Abstract
We prove interior \(H^{2 s - \varepsilon }\) regularity for weak solutions of linear elliptic integro-differential equations close to the fractional s-Laplacian. The result is obtained via intermediate estimates in Nikol’skii spaces, which are in turn carried out by means of an appropriate modification of the classical translation method by Nirenberg.
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1 Introduction
One of the first fundamental achievements in the field of the regularity theory for weak solutions of second-order linear elliptic differential equations is the existence of weak second derivatives. Indeed, let \({\varOmega }\) be an open set of \(\mathbb {R}^n\) and \(u \in H^1({\varOmega })\) a weak solution of
where the \(n \times n\) matrix \(A = [a_{i j}]\) is uniformly elliptic, with entries \(a_{i j} \in C^{0, 1}_\mathrm{loc}({\varOmega })\), and the right-hand term \(f \in L^2({\varOmega })\). Then, one gets that \(u \in H^2_\mathrm{loc}({\varOmega })\) and, for any domain \({\varOmega }' \subset \subset {\varOmega }\),
for some constant \(C > 0\) independent of u and f.
Such result is typically ascribed to Louis Nirenberg, who in [26] obtained higher-order Sobolev regularity for general linear elliptic equations. To do so, he introduced the by now classical translation method. In the setting of Eq. (1.1), the idea is basically to consider the difference quotients
for \(i = 1, \ldots , n\) and \(h \ne 0\) suitably small in modulus, and use the equation itself to recover a uniform bound in h for the gradient of \(D_i^h u\) in \(L^2({\varOmega }')\). A compactness argument then shows that \(u \in H^2_\mathrm{loc}({\varOmega })\). Nice presentations of this technique are for instance contained in [14] and [16].
After this, several generalizations were achieved. For example, the translation method has been successfully adapted to study nonlinear equations and systems (see e.g. [9, 17, 22, 25, 34] and references therein). Furthermore, in Refs. [32] and [10] the authors deduced sharp higher-order regularity in both Sobolev and Besov classes for singular or degenerate equations of p-Laplacian type. See also [23, 24], where similar fractional estimates were obtained in a non-differentiable vectorial setting.
The object of this note is the attempt of a generalization of the above-discussed higher differentiability to a non-local analogue of Eq. (1.1), modelled upon the fractional Laplacian.
Given any open set \({\varOmega }\subset \mathbb {R}^n\), we consider a solution u of the linear equation
where \(f \in L^2({\varOmega })\) and \(\mathcal {E}_K\) is defined by
Here K is a measurable function which is comparable in the small to the kernel of the fractional Laplacian. Indeed, if we take
with \(s \in (0, 1)\), then (1.2) is the weak formulation of the equation
for the fractional Laplace operator of order 2s
On the other hand, more general kernels are admissible as well, possibly not translation invariant. However, if the kernel is not translation invariant, we need to impose on K some sort of joint local \(C^{0, s}\) regularity. We stress that this last hypothesis seems very natural to us. Indeed, while translation invariant kernels correspond in the local framework to the constant coefficient case, asking K to be locally Hölder continuous is a legitimate counterpart to the Lipschitz regularity assumed on the matrix A in (1.1).
Integro-differential equations have been the object of a great variety of studies in recent years. A priori estimates for quite general linear equations were obtained in [2, 31] (Hölder estimates) and in [1] (Schauder estimates). Other fundamental results in what concerns pointwise regularity were achieved by Caffarelli and Silvestre in [7, 8]. The two authors developed there a theory for viscosity solutions, in order to deal with general fully nonlinear equations. The framework considered here is instead that of weak (or energy) solutions. These two notions of solutions are of course very close, as it is discussed in [27] and [30], but, since we have a datum f in \(L^2\), the weak formulation (1.2) seems to us more appropriate.
The literature on the regularity theory for weak solutions is indeed very rich, and it is not possible to provide here an exhaustive account of the many contributions. Just to name a few, Kassmann addressed the validity of a Harnack inequality and established interior Hölder regularity for non-local harmonic functions through the language of Dirichlet forms (see [18–20]). In Ref. [27], the authors obtained Hölder regularity up to the boundary for a Dirichlet problem driven by the fractional Laplacian. Concerning regularity results in Sobolev spaces, \(H^{2 s}\) estimates are proved in [13] for entire translation invariant equations. Also, the very recent [21] provides higher differentiability/integrability in a nonlinear setting quite similar to ours.
Here we show that a solution u of (1.2) has better weak (fractional) differentiability properties in the interior of \({\varOmega }\). By adapting the translation method to this non-local setting, we prove that
Notice that the symbol \(N^{r, p}({\varOmega })\), for \(r > 0\) and \(1 \leqslant p < +\infty \), denotes here the so-called Nikol’skii space.
Since both Nikol’skii and fractional Sobolev spaces are part of the wider class of Besov spaces, standard embedding results within this scale allow us to deduce from (1.3) that
for any \(\varepsilon > 0\).
We do not know whether or not (1.4) is the optimal interior regularity for solutions of (1.2) in the Sobolev class. While one would arguably expect u to belong to \(H^{2 s}_\mathrm{loc}({\varOmega })\), there is no hope in general to extend such regularity up to the boundary, as discussed in Sect. 8. Finally, we stress that the exponent \(2 s - \varepsilon \) still provides Sobolev regularity for the gradient of u, when \(s > 1 / 2\).
We point out that, almost concurrently to the present work and independently from it, a result rather similar to (1.4) has been obtained in [3]. Indeed, the authors address there the problem of establishing higher Sobolev regularity for a nonlinear, superquadratic generalization of Eq. (1.2). When restricted to the linear case, their result is analogous to ours, for \(s \leqslant 1/2\), and slightly weaker, for \(s > 1/2\).
In the upcoming section, we specify the framework in which the model is set. We give formal definitions of the notion of solution and of the class of kernels under consideration. Moreover, we introduce the various functional spaces that are necessary for these purposes. After such preliminary work, we are then in position to give the precise statements of our results.
2 Definitions and formal statements
Let \(n \in \mathbb {N}\) and \(s \in (0, 1)\). The kernel \(K: \mathbb {R}^n \times \mathbb {R}^n \rightarrow [0, +\infty ]\) is assumed to be measurable and symmetricFootnote 1, that is
We also require K to satisfy
for some constants \(\Lambda \geqslant \lambda > 0\), \(\beta , M > 0\), and
for a.a. \(x, y, z \in \mathbb {R}^n\), with \(|x - y|, |z| < 1\), and for some \(\Gamma > 0\).
Condition (2.2a) tells that the kernel K is controlled from above and below by that of the fractional Laplacian when x and y are close. Conversely, when \(|x - y|\) is large, the behaviour of K could be more general, as expressed by (2.2b). Under these hypotheses, a great variety of kernels could be encompassed, as for instance truncated ones or having non-standard decay at infinity. Naturally, these requirements are fulfilled (with \(\beta = 2 s\)) when K is globally comparable to the kernel of the fractional Laplacian, that is when (2.2a) holds a.e. on the whole \(\mathbb {R}^n \times \mathbb {R}^n\).
On the other hand, (2.3) asserts that the map
is locally uniformly \(C^{0, s}\) regular, jointly in the two variables x and y. Clearly, (2.3) is satisfied by translation invariant kernels, i.e. those in the form
for some measurable \(k: \mathbb {R}^n \rightarrow [0, +\infty ]\). But more general choices are possible, as for instance kernels of the type
with \(a \in C^{0, s}(\mathbb {R}^n \times \mathbb {R}^n)\). We also stress that (2.3) may be actually weakened by requiring it to hold only inside the set \({\varOmega }\) where the equation will be valid.
In order to formulate the equation and state our main results, we introduce the following functional framework.
Let \(s > 0\), \(1 \leqslant p < +\infty \) and U be any open set of \(\mathbb {R}^n\). We indicate with \(L^p(U)\) the standard Lebesgue space and with \(W^{s, p}(U)\) the (fractional) Sobolev space as defined, for instance, in the monograph [12]. Of course, \(H^s(U) := W^{s, 2}(U)\).
Restricting ourselves to \(s \in (0, 1)\), we denote with X(U) the set of measurable functions \(u : \mathbb {R}^n \rightarrow \mathbb {R}\) such that
where
Notice that, by (2.2), if \(u \in X(U)\) and V is a bounded open set contained in U, then \(u|_V \in H^s(V)\). In addition, \(X_0(U)\) is the subspace of X(U) composed by the functions which vanish a.e. outside U. We refer the reader to [29, Section 5] for informations on very similar spaces of functions.
As it is customary, given any space F(U) of functions defined on a set U, we say that
Let now \({\varOmega }\) be a fixed open set of \(\mathbb {R}^n\). For \(u \in X({\varOmega })\) and \(\varphi \in X_0({\varOmega })\), it is well defined the bilinear form
Given \(f \in L^2({\varOmega })\), we say that \(u \in X({\varOmega })\) is a solution of
if
We remark that, for instance, when K is symmetric and translation invariant, i.e. as in (2.4) with k even, then (2.7) is the weak formulation of the equation
where the operator \(\mathscr {L}_k\) is defined—for u sufficiently smooth and bounded—by
As a last step towards the first theorem, we introduce a weighted Lebesgue space which we will require the solutions to lie in. Given a measurable function \(w: \mathbb {R}^n \rightarrow [0, +\infty )\), we say that \(u \in L^1_w(\mathbb {R}^n)\) if and only if
In what follows, we consider weights of the form
for \(x_0 \in \mathbb {R}^n\) and \(\beta > 0\) as in (2.2b). We denote the corresponding spaces just with \(L^1_{x_0, \beta }(\mathbb {R}^n)\), and we adopt the same notation for their norms. Also, we simply write \(L^1_\beta (\mathbb {R}^n)\) when \(x_0\) is the origin. Notice that, in fact, the space \(L^1_{x_0, \beta }(\mathbb {R}^n)\) does not depend on \(x_0\) and different choices for the base point \(x_0\) lead to equivalent norms. Lastly, we observe that, in consequence of the fact that \(w_{x_0, \beta } \in L^1(\mathbb {R}^n) \cap L^\infty (\mathbb {R}^n)\), the space \(L^1_\beta (\mathbb {R}^n)\) contains both \(L^\infty (\mathbb {R}^n)\) and \(L^1(\mathbb {R}^n)\).
With all this in hand, we are now ready to state the first and principal result of this note.
Theorem 1
Let \(s \in (0, 1)\), \(\beta > 0\) and \({\varOmega }\subset \mathbb {R}^n\) be an open set. Assume that the kernel K satisfies assumptions (2.1), (2.2) and (2.3). Let \(u \in X({\varOmega }) \cap L^1_\beta (\mathbb {R}^n)\) be a solution of (2.6), with \(f \in L^2({\varOmega })\). Then, \(u \in H_\mathrm{loc}^{2 s - \varepsilon }({\varOmega })\) for any small \(\varepsilon > 0\) and, for any domain \({\varOmega }' \subset \subset {\varOmega }\),
for some constant \(C > 0\) depending on n, s, \(\beta \), \(\lambda \), \(\Lambda \), M, \(\Gamma \), \({\varOmega }\), \({\varOmega }'\) and \(\varepsilon \).
The technique we adopt to prove Theorem 1 is basically the translation method of Nirenberg, suitably adjusted to cope with the difficulties arising in this fractional, non-local framework. However, this strategy does not immediately lead to an estimate in Sobolev spaces. In fact, it provides that the solution belongs to a slightly different functional space, which is well studied in the literature and is often referred to as Nikol’skii space. We briefly introduce such class here below.
Let U be a domain of \(\mathbb {R}^n\). Given \(k \in \mathbb {N}\) and \(z \in \mathbb {R}^n\), let
Observe that, by definition,
For any \(z \in \mathbb {R}^n\), we also define \(\tau _z u(x) := u(x + z)\) and
for any \(x \in U_z\). Sometimes we will need to deal with increments along the diagonal for the kernel K, as previously done in (2.3). With a slight abuse of notation, we write
We also consider increments of higher orders. For any \(k \in \mathbb {N}\), we set
for any \(x \in U_{k z}\), with the convention that \(\Delta _z^0 u = u\). Of course, \(\Delta _z^1 u = {\Delta }{_z}u\). Moreover, notice that by (2.11) all \(\Delta _z^j u\), as \(j = 0, 1, \ldots , k\), are well defined in \(U_{k z}\).
Given two parameters \(s \in (0, 2)\) and \(1 \leqslant p < +\infty \), the Nikol’skii space \(N^{s, p}(U)\) is defined as the space of functions \(u \in L^p(U)\) such that
The norm
makes \(N^{s, p}(U)\) a Banach space. We point out that the restriction to \(s < 2\) is assumed here only to avoid unnecessary complications in the definition of the semi-norm (2.12). By the way, the above range for s is large enough for our scopes, and thus, there is no real need to deal with more general conditions. Nevertheless, such limitation will not be considered anymore in Sect. 3, where a deeper look at the space \(N^{s, p}(U)\) will be given.
Now that the definition of Nikol’skii spaces has been recalled, we may finally head to our second main result.
Theorem 2
Let \(s \in (0, 1)\), \(\beta > 0\) and \({\varOmega }\subset \mathbb {R}^n\) be an open set. Assume that the kernel K satisfies assumptions (2.1), (2.2) and (2.3). Let \(u \in X({\varOmega }) \cap L^1_\beta (\mathbb {R}^n)\) be a solution of (2.6), with \(f \in L^2({\varOmega })\). Then, \(u \in N^{2 s, 2}_\mathrm{loc}({\varOmega })\) and, for any domain \({\varOmega }' \subset \subset {\varOmega }\),
for some constant \(C > 0\) depending on n, s, \(\beta \), \(\lambda \), \(\Lambda \), M, \(\Gamma \), \({\varOmega }\) and \({\varOmega }'\).
In the light of this estimate, Theorem 1 follows almost immediately. To see this, it is helpful to understand Sobolev and Nikol’skii spaces in the context of Besov spaces.
For \(s \in (0, 2)\), \(1 \leqslant p < +\infty \) and \(1 \leqslant \lambda \leqslant +\infty \), the Besov space \(B_{\tiny {\lambda }}^{s, p}(U)\) is the set composed by the functions \(u \in L^p(U)\) such that \([ u ]_{B_{\tiny {\lambda }}^{s, p}(U)} < + \infty \), where Observe that, by definition, \(B_{\tiny {\infty }}^{s, p}(U) = N^{s, p}(U)\), while the equivalence \(B_{\tiny {p}}^{s, p}(U) = W^{s, p}(U)\) is also true, though less trivial. Then, since there exist continuous embeddings
as \(1 \leqslant \lambda \leqslant \nu \leqslant + \infty \) and \(1< r< s < + \infty \), it follows
Consequently, up to some minor details that will be discussed later in Sect. 7, Theorem 1 is a consequence of Theorem 2.
Of course, Theorem 2 and inclusion (2.14) yield estimates in many other Besov spaces for the solution u of (2.6). Basically, u lies in any \(B_{\tiny {\lambda }, \mathrm{loc}}^{2 s - \varepsilon , 2}({\varOmega })\), with \(\varepsilon > 0\) and \(1 \leqslant \lambda \leqslant +\infty \).
We point out here that throughout the paper the same letter c is used to denote a positive constant which may change from line to line and depends on the various parameters involved.
The rest of the paper is organized as follows.
In Sect. 3, we review some basic material on Sobolev and Nikol’skii spaces. To keep a leaner notation, we do not approach Besov spaces in their full generality and restrict in fact to the two classes to which we are interested. Despite every assertion of this section is classical and surely well known to the experts, we choose to include here the few results that will be used afterwards, in order to make the work as self-contained as possible.
The subsequent two sections are devoted to some auxiliary results. Section 4 is concerned with a couple of technical lemmata that deal with a discrete integration by parts formula and an estimate for the defect of two translated balls. In Sect. 5, on the other hand, we prove a non-local version of the classical Caccioppoli inequality.
The main results are proved in Sects. 6 and 7.
Finally, Sect. 8 contains some comments on the possible optimal global regularity for the Dirichlet problem associated with (2.6).
3 Preliminaries on Sobolev and Nikol’skii spaces
We collect here some general facts about fractional Sobolev spaces and Nikol’skii spaces. As said before, we avoid dealing with the wider class of Besov spaces in order not to burden the notation too much. For more complete and exhaustive presentations, we refer the interested reader to the books by Triebel, [35–38].
We remark that the proofs displayed only make use of integration techniques, mostly inspired by [33]. While some results cannot be justified with such elementary arguments, we still provide specific references to the above-mentioned books.
Let \(U \subset \mathbb {R}^n\) be a bounded domain with \(C^\infty \) boundary.Footnote 2 Let \(1 \leqslant p < +\infty \) and \(s > 0\), with \(s \notin \mathbb {N}\). Write \(s = k + \sigma \), with \(k \in \mathbb {N}\cup \{ 0 \}\) and \(\sigma \in (0, 1)\). We recall that the fractional Sobolev space \(W^{s, p}(U)\) is defined as the set of functions
where, for \(v \in L^p(U)\),
Clearly, \(\alpha \) indicates a multi-index, i.e. \(\alpha = (\alpha _1, \ldots , \alpha _n)\) with \(\alpha _i \in \mathbb {N}\cup \{ 0 \}\), and \(|\alpha | = \alpha _1 + \cdots + \alpha _n\) is its modulus. Moreover, \(W^{k, p}({\varOmega })\), for \(k \in \mathbb {N}\), denotes the standard Sobolev space and, when \(k = 0\), we understand \(W^{0, p}(U) = L^p(U)\). The space \(W^{s, p}(U)\) equipped with the norm
is a Banach space.
Notice that, for \(v \in L^p(U)\),
In view of this fact, we have the following characterization for \(W^{s, p}(U)\).
Proposition 1
Let \(1 \leqslant p < +\infty \) and \(s > 0\). Let \(k, l \in \mathbb {Z}\) be such that \(0 \leqslant k < s\) and \(l > s - k\). Then,
is a Banach space norm for \(W^{s, p}(U)\), equivalent to \(\Vert \cdot \Vert _{W^{s, p}(U)}\).
A reference for these equivalences is given by Theorem 4.4.2.1 at page 323 of [37]. Note that the result is valid even if s is an integer.
Remark 1
In what follows, we will be mostly interested in norms with \(k = 0\) and therefore \(l > s\). In such cases, we stress that (3.1) may be replaced with the restricted norm
for any \(\delta > 0\), with no modifications to the space \(W^{s, p}(U)\). Indeed, we have
so that
Consequently, the norms defined by (3.1) and (3.2) are equivalent.
The second class of fractional spaces which we are interested in is the Nikol’skii spaces. For \(s = k + \sigma > 0\), with \(k \in \mathbb {N}\cup \{ 0 \}, \sigma \in (0, 1]\), and \(1 \leqslant p < +\infty \), define
where, for \(v \in L^p(U)\),
It can be showed that \(N^{s, p}(U)\) is a Banach space with respect to the norm
Notice that this definition of Nikol’skii space may seem to differ from that given in Sect. 2. In fact, this is not the case, as \(N^{s, p}(U)\) can be equivalently endowed with any norm of the form
where \(k, l \in \mathbb {Z}\) are such that \(0 \leqslant k < s\) and \(l > s - k\) (see again Theorem 4.4.2.1 of [37]).
Remark 2
As for the Sobolev spaces, we will consider norms with \(k = 0\) for the most of the time. We stress that in such cases (3.3) may be replaced with
for any integer \(l > s\) and any \(\delta > 0\).
In the last part of this section, we study the mutual inclusion properties of \(W^{s, p}(U)\) and \(N^{s, p}(U)\). In order to do this, it will be useful to consider another family of equivalent norms. To this aim, for \(l \in \mathbb {N}\) we introduce the so-called l-th modulus of smoothness of u
defined for any \(\eta > 0\). Then, we have
Proposition 2
Let \(s > 0\) and \(1 \leqslant p < +\infty \). Let \(l > s\) be an integer and \(0 < \delta \leqslant +\infty \). Then,
is a Banach space norm for \(W^{s, p}(U)\), equivalent to \(\Vert \cdot \Vert _{W^{s, p}(U)}\).
The same statement holds true for the norms
and the space \(N^{s, p}(U)\).
Proof
We only deal with the Sobolev space case, the Nikol’skii one being completely analogous and easier. Furthermore, we assume \(\delta = 1\). Then, an argument similar to that presented in Remark 1 shows that the result can be extended to any \(\delta \).
For \(u \in L^p(U)\) let
and
We claim that there exists a constant \(c \geqslant 1\) such that
for all \(u \in L^p(U)\). In view of Proposition 1 and Remark 1, this concludes the proof.
To check the left-hand inequality of (3.4), we first observe that
for any \(z \in \mathbb {R}^n\). Then, using polar coordinates,
Now we focus on the second inequality. In order to show its validity, we need the following auxiliary result. For \(x \in U\), \(\eta > 0\) and \(u \in L^p(U)\), let
and define
Then, by virtue of [38, Theorem 1.118] we infer that
for any \(u \in L^p(U)\).
Applying the generalized Minkowski’s inequality to the right-hand side of (3.5) and observing that
we get
Now, Jensen’s inequality implies that
and hence, (3.7) becomes
We finally switch to polar coordinates to compute
By combining this formula with (3.6), we obtain the right inequality of (3.4). Thus, the proof of the proposition is complete.
We are now in position to prove the main results of this section, concerning the relation between Sobolev and Nikol’skii spaces. First, we have
Proposition 3
Let \(s > 0\) and \(1 \leqslant p < +\infty \). Then,
and there exists a constant \(C > 0\), depending on n, s and p, such that
for any \(u \in L^p(U)\).
Proof
In view of Proposition 2, it is enough to prove that, if \(l \in \mathbb {Z}\) is such that \(l > s\), then
for some \(c > 0\). But this is in turn an immediate consequence of the monotonicity of \(\omega _p^l(u; \cdot )\). Indeed, \(\omega _p^l(u; \eta ) \geqslant \omega _p^l(u; t)\), for any \(\eta \geqslant t\), and so
Inequality (3.8) is then obtained by taking the supremum as \(t > 0\) on the right-hand side.
The following provides a partial converse to the above inclusion.
Proposition 4
Let \(s> r > 0\) and \(1 \leqslant p < +\infty \). Then,
and there exists a constant \(C > 0\), depending on n, r, s and p, such that
for any \(u \in L^p(U)\).
Proof
The result follows by noticing that, for \(l \in \mathbb {Z}\) with \(l > s\),
for any \(u \in L^p(U)\), and recalling Proposition 2.
4 Some auxiliary results
Before we can proceed to Sects. 5 and 6, which contain the core argumentations leading to Theorem 2, we need to prove a couple of subsidiary result.
First, we prove the following discrete version of the standard integration by parts formula.
Lemma 1
Let \(B_R\) be some ball of radius \(R > 0\) in \(\mathbb {R}^n\). Assume that K satisfies assumptions (2.1) and (2.2). Let \(u, v \in H^s(B_{8 R})\), with v supported in \(B_{2 R}\). Then,
for any \(z \in \mathbb {R}^n\) such that \(|z| < R\).
Proof
We first expand the integral on the left-hand side of (4.1), obtaining
Then, we write each term on the right-hand side of (4.2) asFootnote 3
We apply the change of variables \(\tilde{x} := x - i z\), \(\tilde{y} := y - i z\) in the first integral, to get
Writing then for \(i = 1, 2\)
and relabelling the variables \(\tilde{x}, \tilde{y}\) as x, y, formula (4.4) becomes
By using (4.3), (4.5) in (4.2) and noticing that \(\tau _{- i z} \chi _{\mathbb {R}^n \setminus B_{6 R}} = \chi _{\mathbb {R}^n \setminus (B_{6 R} + i z)}\), we finally obtain (4.1).
Then, we have the following result, in which we deduce an upper bound for the measure of the symmetric difference of two translated balls in terms of the modulus of the displacement vector. Although the estimate is almost immediate, we include a proof of it for completeness.
We also refer to [28] for a refined version of this result, holding for general bounded sets.
Lemma 2
Let \(B_R\) be some ball of radius \(R > 0\) in \(\mathbb {R}^n\). Then, for any \(z \in \mathbb {R}^n\),
where \(C > 0\) is a dimensional constant.
Proof
First, we observe that we may restrict ourselves to \(|z| \leqslant R / 2\), being the opposite case trivial. With the change of variables \(y := x / R\), we scale
where \(\hat{z} = z / R\). Then, we easily check that
to obtain
The result then follows, since \(1 - (1 - t)^n \leqslant n t\), for any \(t \geqslant 0\).
5 A Caccioppoli-type inequality
In this section, we present an estimate for the \(H^s\) norm of a solution u of (2.6) reminiscent of the classical one by Caccioppoli. Results of this kind are by now well established also for non-local equations, for instance in [4, 11, 21].
Proposition 5
Let \(s \in (0, 1)\), \(\beta > 0\) and \({\varOmega }\subset \mathbb {R}^n\) be an open set. Fix a point \(x_0 \in {\varOmega }\) and let \(r > 0\) be such that \(B_r(x_0) \subset \subset {\varOmega }\). Assume that K satisfies assumptions (2.1) and (2.2). Let \(u \in X({\varOmega }) \cap L^1_\beta (\mathbb {R}^n)\) be a solution of (2.6), with \(f \in L^2({\varOmega })\). Then,
for some constant \(C > 0\) depending on n, s, \(\beta \), \(\lambda \), \(\Lambda \), M, r and \({\mathrm{dist}}\left( B_r(x_0), \partial {\varOmega }\right) \).
We stress that hypothesis (2.3) is not assumed here. Consequently, Proposition 5 holds for a general measurable K which only satisfies (2.2).
Proof (Proof of Proposition 5)
Our argument follows the lines of those contained in the above- mentioned papers. Anyway, we provide all the details for the reader’s convenience.
First, observe that we may assume \(r < 1 / 2\) for the beginning. The case of a general radius \(r > 0\) will then follow by a covering argument. Take \(R > 0\) in such a way that \(r< R < 1 / 2\) and \(B_R(x_0) \subset {\varOmega }\). To simplify the notation, we write \(B_\rho \) instead of \(B_\rho (x_0)\), for any \(\rho > 0\).
Let \(\eta \in C^\infty _0(\mathbb {R}^n)\) be a cut-off function such that
Testing (2.7) with \(\varphi := \eta ^2 u \in X_0({\varOmega })\), we get
We estimate I. Notice that
and, therefore, using (2.2a),
Applying (5.2) and Young’s inequality, we deduce
which, together with (5.4), leads to
We now deal with J. Let \(x \in \mathbb {R}^n \setminus B_R\) and \(y \in B_{(R + r) / 2}\). Then,
and so
since \(R < 1\). In view of this and (2.2), we have
Moreover, using (5.2) we write
and hence by (5.6) and Young’s inequality we get
Finally, we easily compute
Putting (5.3), (5.5), (5.7) and (5.8) together, we obtain
where the first inequality follows from (5.2). Thus, (5.1) is proved.
6 Proof of Theorem 2
We are finally in position to proceed with the demonstration of our principal contribution.
Proof (Proof of Theorem 2)
Let \(x_0 \in {\varOmega }\) and \(R \in (0, 1 / 56)\) be such that \(B_{56 R}(x_0) \subset \subset {\varOmega }\). In the following, any ball \(B_r\) will always be assumed to be centred at \(x_0\). Let \(\eta \in C^\infty _0(\mathbb {R}^n)\) be a cut-off function satisfying
Fix \(z \in \mathbb {R}^n\), with \(|z| < R\), and plug \(\varphi := \Delta _{-z}^2 \left( \eta ^2 \Delta _z^2 u \right) \in X_0({\varOmega })\) in the weak formulation (2.7). By writing \(U = \Delta _z^2 u\), we have
We apply Lemma 1 to I with \(v = \eta ^2 U\), obtaining
Arguing as we did to obtain (5.4) in Proposition 5, we recover
The term \(I_2\) can be dealt with as follows. Applying (2.3) together with Young’s inequality, we have
with \(\delta > 0\). Taking \(\delta = \varepsilon ^{- 2} |z|^s\), for some small \(\varepsilon > 0\), we get
We now estimate \(I_3\). By adding and subtracting the terms
we see that
On the one hand, using (2.2a) and again the weighted Young’s inequality,
On the other hand
and hence
for any \(\gamma > 0\). In view of Lemma 2, we have
Therefore,
The choices \(\delta = \varepsilon ^{-2} |z|^{2 s}\) and \(\gamma = |z|^{2 \sigma - 1}\), for some
then yield
By combining (6.4) and (6.5) with the above inequality, recalling (6.3) and (6.6) we get
Now, we turn our attention to J. Arguing as in (5.7), we use once again (2.2), (6.1) and Young’s inequality to obtain
for any \(\delta > 0\). Setting again \(\delta = \varepsilon ^{- 2} |z|^{2 s}\), this becomes
Finally, we use Young’s inequality as before to deduce
By combining this last estimation, (6.7), (6.8) with (6.2) and noticing that
we find
In view of Proposition 3, we haveFootnote 4
Moreover,
and hence, recalling (6.1),
Consequently, if we choose \(\varepsilon \) suitably small, by (6.10), (6.11) and Proposition 5, estimate (6.9) becomes
where we also employed (6.6). Applying again Proposition 3,
for any \(w \in \mathbb {R}^n\). Taking \(w = z\), from (2.11), (6.1), (6.6) and (6.12), we then get
Now we consider separately the two cases \(s \in (0, 1 / 2]\) and \(s \in (1 / 2, 1)\).
In the first situation, we set \(\sigma = 1 / 2\). Notice that the choice is compatible with (6.6). By Proposition 3,
Therefore, from (6.13)
and thus \(u \in N^{2 s, 2}(B_R)\).
Now we address the more delicate case \(s \in (1 / 2, 1)\). Here we choose \(\sigma = s\) and first deduce from (6.13) and (6.14) that
Note that such a \(\sigma \) is admissible for (6.6), since \(s > 1 / 2\). Repeating the same argument with \(B_{8 R}\) in place of \(B_R\), we see that \(u \in N^{1 / 2 + s, 2}(B_{8 R})\) with
Consequently,
Using this last estimate in combination with (6.13) and selecting \(\sigma = 1\) there, again in agreement with (6.6), we conclude that \(u \in N^{2 s, 2}(B_R)\) and (6.15) is true also for \(s \in (1 / 2, 1)\).
Finally, we use Proposition 5 to control the Gagliardo semi-norm on the right-hand side of (6.15) and recover
Then, (2.13) follows for a general open \({\varOmega }' \subset \subset {\varOmega }\) by a standard covering argument.Footnote 5
We conclude this section with some brief comments on the technique just displayed.
To achieve the result, we tested the equation with a function modelled on the double increment \(\Delta _z^2 u\), which may seem a little unnatural and artificial. In fact, for \(s \in (0, 1/2]\) the first-order increment would have been sufficient. On the other hand, when \(s > 1 / 2\) this strategy is no more conclusive, basically since it leads to \(u \in N^{1 / 2 + s, 2}_\mathrm{loc}({\varOmega })\) only. In order to take advantage of this intermediate regularity and then gain the extra \(s - 1 / 2\) derivatives, we need the order of the increment to be at least 2.
7 Proof of Theorem 1
As previously discussed in Sect. 2, Theorem 1 essentially follows from Theorem 2, in the light of the embedding of Proposition 4. The only detail left is that the results of Sect. 3—specifically, Proposition 4— are only proved for sets having smooth boundary.
But this is not a big drawback. As a matter of fact, we know that estimate (2.9) holds for any domain \({\varOmega }' \subset \subset {\varOmega }\), with \(\partial {\varOmega }' \in C^\infty \). Then, it can be further extended to any \({\varOmega }'\), by noticing that it is always possible to find \({\varOmega }''\) with \(C^\infty \) boundary, such that \({\varOmega }' \subset \subset {\varOmega }'' \subset \subset {\varOmega }\).
8 Towards the optimal regularity up to the boundary
In this conclusive section, we briefly comment on the global Sobolev regularity for the Dirichlet problem driven by (2.6).
For \(x \in \mathbb {R}^n\), we define \(u_s(x) := (x_n)_+^s\). The function \(u_s\) solves
To see this, we write \(u_s(x) = \mu _s(x_n)\), with \(\mu _s(t) := t_+^s\) as \(t \in \mathbb {R}\), and we compute for \(x \in \mathbb {R}^n_+\)
Note that we use \(x'\) and \(y'\) to indicate the first \(n - 1\) components of x and y, respectively. Changing variables by setting \(z' := |y_n - x_n|^{- 1} (y' - x')\) in the inner integral, we get
where
is a finite constant. Then, the equation in (8.1) follows from the fact that \(\mu _s\) is s-harmonic in the half-line \((0, +\infty )\), as shown, for instance, in [6, 27] or [5].
Of course, the function \(u_s\) is of class \(C^{0, s}_\mathrm{loc}(\mathbb {R}^n)\), but not \(C^{0, \alpha }_\mathrm{loc}(\mathbb {R}^n)\), with \(\alpha > s\). On the other hand, the following proposition sheds some light on which could be the optimal Sobolev regularity of \(u_s\), at least when \(s \geqslant 1 / 2\).
Proposition 6
Let \(s \in [1 / 2, 1)\). Then, \(u_s \notin H^{2 s}_\mathrm{loc}(\overline{\mathbb {R}^n_+})\).
Proof
We focus on the case \(s > 1 / 2\), as when \(s = 1 / 2\) the computation is immediate.
Denoting with \(B'_r(z')\), the \((n - 1)\)-dimensional open ball of radius r and centre \(z'\)—with \(B_r' := B_r'(0)\) as usual—and with Q the cylinder \(B'_1 \times (0, 1)\), we shall prove that
First, setting
we claim that
Assuming for the moment (8.3) to hold, we check that then (8.2) follows. While for \(n = 1\) this is immediate, the case \(n \geqslant 2\) requires some comments. Indeed,
For \(\delta \in (0, 1/2)\), we consider the set
and we estimate its measure by computing
Noticing that on \(S(|x_n - y_n| / 4)\), it holds
and that \(|x_n - y_n| / 4 \leqslant 1 / 2\), we finally obtain
Thus, (8.2) is valid.
To complete the proof of the proposition, we are only left to show that (8.3) is true. To do this, we first note that, for \(t > 0,\)
Accordingly, \(\mu _s'\) is decreasing and for \(0< r< t < 1\) we have
so that
Claim (8.3) then follows, since the integral on the right- hand side of the above inequality does not converge.
We remark that, for \(s \in (0, 1 / 2)\), an almost identical argumentation leads to the conclusion that \(u_s \notin H^{s + 1 / 2}_\mathrm{loc}(\overline{\mathbb {R}^n_+})\).
Notes
We stress that the symmetry hypothesis does not really play much of a role here. Indeed, if one considers instead a non-symmetric kernel K, this can be written as the sum of its symmetric and anti-symmetric parts
$$\begin{aligned} K_\mathrm{{sym}}(x, y) := \frac{K(x, y) + K(x, y)}{2} \quad \text{ and } \quad K_\mathrm{{asym}}(x, y) := \frac{K(x, y) - K(y, x)}{2}. \end{aligned}$$But then, it is easily shown that \(K_\mathrm{{asym}}\) cancels out in (2.5), thus leading to an equation driven by the symmetric kernel \(K_\mathrm{{sym}}\). Hence, we may and do assume K symmetric from the outset.
In this regard, we refer the interested reader to [15], where a class of integro-differential equations with non-symmetric kernels are studied.
Most of the assertions contained in this section should be also true under less restrictive hypotheses on the boundary of the set. Of course, the definitions of the spaces require no assumptions at all on the boundary and other results are extended in the literature to Lipschitz sets. Unfortunately, we have not been able to find completely satisfactory references for Eq. 1, and its counterpart for Nikol’skii spaces, under such weaker assumptions. Anyway, the limitation to \(C^\infty \) domains will not have any influence on our applications.
The symbol \(D + z\), where D is a set and z a vector of \(\mathbb {R}^n\), identifies, as conventional, the set
$$\begin{aligned} \left\{ y \in \mathbb {R}^n : y = x + z \text{ with } x \in D \right\} . \end{aligned}$$In the following formulae, it is applied with D an Euclidean ball \(B_r\). Also, it should not be confused with the notation \((B_r)_z\), which will be used later on in Sect. 6 and has to be understood in the sense of definition (2.10).
Note that the right-hand side of (6.16) depends on the norm \(\Vert \cdot \Vert _{L^1_{x_0, \beta }(\mathbb {R}^n)}\) which in turn varies with \(x_0\). Consequently, while performing the covering argument one should take care that those norms depend on the centres of the covering balls. However, as noted in Sect. 2 such norms are all equivalent. The relative compactness of \({\varOmega }'\) then allows the use of a finite number of balls, thus preventing the blow-up of the constant c.
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The author thanks Enrico Valdinoci for his dedication, advice and encouragement.
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Cozzi, M. Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces. Annali di Matematica 196, 555–578 (2017). https://doi.org/10.1007/s10231-016-0586-3
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DOI: https://doi.org/10.1007/s10231-016-0586-3
Keywords
- Non-local equations
- Fractional Laplacian
- Regularity theory
- Translation method
- Fractional Sobolev spaces
- Nikol’skii spaces