Abstract
We study the fractional p-Laplace equation
for \(0<s<1\) and in the subquadratic case \(1<p<2\). We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when \(p\ge 2\). The arguments are based on a careful Moser-type iteration and a perturbation argument.
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1 Introduction
We study the local regularity of solutions of the nonlinear and nonlocal equation
where
is the fractional p-Laplace operator, where P.V. denotes the principal value. It arises as the first variation of the Sobolev–Slobodeckiĭ seminorm for \(W^{s,p}({\mathbb {R}}^N)\), that is, as the first variation of the functional
This operator has generated vast activities in recent years. The main contribution of our work is to provide Hölder regularity of weak solutions of equation (1.1), with an explicit Hölder exponent. This is done in Theorems 1.1 and 1.2. Our results complement the existing results for the superquadratic case, \(p\ge 2\), obtained in [4]. To the best of our knowledge, this is the first result with an explicit Hölder exponent in the subquadratic case, \(1<p<2,\) even for the homogeneous equation. We seize the moment to mention that we can verify that whenever
the Hölder exponent \(\Theta \) obtained in the inhomogeneous setting for \(f\in L^q\), that is
is the sharp one, see Sect. 1.2 below. Hence, our result is sharp under this assumptions.
1.1 Main results
We now present the main results of the paper. For the details regarding the notation used in the two theorems below, such as \(\textrm{Tail}_{p-1,s\,p}\), we refer to Sect. 2.
Theorem 1.1
(Almost \(sp/(p-1)\)-regularity) Let \(\Omega \subset {\mathbb {R}}^N\) be a bounded and open set and assume that \(1<p<2\) and \(0<s<1\). Suppose \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of
Then \(u\in C^{\Gamma -\varepsilon }_{\textrm{loc}}(\Omega )\) for every \(\varepsilon \in (0,\Gamma )\), where
In particular, for every \(\varepsilon \in (0,\Gamma )\) and for every ball \(B_{2R}(x_0)\Subset \Omega \), there exist constants \(\sigma =\sigma (N,s,p,\varepsilon )\in (0,1)\) and \(C=C(N,s,p,\varepsilon )>0\) such that
Theorem 1.2
Let \(\Omega \subset {\mathbb {R}}^N\) be a bounded and open set and assume that \(1<p<2\) and \(0<s<1\). Suppose \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of
where \(f\in L^q_\text {loc}(\Omega )\) with
Let
Then \(u\in C^{\Theta -\varepsilon }_{\textrm{loc}}(\Omega )\) for every \(\varepsilon \in (0,\Theta )\).
In particular, for every \(\varepsilon \in (0,\Theta )\) and for every ball \(B_{4R}(x_0)\Subset \Omega \), there exists a constant \(C=C(N,s,p,q,\varepsilon )>0\) such that
Remark 1.3
It is worth mentioning that in the case when \(q<\infty \) and \(\Theta =(sp-N/q)/(p-1)\), a careful inspection of the proof of Theorem 1.2 reveals that we obtain the stronger result that \(u\in C^{\Theta }_{\textrm{loc}}(\Omega )\), with a similar estimate.
1.2 Comments on the results
We now discuss the sharpness of our results, in particular Theorem 1.2. Choose N, p, q such that
and pick \(s\in [N/q,1-1/p]\). Then it follows that \(sp <N\), \(q >N/sp\), \(sp\le p-1\) and that \((sp-N/q)/(p-1)\ge s\).
Define for some \(\varepsilon >0\) the function
By the assumptions, \(u\in C^{s+\varepsilon }_{\text {loc}}({\mathbb {R}}^N)\cap W^{s,p}_{\text {loc}}({\mathbb {R}}^N)\cap L_{sp}^{p-1}({\mathbb {R}}^N)\). In addition, by homogeneity and radial symmetry it follows that
where \(f\in L^q_{\text {loc}}({\mathbb {R}}^N)\) if and only if \(\varepsilon >0\). It is clear that \(u\not \in C^{\alpha }(B_1)\) for any \(\alpha >\gamma +\varepsilon \). Therefore, the result is sharp in this region of parameters. Now we comment on the assumptions on q and p in Theorem 1.2. We believe that they are sharp and they do perfectly match the sharp assumptions in the local limit. Indeed, in the local case, that should correspond to the limiting case \(s=1\), the assumptions become \(q>N/p\) when \(p\le N\) and \(q\ge 1\) when \(p>N\). These are the proper conditions for the inhomogeneous p-Laplace equation, see [25, 26].
1.3 Known results
The first appearance of equations similar to the fractional p-Laplacian that we are aware of is in [16]. There existence, uniqueness, and the convergence to the \(p-\)Laplace equation as s goes to 1, are proved in the viscosity setting. The starting point of the regularity theory was [11], where the local Hölder regularity was proved, using a nonlocal De Giorgi-type method. See also [10], for a related Harnack inequality. The paper [5] contains several useful regularity estimates for the inhomogeneous equation.
The literature on related Hölder regularity results is vast and we only mention a fraction. A local regularity result using viscosity methods was obtained in [21]. In [9], nonlocal analogues of the De Giorgi classes are introduced and used to prove regularity results in a very general setting. We also seize the opportunity to mention that fractional De Giorgi classes has been used in the context of local equations in [23].
The regularity up to the boundary has been studied in [14, 15]. Basic Hölder regularity up to the boundary is proved for general p and for \(p\ge 2\) finer regularity results up to the boundary are obtained.
In terms of regularity for the inhomogeneous equation, we mention the papers [4, 12, 18]. In [18], the authors study the regularity for equations of the type (1.1) with a right hand side f belonging to a Lorentz space. Sharp results for when u is continuous are obtained. The paper [4] is the counterpart of the present paper in the superquadratic case. In [12], these results are improved and the authors obtain sharp Hölder regularity results when \(p\ge 2\) and when the right hand side belongs to a Marcinkiewicz space.
We stress that for the subquadratic case \(1<p<2\), none of the above mentioned papers include an explicit Hölder exponent.
In addition to Hölder regularity, there has been quite some development of results in terms of higher Sobolev differentiability. In the linear case \(p=2\), see [1, 2, 8, 19] and [20], where the results are valid for more general kernels. For a general p, this has been studied in [3, 7, 12, 24].
We finally mention that the corresponding results for the p-Laplacian are well known. See for instance [25, 26].
1.4 Plan of the paper
In Sect. 2, we discuss notation, definitions and certain results in function spaces. The most important part of the paper is Sect. 3, where we prove Theorem 1.1, using a Moser-type argument that results in an improved differentiability that can be iterated in an unusual way. Following this, in Sect. 4, we treat the inhomogeneous equation, by means of a perturbation argument, using the regularity obtained for the homogeneous equation. Finally, in the Appendix, we include a list of pointwise inequalities that are used throughout the paper.
2 Preliminaries
In this section we present some auxiliary results needed in the rest of the paper.
2.1 Notation
Throughout the paper, we shall use the following notation: \(B_r(x_0)\) denotes the ball of radius r with center at \(x_0\). When \(x_0=0\), we write \(B_r(0):=B_r\). For a function u, we denote the positive and the negative part of u as \(u_{\pm }=\max \{\pm u,0\}\). The conjugate exponent \(\frac{l}{l-1}\) of \(l>1\) will be denoted by \(l'\). We write c or C to denote a positive constant which may vary from line to line or even in the same line. The dependencies on parameters are written in the parentheses.
For \(1<q<\infty \), we define the function \(J_q:{\mathbb {R}}\rightarrow {\mathbb {R}}\) by
and for \(0<s<1\) and \(1<p<\infty \) we use the notation
Moreover, for \(0\le \delta \le 1\), we use the notation
We also define
and
for functions \(\psi :{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) and \(h\in {\mathbb {R}}^N\). Note that the following discrete product rule holds:
2.2 Function spaces
It will be necessary to introduce two Besov-type spaces. For this reason, let \(1\le q<\infty \) and \(\psi \in L^q({\mathbb {R}}^N)\). For \(0<\beta \le 1\), define
and for \(0<\beta <2\), define
The Besov-type spaces \({\mathcal {N}}^{\beta ,q}_\infty \) and \({\mathcal {B}}^{\beta ,q}_\infty \) are defined by
and
The Sobolev-Slobodeckiĭ space is defined as
where the seminorm \([\,\cdot \,]_{W^{\beta ,q}({\mathbb {R}}^N)}\) is given by
The above spaces are endowed with the norms
and
We also introduce the space \(W^{\beta ,q}(\Omega )\) for a subset \(\Omega \subset {\mathbb {R}}^N\),
where naturally
It will also be convenient to use the following abuse of notation for the Sobolev exponent \(p_s^*\) related to the space \(W^{s,p}\): if \(sp<N\) then
and if \(sp>N\) then
2.3 Embedding inequalities
The following result can be found for example in [6, Lemma 2.3].
Lemma 2.1
The following embedding
is continuous, provided \(0<\beta <1\) and \(1\le q<\infty \). Moreover,
for every \(\psi \in {\mathcal {B}}_\infty ^{\beta ,q}({\mathbb {R}}^N)\), for some constant \(C=C(N,q)>0\).
We have the following embedding result from [4, Theorem 2.8].
Theorem 2.2
Let \(\psi \in {\mathcal {N}}_\infty ^{\beta ,q}({\mathbb {R}}^N)\), where \(0<\beta <1\) and \(1\le q<\infty \) such that \(\beta q>N\). Then for every \(0<\alpha <\beta -\frac{N}{q}\), we have \(\psi \in C^\alpha _{\textrm{loc}}({\mathbb {R}}^N)\). More precisely,
for some positive constant \(C=C(N,q,\alpha ,\beta )\) which blows up as \(\alpha \nearrow \beta -\frac{N}{q}\).
The following result follows from [4, Proposition 2.7].
Lemma 2.3
Let \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set. Assume that \(1<p<\infty \) and \(0<s<1\). Then
holds for every \(u\in W^{s,p}({\mathbb {R}}^N)\) such that \(u=0\) almost everywhere in \({\mathbb {R}}^N\setminus \Omega \), for some positive constant \(C=C(N,p,s)\).
2.4 Tail spaces and weak solutions
For a priori estimates, the so-called tail spaces that takes into account the global behavior are expedient. The tail space is defined as
and the global behavior of a function \(u\in L^q_{\alpha }({\mathbb {R}}^N)\) is measured by the quantity
Here \(x_0\in {\mathbb {R}}^N\), \(R>0,\,\beta >0\).
Definition 2.4
(Local weak solution) Suppose \(\Omega \subset {\mathbb {R}}^N\) is an open and bounded set. Assume that \(1<p<2\) and \(0<s<1\). Let \(f\in L_\text {loc}^q(\Omega )\) with \(q\ge (p_s^{*})'\) if \(sp\ne N\) and \(q>1\) if \(sp=N\). We define \(u\in W_{\textrm{loc}}^{s,p}(\Omega )\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) to be a local weak solution of \((-\Delta _p)^s u=f\) in \(\Omega \), if
for every compactly supported \(\phi \in W^{s,p}(\Omega )\), where \(J_p(t)=|t|^{p-2}t\) and \(d\mu =\frac{dx dy}{|x-y|^{N+ps}}\) are defined in (2.1) and (2.2) respectively.
Now we define the notion of weak solution for the Dirichlet problem associated with \((-\Delta _p)^s\). To this end, for given open and bounded sets \(\Omega \Subset \Omega '\subset {\mathbb {R}}^N\) and \(g\in L_{sp}^{p-1}({\mathbb {R}}^N)\), we define
Definition 2.5
(Dirichlet problem) Suppose \(\Omega \Subset \Omega '\subset {\mathbb {R}}^N\) are two open and bounded sets. Assume that \(1<p<2\) and \(0<s<1\). Let \(f\in L^q(\Omega )\) with \(q\ge (p_s^{*})'\) if \(sp\ne N\) and \(q>1\) if \(sp=N\) and \(g\in L^{p-1}_{sp}({\mathbb {R}}^N)\). We define \(u\in X_g^{s,p}(\Omega ,\Omega ')\) to be a weak solution of the boundary value problem
if for every \(\phi \in X_0^{s,p}(\Omega ,\Omega ')\), Eq. (2.4) holds.
By Proposition 2.12 in [4], there exists a unique weak solution of the Dirichlet problem (2.5) in the sense above, given \(g\in W^{s,p}(\Omega ')\cap L_{sp}^{p-1}({\mathbb {R}}^N)\).
3 The homogeneous equation
In this section, we treat the regularity for the homogeneous equation. This is done through an iteration scheme built on improved Besov-type regularity and improved Hölder regularity.
3.1 Improved Besov-type regularity
The starting point is the following improved Besov-type regularity.
Proposition 3.1
Let \(1<p <2\) and \(0<s<1\). Assume that \(u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of \((-\Delta _p)^s u=0\) in \(B_2\). Suppose that
for some \(\gamma \in [0,1)\). Moreover, suppose that for some \(\alpha \in [0,1)\), \(1\le q<\infty \) and \(0<h_0<\frac{1}{10}\), we have
Then for R such that \(4\,h_0<R\le 1-5\,h_0\), we have
for some positive constant \(C=C(N,s,p,q,h_0,\alpha ,\gamma )\) and \(C\nearrow +\infty \) as \(h_0\searrow 0\).
Proof
We divide the proof into three steps.
Step 1: Discrete differentiation of the equation. We set \( r=R-4h_0\) and recall \(\quad d\mu =\frac{dx dy}{|x-y|^{N+ps}}.\) Take \(\varphi \in W^{s,p}(B_R)\) vanishing outside \(B_{\frac{R+r}{2}}\). Let \(h\in {\mathbb {R}}^N\setminus \{0\}\) be such that \(|h|<h_0\). Testing (2.4) with \(\varphi _{-h}\) and performing a change of variable yields
In what follows, suppose \(\eta \in C^\infty _0(B_R)\) is such that
Testing (3.4) with
we get
We split the above double integral into three pieces:
and
where we used that \(\eta \) vanishes identically outside \(B_{(R+r)/2}\). Thus the Eq. (3.5) can be written as
We estimate \({\mathcal {I}}_j\) for \(j=1,2,3\) separately.
Estimate of \({\mathcal {I}}_1\). We observe that
Therefore, we have
Estimate of \(J_1\): We will now estimate the positive term. With the notation
we have by Lemma A.1 together with the fact that u is locally \(\gamma \)-Hölder continuous (recall (3.1))
Estimate of \(J_2\): We will absorb a part of the term
into the positive term
We write, noticing that \(\delta _h J_p(u(x)-u(y))\) and \(\delta _h u(x)-\delta _h u(y)\) have the same sign
We observe that
which follows from the \((p-1)\)-Hölder regularity for \(J_p\). Using Lemma A.1 and (3.10) in (3.9) yields
Therefore, by the above estimate (3.11) and using Young’s inequality with p and \(p/(p-1)\), we have
where we used that \(\eta ^\frac{p}{p-1}\le \eta ^2\) since \(p/(p-1)\ge 2\). Thus from (3.12), we conclude that
Therefore, using (3.8) and (3.13) in (3.7), we obtain
Thus (3.14) gives
Here we have also used that the factor \(|\eta (x)-\eta (y)|^p\) cancels out the singularity of the kernel in the last term above. We now observe that with
the convexity of \(\tau \mapsto \tau ^2\) implies
Taking the above inequality into account and using (3.15), we get the following lower bound for \({\mathcal {I}}_1\), with \(2\sigma = sp-\gamma (p-2)\):
where we again used that \(\eta \) is Lipschitz. Here \(c=c(p,q)>0\) and \(C=C(N,h_0,p,q,s)>0\). Note that \(\sigma \in (0,1)\) since \(\gamma \in [0,1)\) and \(s\in (0,1)\). By recalling that \({\mathcal {I}}_1+{\mathcal {I}}_2+{\mathcal {I}}_3=0\) from (3.6) and using the estimate for \({\mathcal {I}}_1\), we arrive at
for \(C=C(N,h_0,p,q,s)>0\).
Step 2: Estimates of the nonlocal terms \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\): Both nonlocal terms \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\) can be treated in the same way. We only estimate \({\mathcal {I}}_2\) for simplicity. Using (3.1), since \(|u|\le 1\) in \(B_1\), for every \(x\in B_{(R+r)/2}\) and \(y\in {\mathbb {R}}^N{\setminus } B_R\), we have
where \(C=C(p)>0\). For \(x\in B_{(R+r)/2}\) we have \(B_{(R-r)/2}(x)\subset B_{R}\) and thus
by recalling that \(R-r=4\,h_0\). By using [4, Lemma 2.2 and Lemma 3.3], we get for \(x \in B_{(R+r)/2}\)
In the last estimate we have used the bounds assumed on u in (3.1) and \(4\,h_0< R \le 1\). The term involving \(u_h\) can be estimated similarly. Recall also that \(\eta =0\) outside \(B_{(R+r)/2}\). Hence, we have
Similarly, we get
The combination of (3.16), (3.17) and (3.18) now implies
where \(C=C(N,h_0,p,s,q)>0.\)
Step 3: Transformation to double differences. For \(\xi ,h\in {\mathbb {R}}^N\setminus \{0\}\) such that \(|h|,|\xi |<h_0\), we let
Lemma A.5 implies
Therefore,
where \(C=C(q)>0\). Here we used that \(\eta \equiv 1\) on \(B_r\). By a discrete version of Leibniz rule (see (2.3)),
Inserting (3.21) into (3.20) yields
where \(C=C(q)>0\). For the first term in (3.22), we apply [3, Proposition 2.6] with the choice
and get
where \(C=C(N,h_0,\sigma )>0\). Here we also used that \(\frac{R+r}{2} + 2h_0 = R\).
As for the second term in (3.22), we observe that for every \(0<|\xi |<h_0\)
where \(C=C(N,h_0,s)>0\). Here we have used the estimate of \(\nabla \eta \).
From (3.22), (3.23), and (3.24) we get for any \(0< |\xi | < h_0\)
with \(C=C(N,h_0,s,q,\sigma )>0\). We then choose \(\xi =h\) and take the supremum over h for \(0<|h|< h_0\). Then (3.25) together with (3.19) imply
where \(C=C(N,h_0,p,s,q,\sigma )>0\).
Now we choose \(\theta =\alpha -1/q\) and observe that since \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (3.1), the assumption that \(4h_0<R\le 1-5h_0\) and the fact that \(q\le q+p-1\le q+1\), implies that the first and the second terms in the right hand side of (3.26) can be estimated by the third one. Recalling also that \(2\sigma = sp-\gamma (p-2)\), this yields
where \(C=C(N,h_0,p,s,q,\gamma )>0\). Since \(\alpha <1\), taking into account (3.2) and using the second estimate of [4, Lemma 2.6] we replace the first order difference quotient in the right-hand side of (3.27) with a second order difference quotient. Then (3.27) transforms into the desired inequality (3.3), upon recalling the relations between R, r and \(h_0\). \(\square \)
3.2 Improved Hölder regularity
We can now iterate the improved Besov-type regularity to obtain an improved Hölder regularity.
Proposition 3.2
Assume \(1<p<2\), \(0<s<1\) and \(\gamma \in [0,1)\). Let \(u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) be a local weak solution of \((-\Delta _p)^s u=0\) in \(B_2\). Suppose that
Let \(\tau =\min (sp-\gamma (p-2),1)\). Then for any \(\varepsilon \in (0,\tau )\), we have
Proof
Take \(0<\varepsilon <\tau \) and choose q so that
Then we define the sequence of exponents
where we choose \(i_\infty \ge 1\) such that
Note that this is possible since the sequence of exponents \(\alpha _i\) are increasing towards \(sp-\gamma (p-2)\). Define also
We note that
By applying Proposition 3.1 and with \(R=R_i\) and observing that \(R_i-4\,h_0=R_{i+1}+4\,h_0\), we obtain the iterative scheme of inequalities
and finally
Here \(C=C(N,\varepsilon ,p,s,\gamma )>0\). Also,
Hence, the iterative scheme of inequalities leads us to
Using \(\alpha _{i_\infty }\ge \tau -\varepsilon /2\) in (3.28) implies
Take now \(\chi \in C_0^\infty (B_{5/8})\) such that
In particular we have for all \(|h| > 0\)
We also recall that
Hence, for \(0<|h|< h_0\), using the above properties of \(\chi \) and (3.29), we have
Thus by (3.30) and Lemma 2.1, we have
Finally, thanks to the choice of q we have
We may therefore apply Theorem 2.2 with \(\beta =\tau -\frac{\varepsilon }{2}\) and \(\alpha =\tau -\varepsilon \) to obtain
This concludes the proof. \(\square \)
3.3 Final Hölder regularity
We first prove a normalized version of Theorem 1.1. This is accomplished by iterating the previously obtained improved Hölder regularity.
Theorem 3.3
(Almost \(sp/(p-1)\)-regularity) Let \(1<p<2\) and \(0<s<1\). Suppose \(u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of
such that
Then for any \(\varepsilon \in (0,\Gamma )\), there is \(\sigma (\varepsilon ,N,s,p)>0\) such that \(u\in C^{\Gamma -\varepsilon }(B_\sigma )\), where
Moreover,
Proof
The idea is to apply Proposition 3.2 iteratively. Take \(\varepsilon \in (0,\Gamma )\) and define
Then \(\gamma _i\) is an increasing sequence and \(\gamma _i\rightarrow sp/(p-1)-\varepsilon /2\), as \(i\rightarrow \infty \). Define also \( v_i(x)=u(2^{-i} x) \) and
It is clear that there is \(i_\infty =i_\infty (\varepsilon )\in {\mathbb {N}}\) such that \(\gamma _{i_\infty }\ge \Gamma -\varepsilon \) and \(\gamma _{i_\infty -1}<1\). Now we apply Proposition 3.2 to \(v_i/M_i\) successively with \(\gamma =\gamma _i\) and \(\varepsilon \) replaced by \(\frac{\varepsilon (p-1)}{2}\) and obtain
Note that at every iteration step we get the estimate multiplied by a constant \(C(s,p,\varepsilon ,N)\). Hence, by scaling back we obtain
This is the desired result with \(\sigma = 2^{-i_\infty -1}\). \(\square \)
The proof of the main Hölder regularity now easily follows. We spell out the details.
Proof of Theorem 1.1
By Theorem 1.1 in [11], \(u\in L^{\infty }_{\textrm{loc}}(B_{2R}(x_0))\), so the assumption on the boundedness makes sense. Assume for simplicity that \(x_0=0\) and let
where
Then \(u_R\) is a local weak solution of \((-\Delta _p)^s u=0\) in \(B_2\) and satisfies
By Theorem 3.3, \(u_R\) satisfies the estimate
By scaling back, we obtain the desired estimate. \(\square \)
Remark 3.4
We note that as usual, once a local estimate of the spirit of Theorem 1.1 is obtained, one may obtain a similar estimate for any ball strictly contained in \(\Omega \) by a standard covering argument. See for instance Remark 4.3 in [4] for a proof of such a fact.
4 The inhomogeneous equation
In this section, we treat the regularity for the inhomogeneous equation by approximation.
4.1 Basic regularity for the inhomogeneous equation
For our purpose, we need a uniform Hölder estimate for some exponent \(\alpha \in (0,1)\). The argument used to prove this is inspired by [4, 18].
We begin with a Caccioppoli estimate for solutions to the inhomogeneous equation.
Lemma 4.1
Let \(1<p<2\) and \(0<s<1\). Suppose \(\Omega \subset {\mathbb {R}}^N\) is an open and bounded set such that \(B_r(x_0)\Subset B_R(x_0)\subset \Omega \). For \(f\in L^q_{\textrm{loc}}(\Omega )\), with
we consider a local weak solution \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) of the equation
Then
for some positive constant \(C=C(N,s,p)\).
Proof
We only perform the proof for \(u_+\). Proceeding exactly as in the proof of Corollary 3.6 in [5], we take a smooth function \(\phi \) such that \(\phi =1\) in \(B_r(x_0)\), \(0\le \phi \le 1\) in \(B_\frac{R+r}{2}(x_0)\), \(\phi =0\) outside \(B_{(R+r)/2}(x_0)\) and \(|\nabla \phi |\le C/(R-r)\). Testing the equation with \(\phi ^p u_+\) as in the proof of Corollary 3.6 in [5] we obtain
for some positive constant \(C=C(N,s,p)\). We note that since \(\phi =1\) in \(B_r(x_0)\), the left hand side of (4.1) can be bounded from below by \([u_+]_{W^{s,p}(B_r(x_0))}^p\). In addition, the first two terms can be estimated as
which matches the first two terms in the statement of the lemma. It remains to estimate the term involving f. By Hölder’s inequality, we have
This completes the proof. \(\square \)
The following results provides stability for the inhomogeneous equation.
Lemma 4.2
Let \(1<p<2\), \(0<s<1\) and \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set. Suppose that \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of the equation
where \(f\in L^q_{\textrm{loc}}(\Omega )\), with
Let \(B=B_{\sigma r}\Subset B'=B_r\Subset \Omega \) be a pair of concentric balls and take \(v\in X^{s,p}_u(B,B')\) to be the unique weak solution of
For any \(\varepsilon \in (0,1/2)\) we have
and
whenever \(s\,p\not =N\) and for a constant \(C=C(N,p,s,\sigma )>0\).
If instead \(s\,p=N\), a similar estimate holds with \(N\,p/(N-s\,p)\) replaced by an arbitrary exponent \(m<\infty \) and the constant C depending on m as well.
Proof
We only perform the proof in the case \(sp<N\). We first observe that the existence of v is guaranteed by Theorem 2.12 in [4], since \(u\in W^{s,p}(B')\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\). By using the weak formulations of the equations solved by u and v with the test function \(w=u-v\), we get
Let \(a=u(x), b=u(y), c=v(x)\) and \(d=v(y)\). By Lemma B.4 in [5],
By (4.4) and Hölder’s inequality together with some trivial manipulations, we obtain
Recalling the above choices of a, b, c, d and using Hölder’s inequality together with the localized Sobolev inequality (cf. Proposition 2.3 in [5]), from (4.5) we have
where \(C=C(N,p,s,\sigma )\). By Young’s inequality with exponent 2 this implies
with \(C=C(N,p,s,\sigma )\). Using Young’s inequality with exponents \(1/(p-1)\) and \(1/(2-p)\) we obtain
where \(C=C(N,p,s,\sigma )\).
Using the above estimate and arguing as in the proof of Proposition 2.3 in [5], we have
for some constant \(C=C(N,p,s,\sigma )>0\), which in turn gives (4.2).
Estimate (4.3) now follows by applying Poincaré’s inequality in (4.2), see Lemma 2.3. \(\square \)
In the lemma below we obtain a Campanato estimate. Here we use the notation
to denote the average of u in \(B_r(x_0)\).
Lemma 4.3
(Decay transfer) Let \(1<p<2\), \(0<s<1\) and \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set. Suppose that \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of the equation
where \(f\in L^{q}_{\textrm{loc}}(\Omega )\) with
If \(B_{4R}(x_0)\Subset \Omega \) such that \(0<R\le 1\), then there is \(\alpha \in (0,1)\) such that for any \(\varepsilon \in (0,\frac{1}{2})\) we have
for every \(0<r\le R\). Here
and \(C=C(N,s,p,q,m)>0\).
Proof
The proof is the same as the proof of Lemma 3.5 in [4], except for the last term that appears when applying (4.3) in the present case \(p< 2\). We present some details in the case \(s\,p<N\). In order to estimate this extra term, we use that the boundedness of u together with Lemma 4.1 applied to the balls \(B_{7R/2}(x_0)\) and \(B_{4R}(x_0)\) gives
for some constant \(C=C(N,s,p)\), where we also used Young’s inequality and that \(\frac{N}{q'}+sp-N>0\), \(\gamma >0\) and \(0<r\le R\le 1\). \(\square \)
We are now ready to prove Hölder regularity.
Theorem 4.4
Let \(1<p<2\) and \(0<s<1\). Suppose that \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is a local weak solution of the equation
for \(f\in L^q_{\textrm{loc}}(\Omega )\) with
Then \(u\in C^{\beta }_{\textrm{loc}}(\Omega )\), where
with \(\gamma \) as in (4.6) and \(\alpha \) as in Lemma 4.3.
More precisely, for every ball \(B_{R_0}(z)\Subset \Omega \) we have the estimate
where
Here, the constant C depends only \(N,p,s,q,R_0\) and \(\textrm{dist}(B_{R_0}(z),\partial \Omega )\).
Proof
The proof is almost identical with the proof of Theorem 3.6 of [4]. The only difference is that there is a parameter \(\varepsilon \) and an additional term when applying Lemma 4.3. We take a ball \(B_{R_0}(z)\Subset \Omega \) and set
Choose a point \(x_0\in B_{R_0}(z)\) and consider the ball \(B_{4R}(x_0)\) with \(R<\min \{1,\textrm{d}/8\}\). IfFootnote 1\(s\,p\not =N\), applying Lemma 4.3 and obtain
for every \(0<r\le R<\min \{1,\textrm{d}/8\}\).
As in the proof of Theorem 3.6 in [4] it is straightforward to estimate these terms and obtain
where \(C=C(N,s,p,q)>0\).
To simplify the notation, let
Then the above estimate reads
We will now see that for a specific choice of R and \(\delta \) in terms of r, this implies that
decays in a power fashion. Indeed, let
where
Then
for \(x_0\in B_{R_0}(z)\) and \(r<\min \{1,(\textrm{d}/8)^\frac{1}{\sigma }\}\) where \(\beta =\frac{\alpha \gamma (p-1)}{\gamma (p-1)+(\alpha p+N)}\). This shows that u belongs to the Campanato spaceFootnote 2\({\mathcal {L}}^{p,N+\beta \,p}(B_{R_0}(z))\), which is isomorphic to \(C^{\beta }(\overline{B_{R_0}(z)})\). The proof is complete. \(\square \)
4.2 Final Hölder regularity
In order to prove Theorem 1.2, we first establish the following stability result.
Lemma 4.5
(Stability in \(L^\infty \)) Let \(1<p<2\), \(0<s<1\). Suppose \(\Omega \subset {\mathbb {R}}^N\) is an open and bounded set and \(f\in L^q_{\textrm{loc}}(\Omega )\) with
Consider a local weak solution \(u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) of the equation
Let \(B_{4}\Subset \Omega \) and assume that
Suppose that \(h\in X^{s,p}_u({B_\frac{3}{2},B_{4}})\) weakly solves
Then there is \(\tau _{M}(\eta )\) such that
and \(\tau _{M}(\eta )\) converges to 0 as \(\eta \) goes to 0.
Proof
The existence of a bound of the form (4.7) is a consequence of the triangle inequality and the local \(L^\infty \) estimate for the equation (Theorem 3.8 in [5]). We will now prove that \(\tau _{M}(\eta )\rightarrow 0\) as \(\eta \rightarrow 0\).
We assume towards a contradiction that there exist two sequences \(\{f_n\}_{n\in {\mathbb {N}}}\subset L^{q}(B_2)\) and \(\{u_n\}_{n\in {\mathbb {N}}}\) such that
but
We note that by Lemma 4.1, any u satisfying the assumptions of the lemma also satisfies the bound
Therefore, (4.2) implies that for every \(\varepsilon \in (0,1/2)\), we have
where \(C=C(M,N,p,s)>0\) is a constant. Since this holds for any \(\varepsilon \in (0,1/2)\), we conclude that
This, together with the fractional Sobolev inequality and Theorem 1.1 in [11] implies that \(h_n\) is locally uniformly bounded in \(B_{3/2}\). Theorem 3.1 in [4] or Theorem 1.1 implies that \(h_n\) is uniformly bounded in \(C^{\beta }(B_{5/4})\) and Theorem 4.4 implies that \(u_n\) is uniformly bounded in \(C^{\beta }(B_{5/4})\) for some \(\beta >0\). Therefore, by the Ascoli–Arzelà theorem, we may conclude that \(u_n-h_n\) converges uniformly in \(\overline{B_{5/4}}\), up to a subsequence. By (4.8) we get that
which gives the desired contradiction. \(\square \)
The following proposition is a rescaled version of Theorem 1.2.
Proposition 4.6
Let \(1<p<2\), \(0<s<1\). Take q such that
and define
For every \(0<\varepsilon <\Theta \) there exists \(\eta =\) \(\eta (N,p,q,s,\varepsilon )>0\) such that if \(f\in L^q_{\textrm{loc}}(B_4(x_0))\) and
then every local weak solution \(u\in W^{s,p}_{\textrm{loc}}(B_4(x_0))\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) of the equation
such that
belongs to \(C^{\Theta -\varepsilon }(\overline{B_{1/8}(x_0)})\) with the estimate
for some constant \(C(N,p,q,s,\varepsilon )>0\).
Proof
Without loss of generality, we may assume that \(x_0=0\). We divide the proof in two parts.
Part 1: Regularity at the origin. We claim that for any \(0<\varepsilon <\Theta \) and every \(0<r<1/2\), there exists \(\eta =\eta (N,p,q,s,\varepsilon )>0\) and a constant \(C=C(N,p,q,s,\varepsilon )>0\) such that if f and u are as above, then we have
Without loss of generality, we assume \(u(0)=0\). Let us fix \(0<\varepsilon <\Theta \). Then we remark that it is enough to prove that there exists \(\lambda <1/2\) and \(\eta >0\) (depending on N, p, q, s and \(\varepsilon \)) such that if f and u are as above, then
for every \(k\in {\mathbb {N}}\). Indeed, if this is true, then for every \(0<r<1/2\), there exists \(k\in {\mathbb {N}}\) such that \(2\lambda ^{k+1}< r\le 2\lambda ^k\). Using the first property from (4.10), we deduce that
where \(C=\frac{1}{(2\lambda )^{\Theta -\epsilon }}\) as desired.
We prove (4.10) by an induction argument. First, we note that (4.10) holds true for \(k=0\), using the assumptions in (4.9). Suppose (4.10) is valid up to k. We prove that this is also valid for \(k+1\) assuming that
for small enough \(\eta \), which is independent of k. We define
We observe that by the hypotheses, it follows that
Furthermore,
We notice that
where we have used the hypotheses on f, the definition of \(\Theta \), and again the fact that \(\lambda <1/2\). By Proposition 2.12 in [4], we consider \(h_k\in X_{w_k}^{s,p}(B_\frac{3}{2},B_4)\) to be the weak solution of
From Lemma 4.5, we obtain
where \(\tau _\eta \rightarrow 0\) as \(\eta \rightarrow 0\) and \(\tau _\eta \) is independent of k. Therefore,
To obtain the above estimate, we have also used the fact that \(h_k\) belongs to \(C^{\Theta -\varepsilon /2}(\overline{B_{1}})\), which follows from Theorem 1.1 and Remark 3.4, with the estimate
We obtained the above estimate by observing that the quantities in the right-hand side are uniformly bounded, independently of k. To this end, Lemma 4.5 and (4.11) along with the triangle inequality gives that
For the tail term, by the triangle inequality, the hypothesis on \(w_k\) and (4.3) combined with Lemma 4.1, we obtain
with \(C=C(N,s,p,q)\). We also made use of (4.11) and that \(h_k=w_k\) outside \(B_{3/2}\), by construction. Therefore, the estimate (4.12) is uniform in k. Let
We can transfer estimate (4.12) to \(w_{k+1}\) by choosing \(\eta \) so that \(2\tau _\eta <\lambda ^\Theta \) and \(\lambda \) small enough. Indeed, we observe that
In particular, the above estimate gives that \(\Vert w_{k+1}\Vert _{L^\infty ({B_2})}\le 1\) for \(\lambda \) satisfying
This information, rescaled back to u, gives precisely the first part of (4.10) for \(k+1\). To obtain the second part of (4.10), we use the upper bound for \(|w_{k+1}|\) and the fact that \(\Theta <\frac{sp}{p-1}\), which gives
Since \(|w_k|\le 1\) in \(B_2\), a change of variable gives
In addition, by the integral bound on \(w_k\) in (4.11) and using that \(\text {Tail}_{p-1,sp}(w_k;0,2)\le 1\), we get
The condition \(\lambda <1/2\) and the fact that
is used in both estimates. Here the constants \(C_2\) and \(C_3\) depend on N, p, q, s and \(\varepsilon \) only. By (4.14), (4.15) and (4.16), we get that the second part of (4.10) holds, provided that
Recalling (4.13), we finally obtain that (4.10) holds true at step \(k+1\) as well, when \(\lambda \) and \(\eta \) (depending on N, p, q, s and \(\varepsilon \)) are chosen so that
The induction is complete.
Part 2: We prove the desired regularity in the whole ball \(B_{1/8}\). To this end, we take \(0<\varepsilon <\Theta \) and choose the associated \(\eta \), obtained in Part 1. Take \(z_0\in B_{1}\), let \(L=2^{N+1}\,(1+|B_2|)\) and define
We observe that \(v\in W^{s,p}_{\textrm{loc}}(B_4)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) and that v is a weak solution in \(B_4\) of
with
Moreover, by construction, we have
Observing that \(B_{1}(z_0)\subset B_2\) along with the definition of L and the hypotheses in (4.9), we get
In the above estimate, we have also used Lemma 2.3 in [4] with the balls \(B_{1}(z_0)\subset B_2\). Therefore applying Part 1 to v, we obtain
which in terms of u is same as
We remark that the above estimate holds for any \(z_0\in B_{1}\). We choose any pair \(x,y\in B_{1/8}\) such that \(|x-y|= r\). Then \(r<1/4\). Setting \(z=(x+y)/2\), we apply (4.17) with \(z_0=z\) and obtain
which is the desired result. \(\square \)
We are now ready to give the proof of the final Hölder regularity result.
Proof of Theorem 1.2
Without loss of generality, we may assume \(x_0=0\). We modify u in such a way that it fits into the setting of Proposition 4.6. Let
where we have chosen \(\varepsilon \in (0,\Theta )\) and \(\eta \) as in Proposition 4.6. Note that u is locally bounded by Theorem 3.8 in [5]. By scaling arguments, it is enough to prove that the rescaled function
satisfies the estimate
It is straightforward to see that the choice of \({\mathcal {A}}_R\) implies
Also, \(u_R\) is a local weak solution of
with \(\Vert f_R\Vert _{L^{q}(B_{2})}\le \eta \). Therefore, applying Proposition 4.6 to \(u_R\), we obtain
After scaling back, this concludes the proof. \(\square \)
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Acknowledgements
We are grateful to Lorenzo Brasco for many fruitful discussions related to this manuscript and in particular for his useful suggestions regarding the proof of Theorem 4.4. We also thank Alireza Tavakoli for his suggestions. E. L. has been supported by the Swedish Research Council, grant no. 2023-03471, 2017-03736 and 2016-03639 under the development of this project. Part of this material is based upon work supported by the Swedish Research Council under grant no. 2016-06596 while the second author were participating in the research program “Geometric Aspects of Nonlinear Partial Differential Equations”, at Institut Mittag-Leffler in Djursholm, Sweden, during the fall of 2022.
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Useful Inequalities
Useful Inequalities
The following inequality follows from from [22, (I), Page 95].
Lemma A.1
Let \(1<p<\infty \) and \(a,b,c,d\in {\mathbb {R}}\). Then we have
Lemma A.2
Let \(p\ge 2\) and \(q>1\). For every \(a,b\in {\mathbb {R}}\), we have
This is Lemma A.1 in [4].
Lemma A.3
Let \(1<p<2\) and \(a,c \in {\mathbb {R}}^n\). Then
This is inequality (2.3) in [17].
Lemma A.4
Let \(\gamma \ge 2\), \(1<p<2\) and \(a,b,c,d \in {\mathbb {R}}\). Then
Proof
The proof is just a combination of Lemmas A.2 and A.3. \(\square \)
The following inequality can be found in [4, Lemma A.3].
Lemma A.5
Let \(\gamma \ge 1\). Then for every \(A,B\in {\mathbb {R}}\), we have
for some constant \(C=C(\gamma )>0\).
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Garain, P., Lindgren, E. Higher Hölder regularity for the fractional p-Laplace equation in the subquadratic case. Math. Ann. (2024). https://doi.org/10.1007/s00208-024-02891-z
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DOI: https://doi.org/10.1007/s00208-024-02891-z