in DG_+(\Omega ; q, c, s, f)\) are locally bounded from above in \(\Omega \) under the assumptions of Theorem 3.1. Local boundedness from below for functions \(u \in DG_-(\Omega ; q, c, s, f)\) can be proved in the same way. Finiteness of \({\mathrm {Tail}}_{f'}(u_+;x_0,R/2)\) is a consequence of \(u \in V^{s,f}(\Omega |\mathbb {R}^d)\), see [13, Proposition 3.2].
Proof
We may assume that \(x_0=0\). For \(j \ge 0\), we set
where k is a positive number that will be determined later. Note that \(R_{j+1} < R_j \le 2R_{j+1}\), \(k_j \le {\tilde{k}}_j \le k_{j+1}\) and \(w_{j+1} \le {\tilde{w}}_j \le w_j\). We denote by \(A^+_{h, r}\) the set \(\lbrace x \in B_r: u(x) > h \rbrace \).
Let \(\sigma = \max \lbrace s_0/2, (3s-1)/2 \rbrace \in (0, s)\). Then, it is easy to check that
First, by Hölder’s inequality we have
By applying the fractional Sobolev inequality to \({\tilde{w}}_j/R^s\) in \(B_{j+1}\), we estimate
Note that from monotonicity of \(f'\) and assumption (\(f^{q}\)) it follows:
for any \(a, b \in {\mathbb {R}}\). This inequality applied to \(a={\tilde{w}}_j(x)\), \(b={\tilde{w}}_j(y)\), together with (3.1), yields
We have thus far obtained
where the constant C depends only on d, \(s_0\) and q at this point.
In order to set up a suitable iteration scheme based on (3.2), it remains to estimate the quantity \(\Phi _{W^{s, f}(B_{j+1})}({\tilde{w}}_j)\). Since \(u \in DG_+(\Omega ; q, c, s, f)\), we have
For \(I_1\), we use Lemma 2.1 and the fact that \(R_j \le R\) to deduce
For \(I_2\), using monotonicity of \(f'\) and the assumption (\(f^{q}\)) again, we observe that
Thus, \(I_2\) can be estimated as
If we assume that for some \(\delta \in (0,1)\), \(k \ge k_1 := \delta 2^s {\mathrm {Tail}}_{f'}(u_+;0,R/2)\), it follows:
where we used (2.2), and \(C = C(d,s_0,q,c)\) is a positive constant. Combining (3.3), (3.4) and (3.5):
Since we have by Lemma 2.1
it follows from (3.2), (3.6) and (3.7) that
where \(b=2^{d+3q} > 1\), \(\beta =\sigma /d > 0\), \(C_0=C_0(d, s_0, q, c) > 1\) and
Let us take
where \(\gamma = 2d(d+q)/s_0\), \(C_1 = C_0^{2d/s_0} b^{4d^2/s_0^2}\) and \(k_1\) is as before. This choice provides
where we used that \(\sigma \ge s_0 /2\). Therefore, [28, Lemma 4.7] shows that \(Y_j \rightarrow 0\) as \(j \rightarrow \infty \), which concludes that \(u \le k\) a.e. in \(B_{\rho R}\). By monotonicity of f, it follows that \(f(u/R^s) \le f(k/R^s)\) a.e. in \(B_{\rho R}\), which implies the desired result due to Lemma 2.1 and since \(f(a+b) \le 2^{q}(f(a) + f(b))\). \(\square \)
The following result includes Theorem 3.1 as the special case \(\varepsilon = 1\). It is a direct consequence of Theorem 3.1 and a classical iteration argument.
Corollary 3.3
Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(0< s_0 \le s < 1,\) \(q > 1,\) \(c > 0,\) \(\varepsilon \in (0,1]\) and assume that f satisfies \((f^{q}).\) There exists a constant \(C > 0\) such that if \(u \in DG_+(\Omega ; q, c, s, f),\) then for any \(B_R(x_0) \subset \Omega \) and \(\delta \in (0,1),\) it holds that
where \(\mu = 2d(q-1)/(\varepsilon s_0)\). The constant C depends only on d, \(s_0,\) q, c and \(\varepsilon \).
Proof
We may assume that \(x_0 = 0\). Let \(1/2 \le \rho < \tau \le 1\), \(\delta _0 \in (0,1)\). By applying Theorem 3.1 with \(\delta _0\), we have
where \(C = C(d, s_0, q, c) > 0\), \(\gamma = 2d(d+q)/s_0\) and
Using Young’s inequality, we obtain
for some \(C = C(d, s_0, q, c, \varepsilon ) > 0\), where \(\mu = 2d(q-1)/(\varepsilon s_0)\). Combining (3.9) and (3.10):
for any \(1/2 \le \rho < \tau \le 1\). Therefore, by an iteration lemma, see [21, Lemma 1.1]:
Using the inequality \((a+b)^{\varepsilon } \le a^{\varepsilon } + b^{\varepsilon }\), we obtain
where \(C = C(d, s_0, q, c, \varepsilon ) > 1\). For a given \(\delta \in (0,1)\), the inequality (3.8) follows from (3.11) by setting \(\delta _0 = (\delta /C)^{1/\varepsilon } \in (0,1)\). \(\square \)
4 Weak Harnack inequality
The goal of this section is to prove a weak Harnack inequality for functions \(u \in DG_-(\Omega ; q, c, s, f)\). There exist several possible estimates in the literature, which go under the name “weak Harnack inequality”. They all differ in the aspect that \(\inf u\) is estimated by different Lebesgue-norms of u. We will prove an estimate of the following type since it allows us to deduce a full Harnack inequality by combination with Corollary 3.3.
Theorem 4.1
Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(0< s_0 \le s < 1,\) \(1<p \le q,\) \(c > 0\) and assume that f satisfies \((f_{p}^{q}).\) There exist constants \(C > 0\) and \(\varepsilon \in (0,1)\) such that if \(u \in DG_-(\Omega ; q, c, s, f)\) is nonnegative in \(B_{R}(x_0) \subset \Omega ,\) then
The constants C and \(\varepsilon \) depend only on d, \(s_0,\) q and c.
Before we give the proof of Theorem 4.1, we recall the following growth lemma from [13]:
Lemma 4.2
[13, Theorem 4.1] Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(1<p\le q,\) \(c, H > 0,\) \(R>0,\) \(s_0\in (0,1)\) and assume \(s \in [s_0,1).\) Assume that f satisfies \((f_{p}^{q})\). Suppose that \(B_{4R} = B_{4R}(x_0) \subset \Omega \). Let \(u \in DG_-(\Omega ; q, c, s, g)\) satisfy \(u \ge 0\) in \(B_{4R}\) and
for some \(\gamma \in (0,1)\). There exists \(\delta \in (0,1)\) such that if
then
The constant \(\delta \) depends only on d, \(s_0,\) p, q, c and \(\gamma .\)
Proof of Theorem 4.1
Without loss of generality, we assume that \(x_0 = 0\). Let us define
First of all, we claim that for any \(H > 0\) it holds:
Here, \(a = \frac{\log \frac{1}{2}}{\log \delta }\), where \(\delta \in (0,1)\) is the constant from Lemma 4.2 applied with \(\gamma = \frac{6^{-d}}{2}\) and \(H := t\). The proof of (4.1) is a well-known consequence of Lemma 4.2 and a covering argument due to Krylov and Safonov. It is explained in detail in [14, Lemma 6.7 and (6.41)] and can be adapted to our setup without any changes being necessary.
Let us explain how to deduce the desired result from (4.1). We choose \(\varepsilon = \frac{1}{2}\min (1,\frac{a}{q})\) and compute by Cavalieri’s principle and performing a change of variables
For \(I_1\), by a change of variables,
For \(I_2\), we apply (\(f^{q}\)) and obtain
From integration by parts and again (\(f^{q}\)), we see that
where we used the definition of \(\varepsilon \) and Lemma 2.1(ii) in the last step. It follows that
which yields, upon combining (4.2) and (4.3):
where we used that \(f(a+b) \le 2^q(f(a) + f(b))\) and \((a+b)^{\varepsilon } \le a^{\varepsilon } + b^{\varepsilon }\). The desired result follows by noticing that
which is a direct consequence of Lemma 2.7 applied with \(\lambda = 8^s\). \(\square \)
5 Harnack inequality
In this section, we prove our main result Theorem 1.1. First, we prove the following estimate for \({\mathrm {Tail}}_{f'}(u_+;x_0,R)\).
Lemma 5.1
Let \(\Omega \) be an open subset in \(\mathbb {R}^d.\) Let \(0< s_0 \le s < 1,\) \(1 < p \le q,\) \(c > 0\) and assume that f satisfies \((f_{p}^{q}).\) There exists a constant \(C > 0\) such that if \(u \in DG_-(\Omega ; q, c, s, f)\) is nonnegative in \(B_{R}(x_0) \subset \Omega ,\) then
The constant C depends on d, \(s_0,\) p, q and c.
Proof
Without loss of generality, we may assume \(x_0=0\). Let \(w = u - 2M\), where \(M = \sup _{B_{R/2}} u\). By \(u \in DG_-(\Omega ;q,c,s,f)\):
Note that due to (\(f_{p}\)) it holds \(f'(0) = 0\). This allows us to consider the integral over \(\mathbb {R}^d\) for the term on the left-hand side. Since \(|x-y| \le 2|y|\) for every \(x \in B_{R/4}\), \(y \in B_{R/2}^c\) and by (2.3), we estimate the first term from below by
Note that by monotonicity of \(f'\) and (2.1)
Furthermore, the terms on the right-hand side of (5.1) can be estimated from above by:
using similar arguments. Altogether, we obtain
where we used (2.1) in the last step. Next, we apply \((f')^{-1}\) on both sides of the estimate and multiply with \((R/2)^s\) to obtain
where we applied Lemma 2.6, (2.6) and Lemma 2.7. \(\square \)
We are now ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1
We may assume that \(x_0=0\). Let \(\varepsilon \in (0,1)\) be the constant from Theorem 4.1. By Corollary 3.3 and Lemma 5.1, we have
where \(\mu = 2d(q-1)/(\varepsilon s_0)\). Using (\(f^{q}\)), Lemma 2.1 and \(u\ge 0\) in \(B_R\), we obtain
By taking \(\delta \) sufficiently small so that \(C \delta < 1/2\), we have
Thus, it follows from Theorem 4.1 that
The desired inequality follows by using \((a+b)^{\frac{1}{\varepsilon }} \le 2^{\frac{1}{\varepsilon }-1}(a^{\frac{1}{\varepsilon }} + b^{\frac{1}{\varepsilon }})\), as well as the estimate \(f(a) + f(b) \le 2 f(a+b)\) and Lemma 2.4. \(\square \)
Data availability statement
Data sharing is not applicable to this article as no data sets were generated or analysed.
References
Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: On fractional Orlicz–Sobolev spaces. Anal. Math. Phys. 11(2), Paper No. 84, 21 (2021)
Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963–1999 (2009)
Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837–850 (2005)
Byun, S.-S., Kim, H., Ok, J.: Local Hölder continuity for fractional nonlocal equations with general growth (2021). arXiv:2112.13958
Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002)
Byun, S.-S., Ok, J., Song, K.: Hölder regularity for weak solutions to nonlocal double phase problems (2021). arXiv:2108.09623
Bogdan, K., Sztonyk, P.: Harnack’s inequality for stable Lévy processes. Potential Anal. 22(2), 133–150 (2005)
Bonder, J.F., Salort, A.M.: Fractional order Orlicz–Sobolev spaces. J. Funct. Anal. 277(2), 333–367 (2019)
Bonder, J.F., Salort, A., Vivas, H.: Interior and up to the boundary regularity for the fractional \(g\)-Laplacian: the convex case (2020). arXiv:2008.05543
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Process. Appl. 108(1), 27–62 (2003)
Chaker, J., Kim, M.: Local regularity for nonlocal equations with variable exponents (2021). arXiv:2107.06043
Chen, Z.-Q., Kumagai, T., Wang, J.: Elliptic Harnack inequalities for symmetric non-local Dirichlet forms. J. Math. Pures Appl. 9(125), 1–42 (2019)
Chaker, J., Kim, M., Weidner, M.: Regularity for nonlocal problems with non-standard growth (2021). arXiv:2111.09182
Cozzi, M.: Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes. J. Funct. Anal. 272(11), 4762–4837 (2017)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)
Di Castro, A., Kuusi, T., Palatucci, G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)
De Filippis, C., Palatucci, G.: Hölder regularity for nonlocal double phase equations. J. Differ. Equ. 267(1), 547–586 (2019)
Fang, Y., Zhang, C.: On weak and viscosity solutions of nonlocal double phase equations. Int. Math. Res. Not. (2021)
Fang, Y., Zhang, C.: Harnack inequality for the nonlocal equations with general growth (2022). arXiv:2201.09495
Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)
Goel, D., Kumar, D., Sreenadh, K.: Regularity and multiplicity results for fractional \((p,q)\)-Laplacian equations. Commun. Contemp. Math. 22(8), 1950065, 37 (2020)
Giacomoni, J., Kumar, D., Sreenadh, K.: Interior and boundary regularity results for strongly nonhomogeneous \(p\), \(q\)-fractional problems. Adv. Calc. Var. (2021)
Giacomoni, J., Kumar, D., Sreenadh, K.: Global regularity results for non-homogeneous growth fractional problems. J. Geom. Anal. 32(1), Paper No. 36, 41 (2022)
Kassmann, M.: The classical Harnack inequality fails for non-local operators. SFB 611-Preprint, No. 360 (2007)
Kassmann, M.: Harnack inequalities: an introduction. Bound. Value Probl. 2007, 1–21 (2007). https://doi.org/10.1155/2007/81415
Kassmann, M.: A new formulation of Harnack’s inequality for nonlocal operators. C. R. Math. Acad. Sci. Paris 349(11–12), 637–640 (2011)
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York. Translated from the Russian by Scripta Technica, Inc, Translation editor: Leon Ehrenpreis (1968)
Mascolo, E., Papi, G.: Harnack inequality for minimizers of integral functionals with general growth conditions. NoDEA Nonlinear Differ. Equ. Appl. 3(2), 231–244 (1996)
Ok, J.: Local Hölder regularity for nonlocal equations with variable powers (2021). arXiv:2107.06611
Strömqvist, M.: Harnack’s inequality for parabolic nonlocal equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36(6), 1709–1745 (2019)
Song, R., Vondraček, Z.: Harnack inequality for some classes of Markov processes. Math. Z. 246(1–2), 177–202 (2004)
Stinga, P.R., Zhang, C.: Harnack’s inequality for fractional nonlocal equations. Discret. Contin. Dyn. Syst. 33(7), 3153–3170 (2013)
Funding
Open Access funding enabled and organized by Projekt DEAL. Jamil Chaker gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (SFB 1283/2 2021-317210226). Minhyun Kim and Marvin Weidner gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (GRK 2235/2 2021-282638148).
Author information
Authors and Affiliations
Contributions
All authors developed and discussed the results and contributed to the final manuscript.
Corresponding author
Ethics declarations
Conflict of Interest
All authors state no conflict of interest.
Informed Consent
Informed consent has been obtained from all individuals included in this research work.
Additional information
Communicated by Y. Giga.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chaker, J., Kim, M. & Weidner, M. Harnack inequality for nonlocal problems with non-standard growth. Math. Ann. 386, 533–550 (2023). https://doi.org/10.1007/s00208-022-02405-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-022-02405-9