Abstract
We consider equations involving a combination of local and nonlocal degenerate p-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.
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1 Introduction
1.1 Overview
In this article, we study regularity properties of weak solutions of the mixed local and nonlocal p-Laplace equation
where \(\Omega \) is an open and bounded set in \({\mathbb {R}}^N,\,N\ge 1\). We assume that \(0<s<1,\,2\le p<\infty \) and \(f\in L^q_\textrm{loc}(\Omega )\) for some \(q\ge 1\) (for the precise assumptions, see Sects. 1.2 and 2). Here
is the p-Laplace operator and
is the fractional p-Laplace operator, where P.V. denotes the principal value.
The main objective of this article is to establish Hölder regularity of weak solutions of Eq. (1.1), with an explicit Hölder exponent. This is done in Theorem 1.3 and Theorem 1.4. From this, Hölder regularity of the gradient follows in the case when \(f=0\) and \(sp<(p-1)\). We also establish existence and uniqueness in Theorem 1.1 and local boundedness in Theorem 1.2. Our results are presented in detail in the next section.
1.2 Main results
Here we present the main results of this paper: existence, uniqueness and regularity of weak solutions. For the notion of weak solutions and relevant notation such as \(\textrm{Tail}_{p-1,s\,p,s\,p}\), we refer to Sect. 2. In the theorem below, \(p^*\) refers to the Sobolev exponent, see (2.1).
Theorem 1.1
(Existence and uniqueness) Suppose \(1<p<\infty \), \(0<s<1\) and \(A>0\). Let \(\Omega \Subset \Omega '\subset {\mathbb {R}}^N\) be two open and bounded sets where \(f\in L^q(\Omega )\), with
and \(g\in W^{1,p}(\Omega ')\cap L^{p-1}_{sp}({\mathbb {R}}^N)\). Then there is a unique weak solution \(u\in W_g^{1,p}(\Omega )\) of
Theorem 1.2
(Local boundedness) Suppose \(2\le p<\infty \), \(0<s<1\) and \(0\le A\le 1\). Let \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set and \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) be a weak solution of
where \(f\in L_\text {loc}^{q}(\Omega )\) with
Then \(u^+=\max \{u,0\}\), satisfies \(u^+\in L^\infty _\text {loc}(\Omega )\) and for every \(0<R<1\) such that \(B_{R}(x_0)\Subset \Omega \) and every \(0<\sigma <1\), there holds
where \(C=C(N,s,p,q,\sigma )>0\).
Theorem 1.3
(Almost Lipschitz regularity) Suppose \(2\le p<\infty \), \(0<s<1\) and \(0\le A\le 1\). Let \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set and \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) be a weak solution of
Then \(u\in C^\delta _{\textrm{loc}}(\Omega )\) for every \(0<\delta <1\).
More precisely, for every \(0<\delta <1\) and every ball \(B_{2R}(x_0)\Subset \Omega \) with \(0<R<1\), there exists a constant \(C=C(N,s,p,\delta )>0\) such that
Theorem 1.4
(Higher Hölder regularity) Suppose \(2\le p<\infty \), \(0<s<1\) and \(0\le A\le 1\). Let \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set, and \(f\in L^q_{\text {loc}}(\Omega )\) where
Let \(\Theta =\min \{(p-N/q)/(p-1),\frac{sp}{p-1},1\}\) and \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) be a weak solution of
Then \(u\in C^\delta _{\textrm{loc}}(\Omega )\) for every \(0<\delta <\Theta \).
More precisely, for every \(0<\delta <\Theta \) and every ball \(B_{4R}(x_0)\Subset \Omega \) such that \(R\in (0,1)\), there exists a constant \(C=C(N,s,p,q,\delta )>0\) such that
Corollary 1.5
Suppose \(2\le p<\infty \), \(0<s<1\) and \(sp<(p-1)\). Let \(\Omega \subset {\mathbb {R}}^N\) be an open and bounded set, \(0\le A\le 1\) and \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) be a weak solution of
Then \(u\in C^{1,\alpha }_{\textrm{loc}}(\Omega )\) for some \(\alpha \in (0,1)\).
More precisely, for every ball \(B_{2R}(x_0)\Subset \Omega \) with \(0<R<1\), there exists a constant \(C=C(N,s,p,q,\delta )>0\) such that
where \(\textrm{Tail}_{p-1,s\,p,s\,p}(u;x_0,R)\) is defined in (2.2).
Remark 1.6
The reason for which we have included a constant A in the equation in the above results, is that in the proofs we will consider rescaled solutions. For these, a constant appears in front of the operator \((-\Delta _p)^s\).
1.3 Comments on the results
We first comment on the sharpness of our results, more specifically Theorem 1.4. In general, the results are most likely not sharp. For instance, the results in [17] give \(C^{1,\alpha }\)-regularity for solutions for all \(s\in (0,1)\) and all \(p\in (1,\infty )\), under the additional assumption that \(u\in W^{s,p}({\mathbb {R}}^N)\).
However, our results are almost sharp when \((p-N/q)/(p-1)\le \frac{sp}{p-1}\le 1\). Indeed, assume
and let
for some \(\epsilon >0\). Then
with \(f\in L^q_{\text {loc}({\mathbb {R}}^N)}\) if and only if \(\gamma +\epsilon >(sp-N/q)/(p-1)\). Moreover,
with \(g\in L^q_{\text {loc}({\mathbb {R}}^N)}\) if and only if \(\gamma +\epsilon >(p-N/q)/(p-1)\). It is clear that \(u\not \in C^{\alpha }(B_1)\) for any \(\alpha >\gamma +\epsilon \). This shows that in this regime of parameters, the results of Theorem 1.4 are almost sharp.
Now we turn our attention to the Hölder exponents in Theorems 1.3 and 1.4. Note that even in the case when f is smooth Theorem 1.4 only gives almost Hölder regularity of order \(\min \{sp/(p-1),1\}\), while we for \(f=0\) reach almost Lipschitz regularity in Theorem 1.3. The reason for this discrepancy is that we prove Theorem 1.4 by treating the inhomogeneous equation as a perturbation of the homogeneous one. The restriction of the exponent arises when we need a uniform control of the decay at infinity at different scales, see (5.17). It may be possible to treat this as a perturbation of the homogeneous p-Laplace equation instead, but we were not able to control the decay at infinity in such an approach.
We also make a small comment regarding the assumption \(sp<(p-1)\) in Corollary 1.5. This assumption arises as a condition for when \((-\Delta _p)^s u\) is bounded for almost Lipschitz functions u. The result is then obtained by treating \((-\Delta _p)^s u\) as a bounded term.
1.4 Known results
In the homogeneous setting \(f=0\) and for \(p=2\), Eq. (1.1) reads
Based on the theory of probability and analysis, Eq. (1.6) has been intensely studied in recent years. We mention the work of Foondun [27], where a Harnack inequality and local Hölder continuity are established. We also refer to the Chen et al. [13,14,15], Athreya and Ramachandran [2] and the references therein for related results. For the parabolic problem associated with (1.6), Barlow et al. [3], Chen and Kumagai [16] proved a Harnack inequality and local Hölder continuity.
Recently, the regularity theory has also been developed by a purely analytic approach. For the linear case \(p=2\), existence, local boundedness, interior Sobolev regularity and a strong maximum principle, along with other qualitative properties of solutions have been established by Biagi et al. in [6]. Local boundedness is also established in Dipierro et al. [20]. For existence and nonexistence results, we refer to Abatangelo and Cozzi [1]. We also refer to Biagi et al. [4, 8], Dipierro et al. [22], Dipierro et al. [21], Dipierro and Valdinoci [23] and the references therein.
In the nonlinear setting \(p\ne 2\), for \(f=0\), regularity results of weak solutions in terms of local boundedness, Harnack estimates, local Hölder continuity and semicontinuity results have been obtained in Garain and Kinnunen [28]. In [7], Biagi et al. established boundedness and strong maximum principle in the inhomogeneous case. In the case of a bounded function f, Biagi et al. [5] has obtained local Hölder continuity for globally bounded solutions and Garain-Ukhlov [31] studied existence, uniqueness, local boundedness and further qualitative properties of solutions. Moreover, for more general inhomogeneites, local boundedness is proved in Salort and Vecchi [34]. Very recently, Hölder and gradient regularity were proved by De Filippis and Mingione in [17], where a general type of mixed nonlinear problems are considered. Even a mix of different orders and different homogeneities of the operators is allowed. The results therein that applies to (1.1) are proved under the global assumption that \(u\in W^{s,p}({\mathbb {R}}^N)\). Under this assumption, their results contain ours as a special case.
We also seize the opportunity to mention that very recently, the regularity theory for mixed parabolic equations has gained an increasing amount of attention. In the linear case, a weak Harnack inequality is proved for the parabolic analogue of Eq. (1.6) in Garain and Kinnunen [30]. For the nonlinear case, see Fang et al. [26, 35] and Garain and Kinnunen [29]. Among other things, local boundedness and Hölder continuity have been established.
Finally, we wish to mention [24], where a similar approach using difference quotients has been used to obtain improved regularity for quasilinear subelliptic equations in the Heisenberg group.
1.5 Plan of the paper
In Sect. 2, we introduce relevant notation and definitions and certain standard result in function spaces. In Sect. 3, we establish existence and uniqueness using standard methods from functional analysis. The core of the paper is mainly in Sect. 4, where we prove almost Lipschitz regularity for the homogeneous equation, using a Moser-type argument that results in an improved differentiability that can be iterated. Here we also prove Corollary 1.5. This is followed by Sect. 5, where the local boundedness and higher Hölder regularity for the inhomogeneous equation is established. This is based on approxmation with the homogenous 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. Throughout the paper, we shall use the notation that follows. We denote by \(B_r(x_0)\), the ball of radius r centered at \(x_0\). When \(x_0=0\), we will simply write \(B_r\). It will also be convenient to use the notation \(u^+=\max \{u,0\}\). The monotone and \((p-1)\)-homogeneous function
is expedient when treating equations of p-Laplacian type. Discrete differences play an important role. Therefore, for a measurable function \(\psi :{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) and a vector \(h\in {\mathbb {R}}^N\), we define
2.1 Function spaces
For \(p\in (1,\infty )\) and \(u\in W^{1,p}(\Omega )\), the \(W^{1,p}\)-seminorm is defined by
We also define the critical Sobolev exponent as
Moreover, for \(0<\delta \le 1\), we will employ the \(\delta \)-Hölder seminorm, given by
For \(1\le q<\infty \) and for \(0<\beta <2\), we introduce the Besov-type space
where
Similarly, the Sobolev–Slobodeckiĭ space is defined by
where the seminorm \([\,\cdot \,]_{W^{\beta ,q}({\mathbb {R}}^N)}\) reads
These spaces are endowed with their corresponding norms
and
At times, we will also work with the space \(W^{\beta ,q}(\Omega )\) for a subset \(\Omega \subset {\mathbb {R}}^N\),
where we define
2.2 Tail spaces
In the study of nonlocal equations, the global behavior of solutions comes into play. This is entailed by the tail space
and measured by the quantity
defined for every \(x_0\in {\mathbb {R}}^N\), \(R>0,\,\beta >0\) and \(u\in L^q_{\alpha }({\mathbb {R}}^N)\). We observe that the quantity above is always finite, for a function \(u\in L^q_{\alpha }({\mathbb {R}}^N)\).
2.3 Auxiliary results for functions spaces
The next result asserts that the standard Sobolev space is continuously embedded in the fractional Sobolev space, see [18, Proposition 2.2]. The argument uses the smoothness property of \(\Omega \) so that we can extend functions from \(W^{1,p}(\Omega )\) to \(W^{1,p}({\mathbb {R}}^N)\) and that the extension operator is bounded.
Lemma 2.1
Let \(\Omega \) be a smooth bounded domain in \({\mathbb {R}}^N\), \(1<p<\infty \) and \(0<s<1\). There exists a positive constant \(C=C(N,p,s,\Omega )\) such that \( \Vert u\Vert _{W^{s,p}(\Omega )}\le C\Vert u\Vert _{W^{1,p}(\Omega )} \) for every \(u\in W^{1,p}(\Omega )\).
The following result for the fractional Sobolev spaces with zero boundary value follows from [12, Lemma 2.1]. The main difference compared to Lemma 2.1 is that the result holds for any bounded domain, since for the Sobolev spaces with zero boundary value, we may always extend by zero.
Lemma 2.2
Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^N\), \(1<p<\infty \) and \(0<s<1\). Then there exists a positive constant \(C=C(N,p,s,\Omega )\) such that
for every \(u\in W_0^{1,p}(\Omega )\). Here we consider the zero extension of u to the complement of \(\Omega \).
The following result is a local version of [9, Lemma 2.3].
Lemma 2.3
Let \(\beta \in (0,1)\), \(p\in (1,\infty )\), \(x_0\in {\mathbb {R}}^N\), \(R>0\) and \(h_1>0\). Suppose
Then
Here \(C=C(N,p)>0\).
Proof
Without loss of generality, we assume that \(x_0=0\). Let \(0<|h|<h_1\). Let \(\eta \in C_c^{\infty }(B_{R+\frac{h_1}{2}})\) be such that \(0\le \eta \le 1\), \(|\nabla \eta |\le \frac{C}{h_1}\), \(\Vert D^2 \eta \Vert \le \frac{C}{h_1^{2}}\) in \(B_{R+\frac{h_1}{2}}\) for some constant \(C=C(N,p)>0\) and \(\eta \equiv 1\) in \(B_R\). Then
for some constant \(C=C(N,p)>0\). Note that the functions \(\eta _{2h},\,\delta _h\eta _h\) and \(\delta _h ^2 \eta \) have support inside \(B_{R+\frac{5h_1}{2}}\). Moreover, we obtain
By the hypothesis (2.3) and \(\eta \in C_c^{\infty }(B_{R+\frac{h_1}{2}})\), it follows that \(u\eta \in {\mathcal {B}}_{\infty }^{\beta ,p}({\mathbb {R}}^N)\). Then by [9, Lemma 2.3], we have
Using the above properties of \(\eta \), (2.5)–(2.7) and the fact that \(0<\beta <1\), we have
for some \(C=C(N,p)\). This proves the result. \(\square \)
Our next result is a local version of [9, Proposition 2.4].
Lemma 2.4
Let \(\alpha \in (1,2)\), \(p\in (1,\infty )\), \(R>0\), \(x_0\in {\mathbb {R}}^N\) and \(h_1>0\). Suppose
Then
where \(C=C(N,p)>0\).
Proof
Without loss of generality, we assume that \(x_0=0\). Let \(\eta \in C_c^{\infty }(B_{R+\frac{h_1}{2}})\) be as defined in (2.5). Using the assumption (2.8) and \(\eta \in C_c^{\infty }(B_{R+\frac{h_1}{2}})\), we have \(u\eta \in {\mathcal {B}}_{\infty }^{\alpha ,p}({\mathbb {R}}^N)\). Therefore, by [9, Propsotion 2.4], we get
for some \(C=C(N,p)>0\). Next, using the properties of \(\eta \) from (2.5) and (2.6), we observe that
for some positive constant \(C=C(N,p)>0\). Now we estimate the second integral in the RHS of (2.11). To this end, using (2.8), we get
Since \(0<\alpha -1<1\), by Lemma 2.3, it follows that
for some \(C=C(N,p)\). Combining the estimates (2.13) and (2.11) in (2.10) and noting that \(\eta \equiv 1\) in \(B_{R}\), the result follows. \(\square \)
Lemma 2.5
Suppose \(u\in W^{1,p}({\mathbb {R}}^N)\), where \(p\in (1,\infty )\). Then
Proof
We have
Therefore, by Hölder’s inequality
Upon integrating, we obtain
\(\square \)
We seize the opportunity to mention that a local version of the above lemma can be found in Theorem 3 on page 277 in [25].
2.4 Weak solutions
Below, we define weak solutions of (1.1), allowing also for a factor A that will be needed in the sequel, when treating rescaled solutions.
Definition 2.6
Let \(1<p<\infty ,\,0<s<1\) and \(A\ge 0\). Suppose \(f\in L^q(\Omega )\), with
We say that \(u\in W_{\textrm{loc}}^{1,p}(\Omega )\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) is a weak subsolution (or supersolution) of
if for every \(K\Subset \Omega \) and for every nonnegative \(\phi \in W_{0}^{1,p}(K)\), we have
where
We say that u is a weak solution of (1.1), if equality holds in (2.14) for every \(\phi \in W_0^{1,p}(K)\).
Remark 2.7
By Lemma 2.1 and Lemma 2.2, Definition 2.14 makes sense.
We now detail the notion of weak solutions to the Dirichlet boundary value problem. For that purpose, given \(\Omega \subset {\mathbb {R}}^N\) an open and bounded set, consider a bounded domain \(\Omega ^{'}\) such that \(\Omega \Subset \Omega ^{'}\subset {\mathbb {R}}^N\). Then for \(g\in W^{1,p}(\Omega ')\), we define
When \(u\in W_g^{1,p}(\Omega )\) we will repeatedly identify u as being extended by g outside of \(\Omega \).
Definition 2.8
(Dirichlet problem) Let \(1<p<\infty ,\,0<s<1\) and \(A\ge 0\). Suppose \(\Omega \Subset \Omega '\subset {\mathbb {R}}^N\) be two open and bounded sets, \(f\in L^q(\Omega )\), with
and \(g\in W^{1,p}(\Omega ')\cap L^{p-1}_{sp}({\mathbb {R}}^N)\). We say that \(u\in W_g^{1,p}(\Omega )\) is a weak solution of the boundary value problem
if for every \(\phi \in W_0^{1,p}(\Omega )\), we have
where \(J_p\) and \(d\mu \) are defined in (2.15) above.
Remark 2.9
Note that Definition 2.18 makes sense by Lemma 2.1 and Lemma 2.2, since we may choose a smooth set K such that \(\Omega \Subset K\Subset \Omega '\).
3 Existence and uniqueness
Here we prove existence and uniqueness of solutions of the Dirichlet problem (2.17).
Proof of Theorem 1.1
In what follows, whenever X is a normed vector space, we denote by \(X^*\) its topological dual.
We first note that \(W_0^{1,p}(\Omega )\) is a separable reflexive Banach space. We now introduce the operator \({\mathcal {A}}:W_g^{1,p}(\Omega )\rightarrow (W_0^{1,p}(\Omega ))^*\) defined by
where \(\langle \cdot ,\cdot \rangle \) denotes the relevant duality product. We observe that \({\mathcal {A}}(v)\in (W_0^{1,p}(\Omega ))^*\) for every \(v\in W_g^{1,p}(\Omega )\) (by Lemma 2.1 and [32, Remark 1]). Moreover, as in the proof of [32, Lemma 3], we have that \({\mathcal {A}}\) has the following properties:
-
1.
for every \(v,u\in W_g^{1,p}(\Omega )\), we have
$$\begin{aligned} \langle {\mathcal {A}}(u)-{\mathcal {A}}(v),u-v\rangle \ge 0, \end{aligned}$$with equality if and only if \(u=v\); This follows from applying Lemma A.1 to the nonlocal part and noting that for the local term we have the following inequalities (see [37, Page 11]):
$$\begin{aligned} \langle {\mathcal {A}}(u)-{\mathcal {A}}(v),u-v\rangle \ge {\left\{ \begin{array}{ll} \displaystyle C_1\Big (\int _{\Omega }|\nabla (u-v)|^p\,dx\Big )^\frac{1}{p},\text { if }p\ge 2,\\ \frac{\displaystyle C_2\big (\int _{\Omega }|\nabla (u-v)|^p\,dx\big )^\frac{2}{p}}{\displaystyle \left( \left( \int _{\Omega }|\nabla u|^p\,dx\right) ^\frac{1}{p}+\left( \int _{\Omega }|\nabla v|^p\,dx\right) ^\frac{1}{p}\right) ^{2-p}},\text { if }1<p<2, \end{array}\right. }\nonumber \\ \end{aligned}$$(3.1)for some positive constants \(C_1,\,C_2\).
-
2.
if \(\{u_n\}_{n\in {\mathbb {N}}}\subset W_g^{1,p}(\Omega )\) converges in \(W^{1,p}(\Omega )\) to \(u\in W_g^{1,p}(\Omega )\), then
$$\begin{aligned} \lim _{n\rightarrow \infty } \langle {\mathcal {A}}(u_n)-{\mathcal {A}}(u),v\rangle =0\quad \text {for all }v\in W_0^{1,p}(\Omega ); \end{aligned}$$This follows from the application of Lemma 2.1 together with Hölder’s inequality and the coupling of weak and strong convergence.
-
3.
From (3.1), it follows that
$$\begin{aligned} \lim _{\Vert u\Vert _{W^{1,p}(\Omega )}\rightarrow +\infty } \frac{\langle {\mathcal {A}}(u)-{\mathcal {A}}(g),u-g\rangle }{\Vert u-g\Vert _{W^{1,p}(\Omega )}}=+\infty . \end{aligned}$$
Finally, we introduce the modified functional
We observe that \({\mathcal {A}}_0:W_0^{1,p}(\Omega )\rightarrow (W_0^{1,p}(\Omega ))^*\). Moreover, properties (1), (2) and (3) above imply that \({\mathcal {A}}_0\) is monotone, coercive and hemicontinuous (see [36, Chapter II, Section 2] for the relevant definitions). It is only left to observe that under the standing assumptions, the linear functional
belongs to the topological dual of \(W_0^{1,p}(\Omega )\). Notice that for every \(v\in W_0^{1,p}(\Omega )\) we haveFootnote 1
and the last term can be controlled using the Sobolev embedding \(W^{1,p}({\mathbb {R}}^N)\rightarrow L^{p^*}({\mathbb {R}}^N)\) (see [25]). Then by [36, Corollary 2.2], we obtain the existence of \(v\in W_0^{1,p}(\Omega )\) such that
By definition, this is equivalent to
i.e.
which is the same as (2.18), since \(v=0\) in \({\mathbb {R}}^N\setminus \Omega \) and that
Then \(v+g\) is the desired solution. Uniqueness now follows from the strict monotonicity of the operator \({\mathcal {A}}_0\). \(\square \)
Remark 3.1
(Variational solutions) Under the slightly stronger assumption \(g\in W^{1,p}(\Omega ')\cap L^p_{s\,p}({\mathbb {R}}^N)\), existence of the solution to (2.17) can be obtained by solving the following strictly convex variational problem
where the functional \({\mathcal {F}}\) is defined by
Existence of a minimizer can be obtained using the Direct Methods in the Calculus of Variations.
4 Almost Lipschitz regularity for the homogeneous equation
In this section, we prove the almost Lipschitz regularity for the homogeneous equation. We first start with the result below, where we differentiate the equation discretely and test with powers of \(\delta _h u\). This yields an iteration scheme of Moser-type. This is the core of the paper.
Proposition 4.1
Let \(2\le p<\infty \), \(0<s<1\) and \(0\le A\le 1\). Suppose that \(u\in W^{1,p}_{\textrm{loc}}(B_2(x_0))\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) is a weak solution of \(-\Delta _p u+A(-\Delta _p)^s u=0\) in \(B_2(x_0)\). Assume that
Let \(0<h_0<\frac{1}{10}\) and R be such that \(4h_0<R\le 1-5h_0\) and \(\nabla u\in L^q(B_{R+4h_0}(x_0))\) for some \(q\ge p\). Then
for some constant \(C=C(N,h_0,p,q,s)>0\).
Proof
Without loss of generality, we assume that \(x_0=0\). We divide the proof into five steps.
Step 1: Discrete differentiation of the equation. Let \(r=R-4h_0\) and \(\phi \in W^{1,p}(B_R)\) vanish outside \(B_{\frac{R+r}{2}}\). Since u is a weak solution of \(-\Delta _p u+A(-\Delta _p)^s u=0\) in \(B_2\), from Definition 2.6, we have
Let \(h\in {\mathbb {R}}^n\setminus \{0\}\) be such that \(|h|<h_0\). Choosing \(\phi =\phi _{-h}\) in (4.3) and using a change of variables, we have
Subtracting (4.3) with (4.4) and dividing the resulting equation by |h|, we obtain
for every \(\phi \in W^{1,p}(B_R)\) vanishing outside \(B_{\frac{R+r}{2}}\). Let \(\eta \) be a nonnegative Lipschitz cut-off function such that
for some constant \(C=C(N)>0\). Suppose \(\alpha \ge 1\), \(\theta >0\) and testing (4.5) with
we get
where
and
Step 2: Estimate of the local integral I. We observe that
Estimate of \(I_1\): Since \(p\ge 2\), using Lemma A.2 and that \(\alpha \ge 1\), we get
Moreover, for \(p\ge 2\), using Lemma A.1, we have
Estimate of \(I_2\): Since \(p\ge 2\), using Lemma A.3 and Young’s inequality with exponents 2 and 2, we obtain
for some \(\epsilon \in (0,\frac{4}{p^2})\), where to obtain the last inequality above, we have used the estimate (4.10). Thus, using the estimate (4.12) in (4.9), it follows that
for some positive constants \(c,\,C\) depending on p. Therefore, using the estimate (4.13) in (4.7), we have
for some positive constants \(c,\,C\) depending on \(p,\,\alpha \).
Estimate of \(I_{14}:\) Let \(p>2\), then using the properties of \(\eta \) and Young’s inequality with exponents \(\frac{q}{p-2}\) and \(\frac{q}{q-p+2}\), using that \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (4.1), we have
for some constant \(C=C(N,h_0,p,q)>0\). Note that when \(p=2\), again using that \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (4.1), we have
which gives the estimate (4.15) for \(p=2\).
Estimate of \(I_{15}\): We observe that
Estimates of \(I_{16}\) and \(I_{17}\): If \(p=2\), using the boundedness assumption \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (4.1) and the properties of \(\eta \), we have
for some \(C=C(N,p)>0\). For \(p>2\), using Young’s inequality with exponents \(\frac{q}{p-2}\) and \(\frac{q}{q-p+2}\), we get
for \(C=C(N,h_0,p,q)>0\), where we have again used using the boundedness assumption \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (4.1). Therefore, using (4.17) and (4.18), for any \(p\ge 2\), we obtain
for \(C=C(N,h_0,p,q)>0\). Similarly, we obtain
for \(C=C(N,h_0,p,q)>0\). Combining the estimates (4.19) and (4.20) in (4.16), we have
for \(C=C(N,h_0,p,q)>0\). Using the estimates (4.15) and (4.21) in (4.14) we have
for \(c=c(p,\alpha )>0\) and \(C=C(N,h_0,p,q,\alpha )>0\).
Step 3: Estimate of the nonlocal integral J. First, we notice that
where
and
Estimate of \(J_1\): Proceeding exactly as in the proof of the estimate of \({\mathcal {I}}_1\) in [10, Step 1, pages 813-817], we get
for some constants \(c=c(p,\alpha )>0\) and \(C=C(p,\alpha )>0\), where
and
Proceeding along the lines of the proof of the estimates of \({\mathcal {I}}_{11}\)Footnote 2 and \({\mathcal {I}}_{12}\) in [10, Step 2, pages 817-819], we get
and
where \(C=C(N,h_0,p,s,q)>0\). Therefore, using the estimates (4.25) and (4.26) in (4.24), we have
for some constants \(c=c(p,\alpha )>0\) and \(C=C(N,h_0,p,s,q,\alpha )>0\).
Estimates of \(J_2\) and \(J_3\): Noting the assumptions in (4.1) and then proceeding along the lines of the proof of the estimates of \({\mathcal {I}}_2\) and \({\mathcal {I}}_3\) in [10, Step 3, pages 819-820], it follows that
where \(C=C(N,h_0,s,p)>0\). Combining the estimates (4.27) and (4.28) in (4.23), we have
for some constants \(c=c(p,\alpha )>0\) and \(C=C(N,h_0,p,s,q,\alpha )>0\).
Step 4: Going back to the equation. Inserting the estimates (4.22) and (4.29) in (4.6), it follows that
for some constant \(C=C(N,h_0,p,s,q,\alpha )>0\). Next, we estimate the integral in the left hand side of the above inequality (4.30). Indeed, we observe that following the lines of the proof of the estimate (4.12) in [10, page 821] (one can run the same argument with \(s=1\) there), we have the following estimate
where \(C=C(p,\alpha )>0\). Next, by Lemma 2.5 combined with the fact that \(\eta \) is supported only in \(B_R\)
where \(C=C(N,h_0,p)>0\). Noting the properties of \(\eta \), the fact that \(\Vert u\Vert _{L^\infty (B_1)}\le 1\) from (4.1) and using Young’s inequality as in the proof of the estimate (4.14) in [10, pages 821-822], for any \(0<|\xi |<h_0\), we get
where \(C=C(N,h_0,p)>0\). Combining (4.32) and (4.33) in (4.31), for every \(0<|\xi |<h_0\), we have
where \(C=C(N,h_0,p,\alpha )>0\). Choosing \(\xi =h\) and taking supremum over h for \(0<|h|<h_0\) and then using (4.34) in (4.30), it follows that
where \(C=C(N,h_0,p,q,s,\alpha )>0\).
Step 5: Conclusion. Now we set,
Therefore, we obtain
Plugging these values in (4.35), we finally deduce that
where \(C=C(N,h_0,p,q,s)>0\). In particular, recalling that \(r=R-4h_0\) and using Theorem 3 on page 277 in [25] to estimate the difference quotients, (4.36) gives
where \(C=C(N,h_0,p,q,s)>0\). \(\square \)
Lemma 4.2
(Estimate of the local seminorm) Let \(2\le p<\infty ,\,0<s<1\) and \(0\le A\le 1\). Suppose \(u\in W^{1,p}_{\textrm{loc}}(B_2(x_0))\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) is a weak solution of
satisfying
Then
Proof
Without loss of generality, we assume \(x_0=0\). We only provide the proof for \(w=u^+\), the proof of \(u^-\) is similar. We apply [28, Lemma 3.1] with \(r=1\), \(x_0=0\) and with \(\psi \in C_0^\infty (B_\frac{8}{9})\) such that \(\psi =1\) on \(B_\frac{7}{8}\), \(0\le \psi \le 1\) and \(|\nabla \psi | \le C\) for some \(C=C(N)>0\). By using the properties of \(\psi \) and \(A\in [0,1]\) this yields
Hence the result follows. \(\square \)
We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3
We first observe that \(u\in L^\infty _{\textrm{loc}}(\Omega )\), by [28, Theorem 4.2]. We assume for simplicity that \(x_0=0\), then we set
We point out that it is sufficient to prove that the rescaled function
satisfy the estimate
By scaling back, we would get the desired estimate. Observe that by definition, the function \(u_R\) is a local weak solution of \(-\Delta _p u+A\,R^{p-ps}(-\Delta _p)^s u=0\) in \(B_2\) and satisfies
The last estimate follows from Lemma 4.2. In what follows, we will omit the subscript R and simply write u in place of \(u_R\), in order not to overburden the presentation.
We fix \(0<\delta <1\) and choose \(i_\infty \in {\mathbb {N}}\setminus \{0\}\) such that
Then we define the sequence of exponents
We define also
We note that
By applying Proposition 4.1 withFootnote 3
and by (4.39) along with \(A\in [0,1]\) and \(R\in (0,1)\), we obtain
Noting that \(R_i-10 h_0=R_{i+1}+4h_0\) for every \(i=0,1,\ldots ,i_{\infty }-1\) and using Lemma 2.4 in (4.40), we get
Again, by Proposition 4.1 and applying (4.41), we obtain
Further, using Lemma 2.4 in (4.42), we get
Repeating this procedure, we obtain the iteration scheme
for all \(i=0,1,\ldots ,i_\infty -1\). Choosing \(i=i_{\infty }-1\) in (4.44) and using the facts that \(\Vert u\Vert _{L^{\infty }(B_1)}\le 1,\,\,[u]_{W^{1,p}(B_1)}\le 1\), we obtain
for \(C=C(N,p,s,\delta )>0\). Since \(q_{i_\infty }>N\) and \(R_{i_\infty }+4h_0=\frac{3}{4}\), by Morrey’s embedding theorem, we get \(u\in C^{\delta }_{\textrm{loc}}(B_{\frac{3}{4}})\) and
for \(C=C(N,p,s,\delta )>0\). Since \(\delta \in (0,1)\) is arbitrary, the result follows. \(\square \)
Since the result above implies that the nonlocal term is bounded when \(sp<(p-1)\), we can finally give the proof of Corollary 1.5.
Proof of Corollary 1.5
Upon rescaling as in the proof of Theorem 1.3, it is sufficient to prove that \(\Vert u_R\Vert _{C^{1,\alpha }(B_\frac{1}{8})}\le C\) with \(u_R\) as defined in (4.38) satisfying
Theorem 1.3 implies that there is \(\delta >sp/(p-1)\) such that
Now take any \(x_0\in B_{1/4}\). Then
by the choice of \(\delta \). Moreover,
by (4.45). Hence, \(\Vert (-\Delta _p)^s u_R\Vert _{L^\infty (B_{1/4})}\le C(N,s,p,\delta )\) and therefore also \(\Vert \Delta _p u_R\Vert _{L^\infty (B_{1/4})}\le C(N,s,p,\delta )\) which together with (4.45) and the well known \(C^{1,\alpha }\)-estimates for the p-Laplacian (see for instance the corollary on page 830 in [19]) imply
\(\square \)
5 Regularity for the inhomogeneous equation
In this section, we prove the boundedness and the regularity for the inhomogeneous equation.
5.1 Boundedness
We now address the boundedness, by comparing with the homogeneous equation. The first one is a consequence of Sobolev’s inequality.
Lemma 5.1
Let \(2\le p<\infty ,\,0<s<1\) and \(A\ge 0\). Suppose \(f\in L^q(\Omega )\) for \(q>N/p\) if \(p\le N\) and \(q\ge 1\) otherwise. Assume that \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) is a weak subsolution of
such that \(B_r(x_0)\Subset \Omega \) and that \(v\in W^{1,p}_{u}(B_r(x_0))\) solves
Then
and
Here, \(S_{N,p}\) is the constant in the Sobolev embedding in \(W^{1,p}\).
Proof
By Sobolev’s inequality and Hölder’s inequality we have
We test the difference of the equations for u and v with \((u-v)^+\) and observe that by Lemma A.1 and some manipulations, we have
Therefore, we may throw away the nonlocal term. We obtain from Hölder’s inequality and Lemma A.1
The two inequalities (5.4) and (5.5) together imply (5.1) and (5.2). Finally, using Poincaré’s inequality and (5.2), we obtain (5.3). \(\square \)
We now perform a Moser iteration to obtain the boundedness.
Proposition 5.2
(\(L^\infty \)-estimate) Let \(2\le p<\infty ,\,0<s<1\) and \(A\ge 0\). Suppose \(f\in L^q(\Omega )\) for \(q>N/p\) if \(p\le N\) and \(q\ge 1\) otherwise. Assume that \(u\in W^{1,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{sp}({\mathbb {R}}^N)\) is a weak subsolution of
such that \(B_r(x_0)\Subset \Omega \) and that v solves
Then \((u-v)^+\in L^\infty (B_r(x_0))\), with the following estimate
Proof
For simplicity, we assume that \(x_0=0\). We follow closely the proof of Theorem 3.1 in [11]. We first note that if \(p>N\) then by Morrey’s inequality
This together with Lemma 5.1 implies
which is the desired result.
We now prove the result for the positive part of \(u-v\) in the case \(p< N\) and then comment on how the proof would be changed if \(p=N\). Let \(w=(u-v)^+\), \(\delta >0\) and \(\beta > 1\). We observe that \(u+\delta \) is again a weak subsolution of (5.6). Insert the test functionFootnote 4
in the difference of the equations for \(u+\delta \) and v. The part coming from the nonlocal part will be non-negative. Indeed, this part is given by
by Lemma A.4 and some manipulations. For the local term we will apply Lemma A.1. This gives
By observing that for every \(\beta \ge 1\) we have
we can rewrite the previous estimate as
With \(\vartheta =(\beta +p-1)/p\), the previous inequality is equivalent to
We now proceed using the Sobolev inequality:
where \(p^*=(N\,p)/(N-p)\). By using this inequality in the left-hand side of (5.8), we get
and thus
By the triangle inequality
Therefore, since \(\vartheta =(\beta +p-1)/p\), we obtain
Using that \(\beta \ge 1\) we also have
Therefore,
so that
Now we make the choice
Then we obtain the estimate
or with the notation \(\gamma = \beta q'\) and \(\chi = p^*/(pq')>1\)
Now it is just a matter of following the exact same steps as in the proof of Theorem 3.1 in Brasco-Parini [11] with \(s=1\). Here we make the choices
Then
and
The final estimate becomes
for some constant \(C=C(p)>0\). Therefore
By the choice of \(\delta \) this becomes
By the estimate (5.1) in Lemma 5.1 we obtain
Comment on the case \(p=N\). For the case \(p=N\), we simply replace the Sobolev embedding with the embedding inequality of \(W_0^{1,N}(B_r)\) into \(L^{q}\) for q large. \(\square \)
Proof of Theorem 1.2
We may assume \(x_0=0\). Upon using the rescaling \(x\mapsto Rx\) it is also enough to prove the estimate
for a solution of
in \(B_1\). Take \(\rho =(1-\sigma )/2+\sigma \) and let v be the solution of
By Proposition 5.2
Moreover, by [28, Theorem 4.2] (note that \(R^{p-sp}< 1\), since \(R<1\))
Therefore,
where we used Lemma 5.1 to estimate the \(L^p\)-norm of \(v^+\) in terms of the \(L^p\)-norm of \(u^+\) and the fact that \(u=v\) outside \(B_{\rho }\) to estimate the tail term. This is the desired result. \(\square \)
5.2 Higher Hölder regularity
Here we turn our attention to the regularity of the inhomogenous equation. We first establish the regularity when f is small and then extend this to the desired result.
Proposition 5.3
Let \(2\le p<\infty \), \(0<s<1\) and q be such that
We consider \(\Theta =\Theta (N,p,q)\) the exponent defined as
For every \(0<\varepsilon <\Theta \) there exists \(\eta (N,p,q,s,\varepsilon )>0\) such that if \(f\in L^q_\textrm{loc}(B_4(x_0))\) and
then every weak solution \(u\in W^{1,p}_{\textrm{loc}}(B_4(x_0))\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) of the equation
that satisfy
belongs to \(C^{\Theta -\varepsilon }(\overline{B_{1/8}(x_0)})\) with the estimate
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. Here we prove that for every \(0<\varepsilon <\Theta \) and every \(0<r<1/2\), there exists \(\eta \) 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 \(u(0)=0\). Fix \(0<\varepsilon <\Theta \) and observe that it is sufficient 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, assume this is true. Then for every \(0<r<1/2\), there exists \(k\in {\mathbb {N}}\) such that \(\lambda ^{k+1}< r\le \lambda ^k\). From the first property in (5.10), we obtain
as desired.
We prove (5.10) by induction. For \(k=0\), (5.10) holds true by the assumptions in (5.9). Suppose (5.10) holds up to k, we now show that it also holds for \(k+1\), provided that
with \(\eta \) small enough, but independent of k. Define
By the hypotheses
Moreover
so that
Here we used the hypotheses on f and the definition of \(\Theta \), and again the fact that \(\lambda <1/2\). By Theorem 1.1, we may take \(h_k\) to be the weak solution of
By Proposition 5.2, we have
Then, we have the following estimate
We also used that \(h_k\) is \(C^{\Theta -\varepsilon /2}\) in \((\overline{B_{1/2}})\) thanks to Theorem 1.3, that impliesFootnote 5
Here we have observed that the quantities in the right-hand side are uniformly bounded, independently of k. Indeed, by the triangle inequality, Proposition 5.2 and (5.11) we have
Let
By choosing \(\eta \) so that \(2C\eta ^\frac{1}{p-1}<\lambda ^\Theta \) and \(\lambda \) small enough, we can transfer estimate (5.12) to \(w_{k+1}\). Indeed, we have
The previous estimate implies in particular that \(\Vert w_{k+1}\Vert _{L^\infty (B_1)}\le 1\) for \(\lambda \) satisfying
This information, rescaled back to u, is exactly the first part of (5.10) for \(k+1\). As for the second part of (5.10), the upper bound for \(|w_{k+1}|\) and the fact that \(\Theta <\frac{sp}{p-1}\) imply
By a change of variables and using that \(|w_k|\le 1\) in \(B_1\), we also see that
In addition, by the integral bound on \(w_k\) in (5.11)
In both estimates, we have also used that \(\lambda <1/2\) and the fact that
We observe that the constants \(C_2\) and \(C_3\) depend on N, p, q, s and \(\varepsilon \) only. From (5.14), (5.15) and (5.16), we get that the second part of (5.10) holds, provided that
By taking (5.13) into account, we finally obtain that (5.10) holds true at step \(k+1\) as well, provided that \(\lambda \) and \(\eta \) (depending on N, p, q, s and \(\varepsilon \)) are chosen so that
The induction is complete.
Part 2: We now show the desired regularity in the whole ball \(B_{1/8}\). We choose \(0<\varepsilon <\Theta \) and take the corresponding \(\eta \), obtained in Part 1. Take \(z_0\in B_{1/2}\), let \(L=2^{N+1}\,(1+|B_1|)\) and define
We observe that \(v\in W^{1,p}_{\textrm{loc}}(B_4)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)\) and that v is a weak solution in \(B_4\) of
with
By construction, we also have
and since \(B_{1/2}(z_0)\subset B_1\), it follows that
by the definition of L and the hypotheses in (5.9). Here we have used Lemma 2.3 in [10] with the balls \(B_{1/2}(z_0)\subset B_1\). We may therefore apply Part 1 to v and obtain
In terms of u this is the same as
We note that this holds for any \(z_0\in B_{1/2}\). Now take any pair \(x,y\in B_{1/8}\) and set \(|x-y|= r\). We observe that \(r<1/4\) and we set \(z=(x+y)/2\). Then we apply (5.18) with \(z_0=z\) and obtain
which is the desired result. \(\square \)
We are now in the position to prove Theorem 1.4.
Proof of Theorem 1.4
We may assume \(x_0=0\) without loss of generality. We modify u so that it fits into the setting of Proposition 5.3. We choose \(0<\delta <\Theta \), take \(\eta \) as in Proposition 5.3 with the choice \(\varepsilon =\Theta -\delta \) and set
By scaling arguments, it is sufficient to prove that the rescaled function
satisfies the estimate
It is easily seen that the choice of \({\mathcal {A}}_R\) implies
In addition, \(u_R\) is a weak solution of
with \(\Vert f_R\Vert _{L^{q}(B_{1})}\le \eta \) and \(R^{p-sp}<1\). We may therefore apply Proposition 5.3 with \(\varepsilon =\Theta -\delta \) to \(u_R\) and obtain
This concludes the proof. \(\square \)
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Notes
We assume for simplicity that \(1<p<N\). The cases \(p\ge N\) can be treated in the same manner, we leave the details to the reader.
We observe that by construction we have
$$\begin{aligned} 4\,h_0<R_i\le 1-5\,h_0,\qquad \text{ for } i=0,\dots ,i_\infty . \end{aligned}$$Thus these choices are admissible in Proposition 4.1.
This function is not really admissible but it can be made rigorous by instead taking \(\min (w,M)\) for some \(M>0\) and then letting \(M\rightarrow \infty \).
Note that Theorem 1.3 gives an estimate in \(B_\frac{1}{4}\), but by covering \(B_\frac{1}{2}\) with balls of radius 1/4 this yields an estimate in \(B_\frac{1}{2}\).
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Acknowledgements
E. L. is supported by the Swedish Research Council, Grant No. 2017-03736. 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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A Pointwise inequalities
A Pointwise inequalities
In this section, we list the pointwise inequalities used throughout the whole paper.
The following result can be found in [33, page 97, Inequality (I)].
Lemma A.1
For \(a,b\in {\mathbb {R}}^N\) and \(p\ge 2\), we have
For the following result, see [33, page 99, Inequality (V)].
Lemma A.2
Let \(a,b\in {\mathbb {R}}^N\). Then for any \(p\ge 2\), we have
For the following inequality, see [33, page 100, Inequality (VI)].
Lemma A.3
Let \(a,b\in {\mathbb {R}}^N\). Then, for any \(p\ge 2\), we have
The following is Lemma A.5 in [10].
Lemma A.4
Let \(p\ge 2\), \(\gamma \ge 1\) and \(a,b,c,d\in {\mathbb {R}}\). Then we have
for some \(C=C(p,\gamma )>0\).
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Garain, P., Lindgren, E. Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations. Calc. Var. 62, 67 (2023). https://doi.org/10.1007/s00526-022-02401-6
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DOI: https://doi.org/10.1007/s00526-022-02401-6