Abstract
In this work we study nonlocal operators and corresponding spaces with a focus on operators of order near zero. We investigate the interior regularity of eigenfunctions and of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem and we prove that eigenfunctions are of class \(C^{\infty }\) if the kernel satisfies this property away from its singularity. Similarly in this case, if in the Poisson problem the right-hand is of class \(C^{\infty }\), then also any weak solution is of class \(C^{\infty }\).
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1 Introduction and main results
A crucial role in the investigation of differential operators is the study of eigenfunctions and corresponding eigenvalues, if they exist. In the classical case of the Laplacian \(-\Delta \) in a bounded domain \(\Omega \) in \(\mathbb {R}^N\), it is well known that there exists a sequence of functions \(u_n\in H^1_0(\Omega )\), \(n\in \mathbb {N}\) and corresponding values \(\lambda _{n}>0\) such that
Here, \(H^1_0(\Omega )\) is as usual the closure of \(C^{\infty }_c(\Omega )\) with respect to the norm \(u\mapsto \Big (\Vert u\Vert _{L^2(\Omega )}^2+\Vert |\nabla u|\Vert _{L^2(\Omega )}^2\Big )^{\frac{1}{2}}\).
With the well-known De Giorgi iteration in combination with the Sobolev embedding, it follows that \(u_n\) must be bounded and by a boot strapping argument using the regularity theory of the Laplacian it follows that \(u_n\) is smooth in \(\Omega \). In the model case of a nonlocal problem, one usually studies the fractional Laplacian \((-\Delta )^s\) with \(s\in (0,1)\). This operator can be defined via its Fourier symbol \(|\cdot |^{2s}\) and it can be shown that for \(\phi \in C^{\infty }_c(\mathbb {R}^N)\) we have
with a suitable normalization constant \(c_{N,s}>0\). As in the above classical case, it can be shown that there exists a sequence of functions \(u_n\in {\mathscr {H}}^s_0(\Omega )\), \(n\in \mathbb {N}\) and corresponding values \(\lambda _{s,n}>0\) such that
Here, the space \({\mathscr {H}}^s_0(\Omega )\) is given by the closure of \(C^{\infty }_c(\Omega )\)—understood as functions on \(\mathbb {R}^N\)—with respect to the norm \(u\mapsto \Big (\Vert u\Vert _{L^2(\Omega )}^2+{\mathscr {E}}_s(u,u)\Big )^{\frac{1}{2}}\), where for \(u,v\in C^{\infty }_c(\mathbb {R}^N)\) we set
With similar methods as in the classical case, it follows that \(u_n\) is smooth in the interior of \(\Omega \).
In the following we investigate the above discussion to the case where the kernel function \(z\mapsto c_{N,s}|z|^{-N-2s}\) is replaced by a measurable function \(j:\mathbb {R}^N\rightarrow [0,\infty ]\) such that for some \(\sigma \in (0,2]\) we have
With \(\sigma =2\), the above yields that \(j\,\textrm{d}z\) is a Lévy measure and the associated operator to this choice of kernel is of order below 2. In the following we focus on the case where the singularity of j is not too large, that is on the case \(\sigma <1\) so that the associated operator is of order strictly below one. We call the operator in this case also of small order.
Motivated by applications using nonlocal models, where a small order of the operator captures the optimal efficiency of the model [1, 24], nonlocal operators with possibly order near zero, i.e. if (1.1) is satisfied for all \(\sigma >0\), have been studied in linear and nonlinear integro-differential equations, see [5, 6, 11,12,13, 17, 18, 25] and the references in there. From a stochastic point of view, general classes of nonlocal operators appear as the generator of jump processes, where the jump behavior is modeled through types of Lévy measures and properties of associated harmonic functions have been studied, see [14, 16, 21, 23] and the references in there. In particular, operators of the form \(\phi (-\Delta )\) for a certain class of functions \(\phi \) are of interest from a stochastic and analytic point of view, see e.g. [2, 3] and the references in there.
In the following, we aim at investigating properties of bilinear forms and operators associated to a kernel j satisfying (1.1) for some \(\sigma \in (0,2]\) from a variational point of view. For this, some further assumptions on j are needed in our method and we present certain explicit examples at the end of this introduction, where our results apply.
Let \(\Omega \subset \mathbb {R}^N\) open, \(u,v\in C^{0,1}_c(\Omega )\) understood as functions defined on \(\mathbb {R}^N\), and consider the bilinear form
where we also write \(b_{j}(u,v):=b_{j,\mathbb {R}^N}(u,v)\) and \(b_{j,\Omega }(u):=b_{j,\Omega }(u,u)\), \(b_j(u)=b_{j}(u,u)\) resp. We denote
Associated to \(b_j\) there is a nonlocal self-adjoint operator \(I_j\) which for \(u,v\in C^{0,1}_c(\Omega )\) satisfies
To investigate the eigenvalue problem for \(I_j\), we further need the space
Note that clearly \({\mathscr {D}}^j(\mathbb {R}^N)=D^j(\mathbb {R}^N)\) and both \(D^j(\Omega )\) and \({\mathscr {D}}^j(\Omega )\) are Hilbert spaces with scalar products
Our first result concerns the eigenfunctions for the operator \(I_j\).
Theorem 1.1
Let (1.1) hold with \(\sigma =2\) and assume j satisfies additionally
Let \(\Omega \subset \mathbb {R}^N\) be a bounded open set. Then there exists a sequence \(u_n\in {\mathscr {D}}^j(\Omega )\), \(n\in \mathbb {N}\) and values \(\lambda _n\) such that
Here, we have
and \(u_1\) is unique up to a multiplicative constant—that is, \(\lambda _1\) is simple. Moreover, \(u_1\) can be chosen to be positive in \(\Omega \). Furthermore, the following statements hold.
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(1)
If in addition j satisfies
$$\begin{aligned} \int _{B_R(0){\setminus } B_r(0)} j^2(z)\ \textrm{d}z<\infty ~\text {for all}~ 0<r<R, \end{aligned}$$(1.7)then \(u_n\in L^{\infty }(\Omega )\) for every \(n\in \mathbb {N}\) and there is \(C=C(\Omega ,j,n)>0\) such that
$$\begin{aligned} \Vert u_n\Vert _{L^{\infty }(\Omega )}\le C\Vert u_n\Vert _{L^2(\Omega )}. \end{aligned}$$ -
(2)
Assume (1.1) holds with \(\sigma <\frac{1}{2}\), (1.7) holds, and there is \(m\in \mathbb {N}\cup \{\infty \}\) such that the following holds: We have \(j\in W^{l,1}(\mathbb {R}^N{\setminus } B_{\epsilon }(0)\) for every \(l\in \mathbb {N}\) with \(l\le 2m\), and there is some constant \(C_j>0\) such that
$$\begin{aligned} |\nabla j(z)|\le C_j|z|^{-1-\sigma -N}~\text {for all}~ 0<|z|\le 3. \end{aligned}$$Then \(u_n\in H^m_{loc}(\Omega )\). In particular, \(u_n\in C^{\infty }(\Omega )\) if \(m=\infty \).
For the definition of the Sobolev spaces \(W^{l,1}\) and \(H^m\) we refer to Sect. 2.1. The first part of Theorem 1.1 indeed follows immediately from the results of [19]. To show the boundedness, we emphasize that in our setting, there are no Sobolev embeddings available and thus it is not clear how to implement the approach via the De Giorgi iteration. We circumvent this, by generalizing the \(\delta \)-decomposition introduced in [13]. The proof of the regularity statement is inspired by the approach used in [7], where the author studies regularity of solutions to equations involving nonlocal operators which are in some sense comparable to the fractional Laplacian and uses Nikol’skii spaces. We emphasize that some of our methods generalize to the situation where the operator is not translation invariant and maybe perturbed by a convolution type operator. We treat these in the present work, too, see e.g. Theorem 4.3 below. Using a probabilistic and potential theoretic approach, a local smoothness of bounded harmonic solutions solving in a certain very weak sense \(I_ju=0\) in \(\Omega \), have been obtained in [16, Theorem 1.7] for radial kernel functions using the same regularity as we impose in statement (2) of Theorem 1.1 (see also [14, 23]). See also [15] for related regularity properties of solutions.
To present our generalization of the above mentioned \(\delta \)-decomposition, let us first note that the first equality in (1.4) can be extended, see Sect. 2. For this, let \({\mathscr {V}}^j(\Omega )\) denote the space of those functions \(u:\mathbb {R}^N\rightarrow \mathbb {R}\) such that \(u|_{\Omega }\in D^j(\Omega )\) and
Given \(f\in L^2_{loc}(\Omega )\), we then call \(u\in {\mathscr {V}}^j(\Omega )\) a (weak) supersolution of \(I_ju=f\) in \(\Omega \), if
In this situation, we also say that u satisfies in weak sense \(I_ju\ge f\) in \(\Omega \). Similarly, we define subsolutions and solutions.
We emphasize that this definition of supersolution is larger than the one considered in [19]. In the case where \(\sigma <1\) in (1.1) this allows via a density result to extend the weak maximum principles presented in [19] as follows.
Proposition 1.2
(Weak maximum principle) Assume (1.1) is satisfied with \(\sigma <1\) and assume that
Let \(\Omega \subset \mathbb {R}^N\) open and with Lipschitz boundary, \(c\in L^{\infty }_{loc}(\Omega )\), and assume either
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(1)
\(c\le 0\) or
-
(2)
\(\Omega \) and c are such that \(\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\inf _{x\in \Omega } \int _{\mathbb {R}^N{\setminus } \Omega }j(x-y)\ \textrm{d}y\).
If \(u\in {\mathscr {V}}^j(\Omega )\) satisfies in weak sense
then \(u\ge 0\) almost everywhere in \(\mathbb {R}^N\).
Remark 1.3
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The Lipschitz boundary assumption on \(\Omega \) is a technical assumption for the approximation argument.
-
(2)
Note that Assumption (1.10) readily implies the positivity \(\Lambda _1(\Omega )\) defined in (1.6), whenever \(\Omega \) is an open set in \(\mathbb {R}^N\) which is bounded in one direction, that is \(\Omega \) is contained (after a rotation) in a strip \((-a,a)\times \mathbb {R}^{N-1}\) for some \(a>0\) (see [11, 20]).
Up to our knowledge Proposition 1.2 is even new in the case of \(j=|\cdot |^{-N-2s}\), that is, the case of the fractional Laplacian (up to a multiplicative constant). In this situation, it holds \({\mathscr {V}}^j(\Omega )=H^s(\Omega )\cap {\mathscr {L}}^1_s\), where
and the proposition can be reformulated as follows.
Corollary 1.4
Let \(s\in (0,\frac{1}{2})\), \(\Omega \subset \mathbb {R}^N\) open and with Lipschitz boundary, \(c\in L^{\infty }_{loc}(\Omega )\), and assume either
-
(1)
\(c\le 0\) or
-
(2)
\(\Omega \) and c are such that \(\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\Lambda _1(\Omega )\).
If \(u\in H^s(\Omega )\cap {\mathscr {L}}^1_s\) satisfies in weak sense
then \(u\ge 0\) almost everywhere in \(\mathbb {R}^N\).
Similarly to the extension of the weak maximum principle, we have also the following extension of the strong maximum principle presented in [19] in the case \(\sigma <1\).
Proposition 1.5
(Strong maximum principle) Assume (1.1) is satisfied with \(\sigma <1\) and assume that j satisfies additionally (1.5). Let \(\Omega \subset \mathbb {R}^N\) open and \(c\in L^{\infty }_{loc}(\Omega )\) with \(\Vert c^+\Vert _{L^{\infty }(\Omega )}<\infty \). Moreover, let \(u\in {\mathscr {V}}^j(\Omega )\), \(u\ge 0\) satisfy in weak sense \(I_ju\ge c(x) u\) in \(\Omega \). Then the following holds.
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(1)
If \(\Omega \) is connected, then either \(u\equiv 0\) in \(\Omega \) or \({{\,\textrm{essinf}\,}}_Ku>0\) for any \(K\subset \subset \Omega \).
-
(2)
j given in (2.2) satisfies \({{\,\textrm{essinf}\,}}_{B_r(0)}j>0\) for any \(r>0\), then either \(u\equiv 0\) in \(\mathbb {R}^N\) or \({{\,\textrm{essinf}\,}}_Ku>0\) for any \(K\subset \subset \Omega \).
Our last results concern the Poisson problem associated to the operator \(I_j\).
Theorem 1.6
Let (1.1) hold with \(\sigma =2\) and assume j satisfies additionally (1.10). Let \(\Omega \subset \mathbb {R}^N\) be a bounded open set. Then for any \(f\in L^2(\Omega )\) there is a unique solution \(u\in {\mathscr {D}}^j(\Omega )\) of \(I_ju=f\). Moreover, if j satisfies (1.7) and \(f\in L^{\infty }(\Omega )\), then also \(u\in L^{\infty }(\Omega )\) and there is \(C=C(\Omega ,j)>0\) such that
Additionally, if (1.1) holds with \(\sigma <\frac{1}{2}\), there is \(m\in \mathbb {N}\cup \{\infty \}\) such that j satisfies the assumptions in Theorem 1.1(2), and it holds \(f\in C^{2m}(\overline{\Omega })\), then \(u\in H^m_{loc}(\Omega )\). More precisely in this case, for every \(\beta \in \mathbb {N}_0^N\), \(|\beta |\le m\) and \(\Omega '\subset \subset \Omega \) there is \(C=C(\Omega ,\Omega ',j,\beta )>0\) such that
In particular, if \(m=\infty \) and \(f\in C^{\infty }(\Omega )\), then \(u\in C^{\infty }(\Omega )\).
Our approach to prove Theorems 1.1 and 1.6 is uniformly by considering an equation of the form
for \(f\in L^2(\Omega )\), \(h\in L^{1}(\mathbb {R}^N)\cap L^2(\mathbb {R}^N)\), and \(\lambda \in \mathbb {R}\). Moreover, several of our results need \(I_j\) not to be translation invariant and this setup is discussed in Sect. 2.
1.1 Examples
We close this introduction with some classes of operators covered by our results.
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(1)
As introduced in [5, 13, 18] the logarithmic Laplacian
$$\begin{aligned} L_{{ \Delta \,}}\phi (x)=c_NP.V.\int _{B_1(0)}\frac{\phi (x)-\phi (x+y)}{|y|^{N}}\ \textrm{d}y - c_N\int _{\mathbb {R}^N{\setminus } B_1(0)}\frac{\phi (x+y)}{|y|^{N}}\ \textrm{d}y+\rho _N\phi (x),\nonumber \\ \end{aligned}$$(1.11)appears as the operator with Fourier-symbol \(-2\ln (|\cdot |)\) and can be seen as the formal derivative in s of \((-\Delta )^s\) at \(s=0\). Here
$$\begin{aligned} c_N=\frac{\Gamma (\frac{N}{2})}{\pi ^{N/2}}= \frac{2}{|S^{N-1}|} ~ \text {and}~ \rho _N :=2\ln (2)+\psi \left( \frac{N}{2}\right) - \gamma \end{aligned}$$(1.12)where \(\psi := \frac{\Gamma '}{\Gamma }\) denotes the digamma function and \(\gamma := -\psi (1)=-\Gamma '(1)\) is the Euler-Mascheroni constant. With \(j(z)=c_N1_{B_1(0)}(z)|z|^{-N}\) the operator \(L_{{ \Delta \,}}\) can be seen as a bounded perturbation of the operator class discussed in the introduction. The following sections cover in particular this operator.
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(2)
The logarithmic Schrödinger operator \((I-\Delta )^{\log }\) as in [12] is an integro-differential operator with Fourier-symbol \(\log (1+|\cdot |^2)\) and also appears as the formal derivative in s of the relativistic Schrödinger operator \((I-\Delta )^s\) at \(s=0\),
$$\begin{aligned} (I-\Delta )^{\log }u(x)= d_{N}P.V.\int _{\mathbb {R}^N}\frac{u(x)-u(x+y)}{|y|^{N}}\omega (|y|)\ \textrm{d}y, \end{aligned}$$where \(d_N=\pi ^{-\frac{N}{2}}\), \(\omega (r)=2^{1-\frac{N}{2}}r^{\frac{N}{2}}K_{\frac{N}{2}}(r)\) and \(K_{\nu }\) is the modified Bessel function of the second kind with index \(\nu \). More generally, operators with symbol \(\log (1+|\cdot |^{\beta })\) for some \(\beta \in (0,2]\) are studied in [22].
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(3)
Finally, also nonradial kernels of the type considered in [19] satisfy in particular the assumptions (2.1) and (4.1). See also also [14, 22, 23] and references in there.
The paper is organized as follows. In Sect. 2 we collect some general results concerning the spaces used in this paper and resulting definitions of weak sub- and supersolutions. Section 3 is devoted to show several density results, which then are used to show the Propositions 1.2 and 1.5. In Sect. 4 we present a general approach to show boundedness of solutions and in Sect. 5 we give the proof of an interior \(H^1\)-regularity estimate for solutions from which we then deduce the interior regularity statement as claimed in Theorems 1.1(2) and 1.6.
Notation In the remainder of the paper, we use the following notation. Let \(U,V\subset \mathbb {R}^N\) be nonempty measurable sets, \(x \in \mathbb {R}^N\) and \(r>0\). We denote by \(1_U: \mathbb {R}^N \rightarrow \mathbb {R}\) the characteristic function, |U| the Lebesgue measure, and \(\text {diam}(U)\) the diameter of U. The notation \(V \subset \subset U\) means that \({\overline{V}}\) is compact and contained in the interior of U. The distance between V and U is given by \(\text {dist}(V,U):= \inf \{|x-y|\,:\, x \in V,\, y \in U\}\). Note that this notation does not stand for the usual Hausdorff distance. If \(V= \{x\}\) we simply write \(\text {dist}(x,U)\). We let \(B_r(U):=\{x\in \mathbb {R}^N\;:\; \text {dist}(x,U)<r\}\), so that \(B_r(x):=B_r(\{x\})\) is the open ball centered at x with radius r. We also put \(B:=B_1(0)\) and \(\omega _N:=|B|\). Finally, given a function \(u: U\rightarrow \mathbb {R}\), \(U \subset \mathbb {R}^N\), we let \(u^+:= \max \{u,0\}\) and \(u^-:=-\min \{u,0\}\) denote the positive and negative part of u, and we write \(\text {supp}\ u\) for the support of u given as the closure in \(\mathbb {R}^N\) of the set \({\{ x\in U\;:\; u(x)\ne 0\}}\).
2 Preliminaries
In the following, we generalize the translation invariant setting of the introduction. For this and from now on, let \(k:\mathbb {R}^N\times \mathbb {R}^N\rightarrow [0,\infty ]\) be a measurable function satisfying for some \(\sigma \in (0,2]\)
We let \(j:\mathbb {R}^N\rightarrow [0,\infty ]\) be the symmetric lower bound of k given by
For \(\Omega \subset \mathbb {R}^N\) open let
where, if \(u=v\) we put
and we drop the index \(\Omega \), if \(\Omega =\mathbb {R}^N\). Note that we have for any fixed \(x\in \Omega \) that \(\kappa _{k,\Omega }(x)<\infty \) by (2.1). We consider the function spaces
Lemma 2.1
Let \(U\subset \Omega \subset \mathbb {R}^N\) open and \(u:\mathbb {R}^N\rightarrow \mathbb {R}\). Then the following hold:
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(1)
\(u\in {\mathscr {D}}^k(\Omega )\ \Rightarrow \ u|_{\Omega }\in D^k(\Omega )\).
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(2)
\({\mathscr {D}}^k(U)\subset {\mathscr {D}}^k(\Omega )\subset {\mathscr {V}}^k(\Omega )\subset {\mathscr {V}}^k(U)\subset {\mathscr {V}}^k_{loc}(U)\).
Proof
This follows immediately from the definitions (see also [19, Section 3]). \(\square \)
Lemma 2.2
(See Proposition 3.3 in [19] or Proposition 1.7 in [20]) For \(\Omega \subset \mathbb {R}^N\) open, let \(\Lambda _1(\Omega )\) be given by (c.f. (1.6))
and let
Then \(\lim \limits _{r\rightarrow \infty }\lambda (r)\ge \int _{\mathbb {R}^N}j(z)\ \textrm{d}z\) with \(j(z):={{\,\textrm{essinf}\,}}\{k(x,x\pm z)\;:\; z\in \mathbb {R}^N\}\) for \(z\in \mathbb {R}^N\) as in (2.2).
Lemma 2.3
Let \(\Omega \subset \mathbb {R}^N\) open and let X be any of the above function spaces. Then the following hold:
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(1)
\(b_{k,\Omega }\) is a bilinear form and in particular we have \(b_{k,\Omega }(u,v)\le b_{k,\Omega }^{1/2}(u)b_{k,\Omega }^{1/2}(v)\). Moreover, \(D^k(\Omega )\) and \({\mathscr {D}}^k(\Omega )\) are Hilbert spaces with scalar products
$$\begin{aligned} \langle u,v\rangle _{D^k(\Omega )}&=\langle u,v\rangle _{L^2(\Omega )}+b_{k,\Omega }(u,v),\\ \langle u,v\rangle _{{\mathscr {D}}^k(\Omega )}&=\langle u,v\rangle _{L^2(\Omega )}+b_{k,\mathbb {R}^N}(u,v). \end{aligned}$$ -
(2)
If \(u\in X\), then \(u^{\pm },|u|\in X\) and we have \(b_{k,\Omega '}(u^+,u^-)\le 0\) for all \(\Omega '\subset \Omega \) with \(b_{k,\Omega '}(u)<\infty \).
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(3)
If \(g\in C^{0,1}(\mathbb {R}^N)\), \(u\in X\), then \(g\circ u\in X\).
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(4)
\(C^{0,1}_c(\Omega )\subset X\).
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(5)
\(\phi \in C^{0,1}_c(\Omega )\), \(u\in X\), then \(\phi u\in {\mathscr {D}}^k(\Omega )\), where if necessary we extend u trivially to a function on \(\mathbb {R}^N\). Moreover, there is \(C=C(N,k,\Vert \phi \Vert _{C^{0,1}(\Omega )})>0\) such that
$$\begin{aligned} b_{k,\mathbb {R}^N}(\phi u)\le C\Big (\Vert u\Vert _{L^2(\Omega ')}^2+b_{k,\Omega '}(u)\Big ) \end{aligned}$$for any \(\Omega '\subset \Omega \) with \(\text {supp}\,\phi \subset \subset \Omega '\).
Proof
Theses statements follow directly from the definition (c.f. [19, Section 3]). To be precise in the last part, let \(\phi \in C^{0,1}_c(\Omega )\) and fix \(L:=\Vert \phi \Vert _{C^{0,1}(\Omega )}\). That is, we have
Then using the inequality for \(x,y\in \mathbb {R}^N\)
we find by the assumptions (2.1)
\(\square \)
Remark 2.4
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(1)
Note that for \(u,v\in {\mathscr {D}}^k(\Omega )\) we have
$$\begin{aligned} b_{k}(u,v)=b_{k,\mathbb {R}^N}(u,v)=b_{k,\Omega }(u,v)+K_{k,\Omega }(u,v). \end{aligned}$$ -
(2)
It follows in particular that there is a nonnegative self-adjoint operator \(I_k\) associated to \(b_{k,\mathbb {R}^{N}}=b_k\) as mentioned in the introduction.
Lemma 2.5
Let \(\Omega \subset \mathbb {R}^N\) open and \(u\in {\mathscr {V}}_{loc}^k(\Omega )\). Then \(b_{k}(u,\phi )\) is well-defined for any \(\phi \in C^{\infty }_c(\Omega )\).
Proof
Let \(\phi \in C^{\infty }_c(\Omega )\) and fix \(U\subset \subset \Omega \) such that \(\text {supp}\ \phi \subset \subset U\). Then with the symmetry of k
where \(\epsilon =\text {dist}(\text {supp}\ \phi ,\mathbb {R}^N{\setminus } U)>0\). \(\square \)
Definition 2.6
Let \(\Omega \subset \mathbb {R}^N\) open and \(f\in L^1_{loc}(\Omega )\). Then \(u\in {\mathscr {V}}_{loc}^k(\Omega )\) is called a weak supersolution of \(I_ku=f\) in \(\Omega \), if
We also say that u satisfies \(I_ku\ge f\) weakly in \(\Omega \).
Similarly, we define weak subsolutions and solutions.
Remark 2.7
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(1)
We note that by Assumption 2.1, it follows that for any function \(u\in {\mathscr {V}}^k_{loc}(\Omega )\) with \(u|_{\Omega }\in C^{0,1}(\Omega )\) for \(\Omega \subset \mathbb {R}^N\) open, we have \(I_ku|_U\in L^{\infty }(U)\) for any \(U\subset \subset \Omega \) and
$$\begin{aligned} I_ku(x)=\int _{\mathbb {R}^N}(u(x)-u(y))k(x,y)\ \textrm{d}y~\text {for} ~x\in \Omega . \end{aligned}$$This follows similarly to the proof of the statements in Lemma 2.3.
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(2)
If \(u\in {\mathscr {V}}^k(\Omega )\), then indeed also \(b_k(u,\phi )\) is well-defined for all \(\phi \in {\mathscr {D}}^k(\Omega )\). Hence also \(b_k\) is well defined on \({\mathscr {V}}^k_{loc}(\Omega )\times {\mathscr {D}}^k(U)\) for all \(U\subset \subset \Omega \). In some of our results the statements need a Lipschitz-boundary of \(\Omega \), which comes into play due to approximation with \(C^{\infty }_c(\Omega )\)-functions (see Section 3 below). However, this can be weakened, if \(u\in {\mathscr {V}}^k(\Omega )\) and the space of test-functions is adjusted.
Lemma 2.8
Let \(\Omega \subset \mathbb {R}^N\) open. Let \(\Omega _1\subset \subset \Omega _2\subset \subset \Omega _3\subset \subset \Omega \). Let \(\eta \in C^{0,1}_c(\mathbb {R}^N)\) such that \(0\le \eta \le 1\) in \(\mathbb {R}^N\) and we have
Let \(f\in L^1_{loc}(\Omega )\) and let \(u\in {\mathscr {V}}^k_{loc}(\Omega )\) satisfy in weak sense \(Iu\ge f\) in \(\Omega \). Then the function \(v=\eta u\in {\mathscr {D}}^k(\Omega _3)\) satisfies in weak sense \(Iv\ge f+g_{\eta ,u}(x)\) in \(\Omega _1\), where
Proof
The fact, that \(v\in {\mathscr {D}}^k(\Omega _3)\) follows from Lemma 2.3. Let \(\phi \in C^{\infty }_c(\Omega _1)\), then
Here, since \((1-\eta )u\equiv 0\) on \(\Omega _2\), we have
Thus the claim follows. \(\square \)
Remark 2.9
The same result as in Lemma 2.8 also holds if “\(\ge \)” in the solution type is replaced by “\(\le \)” or “\(=\)”.
In the following, it is useful to understand functions \(u\in D^k(\Omega )\) satisfying \(b_{k,\Omega }(u)=0\).
Proposition 2.10
Assume that the symmetric lower bound j of k defined in (2.2) satisfies \(\int _{\mathbb {R}^N} j(z)\ \textrm{d}z=\infty \). Let \(\Omega \subset \mathbb {R}^N\) open and bounded and let \(u\in D^k(\Omega )\) such that \(b_{k,\Omega }(u)=0\). Then u is constant.
Proof
Let \(x_0\in \Omega \) and fix \(r>0\) such that \(B_{2r}(x_0)\subset \Omega \). Denote \(q(z):=\min \{c,j(z)\}1_{B_r(0)}(z)\), where we may fix \(c>0\) such that \(|\{q>0\}|>0\) due to the assumption on j. Then by Lemma A.1 we have
where \(a*b=\int _{\mathbb {R}^N}a(\cdot -y)b(y)\ \textrm{d}y\) denotes as usual the convolution. Note that since \(q\in L^1(\mathbb {R}^N)\cap L^{\infty }(\mathbb {R}^N)\) with \(q=0\) on \(\mathbb {R}^N{\setminus } B_r(0)\), it follows that \(q*q\in C(\mathbb {R}^N)\) with support in \(B_{2r}(0)\) and we have
by the assumption on j. Hence there is \(R>0\) with \(q*q\ge \epsilon \) for some \(\epsilon >0\) and thus we have
for any \(\rho \in (0,\frac{R}{2}]\). But then \(u(x)=u(y)\) for almost every \(x,y\in B_{R/2}(x_0)\) so that u is constant a.e. in \(B_{\rho }(x_0)\). Since \(b_{k,\Omega }(u)=b_{k,\Omega }(u-m)\) for any \(m\in \mathbb {R}\), we may next assume that \(u=0\) in \(B_{R/2}(x_0)\) and show that indeed we have \(u=0\) a.e. in \(\Omega \). Denote by W the set of points \(x\in \Omega \) such that there is \(r>0\) with \(u=0\) a.e. in \(B_r(x)\). By definition W is open and the above shows that W is nonempty. Next, let \((x_n)_n\subset W\) be a sequence with \(x_n\rightarrow x\in \Omega \) for \(n\rightarrow \infty \). Then there is \(r_x>0\) such that \(B_{4r_x}(x)\subset \Omega \) and we can find \(n_0\in \mathbb {N}\) such that \(x\in B_{r_x}(x_n)\subset B_{2r_x}(x_n)\subset \Omega \) for \(n\ge n_0\). Repeating the above argument, it follows that u must be zero in \(B_{r_x}(x_n)\) and thus \(x\in W\). Hence, W is relatively open and closed in \(\Omega \) and since W is nonempty, we have \(W=\Omega \). That is \(u=0\) in \(\Omega \). \(\square \)
2.1 On Sobolev and Nikol’skii spaces
We recall here the notations and properties of Sobolev and Nikol’skii spaces as introduced in [7, 26]. In the following, let \(p\in [1,\infty )\) and \(\Omega \subset \mathbb {R}^N\) open.
2.1.1 Sobolev spaces
If \(k\in \mathbb {N}_0\), we set as usual
for the Banach space of k-times (weakly) differentialable functions in \(L^p(\Omega )\). Moreover, as usual, for \(\sigma \in (0,1)\), \(p\in [1,\infty )\) we set
With the norm
the space \(W^{\sigma ,p}(\Omega )\) is a Banach space. For general \(s=k+\sigma \), \(k\in \mathbb {N}_0\), \(\sigma \in [0,1)\) the Sobolev space is defined as
Finally, in the particular case \(p=2\) the space \(H^{s}(\Omega ):=W^{s,2}(\Omega )\) is a Hilbert space.
2.1.2 Nikol’skii spaces
For \(u:\Omega \rightarrow \mathbb {R}\) and \(h\in \mathbb {R}\), let \(\Omega _h:=\{x\in \Omega \;:\; \text {dist}(x,\partial \Omega )>h\}\) and, with \(e\in \partial B_1(0)\), we let
Moreover, for \(l\in \mathbb {N}\), \(l>1\) let
For \(s=k+\sigma >0\) with \(k\in \mathbb {N}_0\) and \(\sigma \in (0,1]\) define
where
It follows that \(N^{s,p}(\Omega )\) is a Banach space with norm \(\Vert u\Vert _{N^{s,p}(\Omega )}:=\Vert u\Vert _{W^{k,p}(\Omega )}+\sum _{|\alpha |=k}[\partial ^{\alpha }u]_{N^{\sigma ,p}(\Omega )}\). It can be shown that this norm is equivalent to
for any fixed \(m,l\in \mathbb {N}_0\) with \(m<\sigma \) and \(l>\sigma -m\) (see [26, Theorem 4.4.2.1]).
Proposition 2.11
(See e.g. Propositions 3 and 4 in [7]) Let \(\Omega \subset \mathbb {R}^N\) open and with \(C^{\infty }\) boundary. Moreover, let \(t>s>0\) and \(1\le p<\infty \). Then
3 Density results and maximum principles
The main goal of this section is to show the following.
Theorem 3.1
Let either \(\Omega =\mathbb {R}^N\) or \(\Omega \subset \mathbb {R}^N\) open and bounded with Lipschitz boundary. In the following, let \(X(\Omega ):={\mathscr {D}}^k(\Omega )\) or \(D^k(\Omega )\). Then \(C^{\infty }_c(\Omega )\) is dense in \(X(\Omega )\). Moreover, if \(u\in X(\Omega )\) is nonnegative, then we have
-
(1)
There exists a sequence \((u_n)_n\subset X(\Omega )\cap L^{\infty }(\Omega )\) with \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \(X(\Omega )\) satisfying that for every \(n\in \mathbb {N}\) there is \(\Omega '_n\subset \subset \Omega \) with \(u_n=0\) on \(\Omega {\setminus } \Omega '_n\) and \(0\le u_n\le u_{n+1}\le u\).
-
(2)
There exists a sequence \((u_n)_n\subset C^{\infty }_c(\Omega )\) with \(u_n\ge 0\) for every \(n\in \mathbb {N}\) and \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \(X(\Omega )\).
Remark 3.2
To put Theorem 3.1 into perspective, we consider the following examples.
-
(1)
In the case \(k(x,y)=|x-y|^{-2s-N}\) for some \(s\in (0,\frac{1}{2})\), the above Theorem is well-known and leads to the interesting property that for any open, bounded Lipschitz set \(\Omega \subset \mathbb {R}^N\) we have
$$\begin{aligned} D^k(\Omega )=H^s(\Omega )=H^s_0(\Omega ). \end{aligned}$$We emphasize that the above equality also holds for \(s=\frac{1}{2}\). Moreover, if \(s<\frac{1}{2}\), it also holds \(H^s(\Omega )=\{u\in H^s(\mathbb {R}^N)\;:\; 1_{\mathbb {R}^N{\setminus }\Omega }u\equiv 0\}\).
-
(2)
If \(k(x,y)=1_{B_1(0)}(x-y)|x-y|^{-N}\), \(D^k(\Omega )\) is associated to the function space of the localized logarithmic Laplacian (see [5]).
The proof is split into several smaller steps. Recall that \({\mathscr {D}}^k(\mathbb {R}^N)=D^k(\mathbb {R}^N)\) by definition.
Lemma 3.3
Let \(u\in D^k(\mathbb {R}^N)\). Then there is a sequence \((u_n)_n\subset D^k(\mathbb {R}^N)\) with \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \(D^k(\mathbb {R}^N)\) satisfying that for every \(n\in \mathbb {N}\) there is \(\Omega _n\subset \subset \mathbb {R}^N\) with \(u_n=0\) on \(\mathbb {R}^N{\setminus } \Omega _n\). Moreover, if \(u\ge 0\), then \((u_n)_n\) can be chosen to satisfy in addition \(0\le u_n\le u_{n+1}\le u\).
Proof
For \(n\in \mathbb {N}\) let \(\phi _n\in C^{0,1}_c(\mathbb {R}^N)\) be radially symmetric and such that \(\phi _n\equiv 1\) on \(B_n(0)\), \(\phi _n\equiv 0\) on \(B_{n+1}(0)^c\). Clearly, we may assume that \([\phi _n]_{C^{0,1}(\mathbb {R}^N)}=1\). By Lemma 2.3 there is hence some \(C=C(N,k)>0\) with \(b_{k,\mathbb {R}^N}(\phi _n u)\le C\Vert u\Vert _{D^k(\mathbb {R}^N)}\) for all \(n\in \mathbb {N}\). In the following, let \(u_n:=\phi _nu\) and without loss of generality we may assume \(u\ge 0\). Since then \(0\le u-u_n\le u\) on \(\mathbb {R}^N\) and \(u-u_n=0\) on \(B_n\), by dominated convergence we have \(\lim \limits _{n\rightarrow \infty }\Vert u-u_n\Vert _2=0\). Moreover, by choice of \(\phi _n\) we have for \(x,y\in \mathbb {R}^N\)
Here, \(U(x,y)\in L^2(\mathbb {R}^N\times \mathbb {R}^N,k(x,y)\ d(x,y))\), since
Thus \(\lim \limits _{n\rightarrow \infty }b_{k,\mathbb {R}^N}(u-u_n)=0\) by the dominated convergence Theorem. \(\square \)
Proposition 3.4
We have that \(C^{\infty }_c(\mathbb {R}^N)\) is dense in \(D^k(\mathbb {R}^N)\). Moreover, if \(u\in D^k(\mathbb {R}^N)\) is nonnegative, then there exists \((\phi _n)_n\subset C^{\infty }_c(\mathbb {R}^N)\) with \(\phi _n\ge 0\) for every \(n\in \mathbb {N}\) and \(\lim \limits _{n\rightarrow \infty }\phi _n= u\) in \(D^k(\mathbb {R}^N)\).
Proof
Let \(u\in D^k(\mathbb {R}^N)\). Moreover, let \(\phi _n\in C^{0,1}_c(\mathbb {R}^N)\) for \(n\in \mathbb {N}\) be given by Lemma 3.3 such that \(\Vert u-\phi _n u\Vert _{s,p}<\frac{1}{n}\). Then \(v_n:=\phi _n u\in D^k(\mathbb {R}^N)\) and there is \(R_n>0\) with \(v_n\equiv 0\) on \(\mathbb {R}^N{\setminus } B_{R_n}(0)\). Next, let \((\rho _\epsilon )_{\epsilon \in (0,1]}\) by a Dirac sequence and denote \(v_{n,\epsilon }:=\rho _\epsilon *v_n\). Then \(v_{n\epsilon }\in C^{\infty }_c(\mathbb {R}^N)\) for all \(n\in \mathbb {N}\), \(\epsilon \in (0,1]\) and
It is hence enough to show that \(v_{n,\epsilon }\rightarrow v_n \) in \(D^k(\mathbb {R}^N)\) for \(\epsilon \rightarrow 0\). In the following, we write v in place of \(v_n\) and \(v_\epsilon =\rho _\epsilon *v\) in place of \(v_{n,\epsilon }\) for \(\epsilon \in (0,1]\). Moreover, let \(R=R_n>0\) with \(v=v_n=0\) on \(\mathbb {R}^N{\setminus } B_R(0)\). Clearly, \(v_{\epsilon }\rightarrow v\) in \(L^2(\mathbb {R}^N)\) for \(\epsilon \rightarrow 0\) and this convergence is also pointwise almost everywhere. Hence it is enough to analyze the convergence of \(b_{k,\mathbb {R}^N}(v-v_{\epsilon })\) as \(\epsilon \rightarrow 0\). From here, the proof follows along the lines of [19, Proposition 4.1] noting that there it is not used that k only depends on the difference of x and y. Note here, that if u is nonnegative then the above constructed sequence is also nonnegative. \(\square \)
Lemma 3.5
Let \(\Omega \subset \mathbb {R}^N\) open and such that \(\partial \Omega \) is bounded. Denote \(\delta (x):=\text {dist}(x,\mathbb {R}^N{\setminus } \Omega )\). Then the following is true.
-
(1)
There is \(C=C(N,\Omega ,k)>0\) such that \(\kappa _{k,\Omega }(x)\le C\delta ^{-\sigma }(x)\ \) for \(x\in \Omega \).
-
(2)
If \(\Omega \) is bounded, then \(1_{\Omega }\in D^k(\mathbb {R}^N)\).
Proof
Let \(C=C(N,\Omega ,k)>0\) be constants varying from line to line and denote \(U:=\{x\in \mathbb {R}^N\;:\; \text {dist}(x,\Omega )\le 1\}\). To see item 1., let \(x\in \Omega \) and fix \(p\in \partial \Omega \) such that \(\delta (x)=|x-p|\). Then
where we have used that \(|x-p|\le |x-y|\) for \(y\in \mathbb {R}^N{\setminus } \Omega \). Now 2. follows immediately from 1., since we have
\(\square \)
Theorem 3.6
(See Theorem 3.1) Let \(\Omega \subset \mathbb {R}^N\) be an open bounded set with Lipschitz boundary. Then \(C^{\infty }_c(\Omega )\) is dense in \({\mathscr {D}}^k(\Omega )\). Moreover, if \(u\in {\mathscr {D}}^k(\Omega )\) is nonnegative, then we have
-
(1)
There exists a sequence \((u_n)_n\subset {\mathscr {D}}^k(\Omega )\) with \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \({\mathscr {D}}^k(\Omega )\) satisfying that for every \(n\in \mathbb {N}\) there is \(\Omega '_n\subset \subset \Omega \) with \(u_n=0\) on \(\mathbb {R}^N{\setminus } \Omega '_n\) and \(0\le u_n\le u_{n+1}\le u\).
-
(2)
There exists a sequence \((u_n)_n\subset C^{\infty }_c(\Omega )\) with \(u_n\ge 0\) for every \(n\in \mathbb {N}\) and \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \({\mathscr {D}}^k(\Omega )\).
Proof
Note that the second claim follows immediately from the first one using [19, Proposition 4.1] as in the proof of Proposition 3.4. Then also the main claim follows by considering \(u^{\pm }\) separately. Hence it is enough to show 1. We proceed similar to [5, Theorem 3.1]. Denote \(\delta (x):=\text {dist}(x,\mathbb {R}^N{\setminus } \Omega )\). For \(r>0\), define the Lipschitz map
Note that we have \(\phi _s\le \phi _r\) for \(0<s\le r\). We show
Note that once this is shown, we have \(u(1-\phi _r)\in {\mathscr {D}}^k(\Omega )\) for \(r>0\) sufficiently small and \(u(1-\phi _r)\rightarrow u\) for \(r\rightarrow 0\). Since also \(0\le u(1-\phi _r)\le u(1-\phi _s)\) for \(0<s\le r\) and \(u(1-\phi _r)=0\) for \(x\in \mathbb {R}^N\) with \(\delta (x)\le r\), it follows that (3.1) implies 1.
The remainder of the proof is to show (3.1). For this, let \(C=C(N,\Omega ,k)>0\) be a constant which may vary from line to line. Let \(A_t:=\{x\in \Omega \;:\; \delta (x)\le t\}\). Note that \(u\phi _r\) vanishes on \(\mathbb {R}^N{\setminus } A_{2r}\), we have \(0\le \phi _r\le 1\) and, moreover,
Then proceeding similarly to the proof of Lemma 2.3.(5) we find for r small enough
Note here, since \(u\in D^k(\mathbb {R}^N)\), we have \(\int _{A_{4r}}u(x)^2 \textrm{d}x +b_{k,A_{4r}}(u)\rightarrow 0\) for \(r\rightarrow 0\). Moreover, we have by Lebesgue’s differentiation theorem
Finally, since
and, by Lemma 3.5, we have
for \(x\in A_{2r}\), so that also \(\int _{A_{2r}}u^2(x)\kappa _{k,A_{4r}}(x)\ \textrm{d}x\rightarrow 0\) for \(r\rightarrow 0\) with a similar argument. \(\square \)
Proof of Theorem 3.1
for \(X(\Omega )={\mathscr {D}}^k(\Omega )\) This statement now follows from Theorem 3.6, Lemma 3.3, and Proposition 3.4. \(\square \)
Theorem 3.7
Let \(\Omega \subset \mathbb {R}^N\) be an open bounded set with Lipschitz boundary. Then \(C^{\infty }_c(\Omega )\) is dense in \(D^k(\Omega )\). Moreover, if \(u\in D^k(\Omega )\) is nonnegative, then we have
-
(1)
There exists a sequence \((u_n)_n\subset D^k(\Omega )\cap L^{\infty }(\Omega )\) with \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \(D^k(\Omega )\) satisfying that for every \(n\in \mathbb {N}\) there is \(\Omega '_n\subset \subset \Omega \) with \(u_n=0\) on \(\Omega {\setminus } \Omega '_n\) and \(0\le u_n\le u_{n+1}\le u\).
-
(2)
There exists a sequence \((u_n)_n\subset C^{\infty }_c(\Omega )\) with \(u_n\ge 0\) for every \(n\in \mathbb {N}\) and \(\lim \limits _{n\rightarrow \infty }u_n= u\) in \(D^k(\Omega )\).
Proof
Consider the Lipschitz map
Then \(v_n:=g_n(u)\in D^k(\Omega )\cap L^{\infty }(\Omega )\) and we have with \(\phi _r\) as in the proof of Proposition 3.6
Clearly, \(b_{k,\Omega }(u-v_n)\rightarrow 0\) for \(n\rightarrow \infty \) by dominated convergence and \(b_{k,\Omega }( \phi _rv_n)\rightarrow 0\) for \(r\rightarrow 0\) analogously to the proof of Proposition 3.6, noting that the term in (3.2) reads in this case
In particular, statement (1) follows. Now statement (2) and the density statement follow analogously, again, to the proof of Proposition 3.6. \(\square \)
Proof of Theorem 3.1
for \(X(\Omega )=D^k(\Omega )\) This statement now follows from Theorem 3.7, Lemma 3.3, and Proposition 3.4. \(\square \)
Remark 3.8
It is tempting to conjecture the following type of Hardy inequality: There is \(C>0\) such that
if \(\Omega \) is a bounded Lipschitz set and k is such that its symmetric lower bound j is not in \(L^1(\mathbb {R}^N)\). Let us mention that for \(k(x,y)=|x-y|^{-2s-N}\) this holds for \(s\in (0,1)\), \(s\ne \frac{1}{2}\), see [4, 8]. Moreover, for \(k(x,y)=1_{B_{1}(0)}(x-y)|x-y|^{-N}\), this has been shown in [5]. In the general framework presented here, however, it is not clear if this is true.
Remark 3.9
With the above density results, we can now note that our definition of weak supersolutions (and similarly of weak subsolutions and solutions), see Definition 2.6, can be extended slightly:
Let \(u\in {\mathscr {V}}^k_{loc}(\Omega )\) satisfy weakly \(I_ku\ge f\) in \(\Omega \) for some \(f\in L^1_{loc}(\Omega )\) and \(\Omega \subset \mathbb {R}^N\) open and bounded with Lipschitz boundary.
-
(1)
If \(f\in L^2_{loc}(\Omega )\), then by density it also holds
$$\begin{aligned} b_k(u,v)\ge \int _{U}f(x)v(x)\ \textrm{d}x~\text {for all nonnegative} ~v\in {\mathscr {D}}^k(U), U\subset \subset \Omega . \end{aligned}$$(3.3) -
(2)
If \(u\in {\mathscr {V}}^k(\Omega )\cap L^{\infty }(\mathbb {R}^N)\) and \(f\in L^2(\Omega )\), then by density it also holds
$$\begin{aligned} b_k(u,v)\ge \int _{\Omega }f(x)v(x)\ \textrm{d}x~\text {for all nonnegative}~ v\in {\mathscr {D}}^k(\Omega ). \end{aligned}$$(3.4)
Finally note that if \(u:\mathbb {R}^N\rightarrow \mathbb {R}\) satisfies \(u1_{U}\in D^k(U)\) for some \(U\subset \subset \mathbb {R}^N\) and \(u\in L^{\infty }(\mathbb {R}^N{\setminus } U)\), then \(u\in {\mathscr {V}}^k_{loc}(U)\).
Proposition 3.10
(Weak maximum principle) Assume that the symmetric lower bound j of k defined in (2.2) satisfies
Let \(\Omega \subset \mathbb {R}^N\) open and with Lipschitz boundary, \(c\in L^{\infty }_{loc}(\Omega )\), and assume either
-
(1)
\(c\le 0\) or
-
(2)
\(\Omega \) and c are such that \(\Vert c^{+}\Vert _{L^{\infty }(\Omega )}<\inf _{x\in \Omega } \int _{\mathbb {R}^N{\setminus } \Omega }k(x,y)\ \textrm{d}y\).
If \(u\in {\mathscr {V}}^k(\Omega )\) satisfies in weak sense
then \(u\ge 0\) almost everywhere in \(\mathbb {R}^N\).
Proof
Note that also \(u^-\in {\mathscr {V}}^k(\Omega )\) and in particular \(u^-\in D^k(\Omega )\). Hence, we can find \((v_n)_n\subset C^{\infty }_c(\Omega )\) with \(v_n\rightarrow u^-\) in \(D^k(\Omega )\) for \(n\rightarrow \infty \) with \(0\le v_n\le v_{n+1}\le u^-\) by Proposition 3.7. Then
On the other hand, since \(u^+v_n=0\) for all \(n\in \mathbb {N}\) and \(u\ge 0\) almost everywhere in \(\mathbb {R}^{N}{\setminus } \Omega \), we find
Hence
Since \(v_n\rightarrow u^-\) in \(D^k(\Omega )\), it follows that \(b_{k,\Omega }(u^-,u^-)=0\), but then \(u^-\) is constant by Proposition 2.10 in \(\Omega \). Assume by contradiction that \(u^-=m>0\). Then the above calculation gives
which is in both cases a contradiction: If in case 1. \(c\le 0\), then since \(\kappa _{k,\Omega }(x)\not \equiv 0\) and since \(v_n\rightarrow m\) in \(D^k(\Omega )\) the right-hand side of (3.6) is negative.
In case 2. this contradiction is immediate in a similar way. \(\square \)
Proof of Proposition 1.2
The statement follows immediately from Proposition 3.10. \(\square \)
Remark 3.11
Usually, the weak maximum principle is stated with an assumption on the first eigenvalue \(\Lambda _1(\Omega )\) in place of \(\inf _{x\in \Omega }\kappa _{k,\Omega }(x)\). This can be done once the Hardy inequality in Remark 3.8 is shown. Indeed, following the proof of Proposition 3.10 gives
With \(n\rightarrow \infty \) and using that by the Hardy inequality it holds \({\mathscr {D}}^k(\Omega )=D^k(\Omega )\), it follows that
and the conclusion follows similarly.
Proof of Corollary 1.4
This statement now follows from Remark 3.11 using Remark 3.8. \(\square \)
Proposition 3.12
(Strong maximum principle) Assume k satisfies additionally (4.1). Let \(\Omega \subset \mathbb {R}^N\) open and \(c\in L^{\infty }_{loc}(\Omega )\) with \(\Vert c^+\Vert _{L^{\infty }(\Omega )}<\infty \). Moreover, let \(u\in {\mathscr {V}}^k(\Omega )\), \(u\ge 0\) satisfy in weak sense \(I_ku\ge c(x) u\) in \(\Omega \).
-
(1)
If \(\Omega \) is connected, then either \(u\equiv 0\) in \(\Omega \) or \({{\,\textrm{essinf}\,}}_Ku>0\) for any \(K\subset \subset \Omega \).
-
(2)
If j given in (2.2) satisfies \({{\,\textrm{essinf}\,}}_{B_r(0)}j>0\) for any \(r>0\), then either \(u\equiv 0\) in \(\mathbb {R}^N\) or \({{\,\textrm{essinf}\,}}_Ku>0\) for any \(K\subset \subset \Omega \).
Proof
This statement follows by approximation from [19, Theorem 2.5 and 2.6]. Here, the statement \(j\notin L^1(\mathbb {R}^N)\) comes into play since we need
to conclude the statement for arbitrary c as stated. \(\square \)
Proof of Proposition 1.5
The statement follows immediately from Proposition 3.12. \(\square \)
4 On boundedness
In the following, let \(h*u(x)=\int _{\mathbb {R}^N} h(x-y)u(y)\ \textrm{d}y\) as usual denote the convolution of two functions.
Theorem 4.1
Assume k is such that
and it holds
Let \(\Omega \subset \mathbb {R}^N\) be an open set. Let \(f\in L^{\infty }(\Omega )\), \(h\in L^1(\mathbb {R}^N)\cap L^{2}(\mathbb {R}^N)\), and let \(u\in {\mathscr {V}}^k_{loc}(\Omega )\) satisfy in weak sense
If \(u^+\in L^{\infty }(\mathbb {R}^N{\setminus } \Omega ')\) for some \(\Omega '\subset \subset \Omega \), then \(u^+\in L^{\infty }(\mathbb {R}^N)\) and there is \(C=C(\Omega ,\Omega ',k,h,\lambda )>0\) such that
Proof
Let \(\Omega _1,\Omega _2,\Omega _3\subset \mathbb {R}^N\) be with Lipschitz boundary and such that
Let \(\eta \in C^{0,1}_c(\Omega _3)\) such that \(0\le \eta \le 1\) and \(\eta =1\) on \(\Omega _2\). Put \(v=\eta u\) and, for \(\delta >0\), denote \(J_{\delta }(x,y):=1_{B_{\delta }(0)}(x-y)k(x,y)\) and \(k_{\delta }(x,y)=k(x,y)-J_{\delta }(x,y)\). Note that by Assumption (2.1) it follows that \(y\mapsto k_{\delta }(x,y)\in L^1(\mathbb {R}^N)\) for all \(x\in \mathbb {R}^N\). Moreover, by Assumption (4.1)
Hence, we may fix \(\delta >0\) such that
In the following, \(C_i>0\), \(i=1,\ldots \) denote constants depending on \(\Omega '\), \(\Omega _i\), \(\lambda \), \(\delta \), \(\Omega \), \(\eta \), k, and h but may vary from line to line—clearly, by the choices of these dependencies are actually only through \(\lambda \), \(\Omega \), \(\Omega '\), \(\eta \), k, and h. First note that by Lemma 2.8 we have in weak sense
In the following, put
Then note that for \(x\in \mathbb {R}^N\) we have
and, since \(\sup \limits _{x\in \Omega _1}\int _{\mathbb {R}^N{\setminus } \Omega _2}(1-\eta (y))k(x,y)\ \textrm{d}y\le C_2\sup \limits _{x\in \Omega _1}\int _{\mathbb {R}^N{\setminus } \Omega _2}\min \{1,|x-y|^{\sigma }\}k(x,y)\ \textrm{d}y<\infty \), it also holds that
Whence, since \(u=v\) in \(\Omega _1\), we have in weak sense
Next, let \(\mu \in C^{\infty }_c(\Omega '')\) for some \(\Omega '\subset \subset \Omega ''\subset \subset \Omega _1\) such that \(0\le \mu \le 1\), \(\mu =1\) on \(\Omega '\), and \(\mu =0\) on \(\mathbb {R}^N{\setminus } \Omega ''\). Let \(\phi _t=\mu ^2(v-t)^+\in {\mathscr {D}}^k(\Omega '')\) for \(t>0\) and note that
Fix \(t>0\) such that
That is, we fix
Then with (4.3)
Note here, that for \(x\in \mathbb {R}^N\) we have by the integrability assumptions on \(k_{\delta }\) and k
so that using that \(v\ge t\) in \(\text {supp}\,\phi _t\) we have
On the other hand, with \(v_t(x)=v(x)-t\), we have
Whence with Poincaré’s inequality, using that by Assumption 4.1 there is for any \(K\subset \mathbb {R}^N\) open and bounded, some \(C>0\) such that \(b_{J_{\delta }}(u)\ge C\Vert u\Vert _{L^2(\mathbb {R}^N)}^2\) for \(u\in {\mathscr {D}}^{J_{\delta }}(K)\), we find for some constant \(C_7\)
Combining (4.7) and (4.4) we have
Whence \(v_t^+=0\) in \(\Omega '\) and thus \(u=v\le t=A\cdot C_{8}\) in \(\Omega '\) as claimed. \(\square \)
Corollary 4.2
If in the situation of Theorem 4.1 we have in weak sense \(I_ku=\lambda u+h*u+f\) in \(\Omega \), then we have \(u\in L^{\infty }(\Omega ')\) and there is \(C=C(\Omega ,\Omega ',k,\lambda ,h)>0\) such that
Proof
This follows by replacing u with \(-u\) (and f with \(-f\)) in the statement of Theorem 4.1. \(\square \)
Theorem 4.3
If in the situation of Theorem 4.1 we have in weak sense \(I_ku=\lambda u+h*u+f\) in \(\Omega \) and \(u\in {\mathscr {D}}^k(\Omega )\), then we have \(u\in L^{\infty }(\Omega )\) and there is \(C=C(\Omega ,k,\lambda ,h)>0\) such that
Proof
Using in the proof of Theorem 4.1 the test-function \(u_t^+\) instead of \(\phi _t\) (and similarly for Corollary 4.2), we find
as claimed. \(\square \)
5 On differentiability of solutions
In the following, \(\Omega \subset \mathbb {R}^N\) is an open bounded set and k satisfies through out the assumptions (2.1) with some \(\sigma <\frac{1}{2}\), (4.1), and (4.2). Moreover, we assume that there is \(j:\mathbb {R}^N\rightarrow [0,\infty ]\) such that \(k(x,y)=j(x-y)\) for \(x,y\in \mathbb {R}^N\) and that for some \(m\in \mathbb {N}\cup \{\infty \}\) the following holds: We have \(j\in W^{l,1}(\mathbb {R}^N{\setminus } B_{\epsilon }(0)\) for every \(l\in \mathbb {N}\) with \(l\le 2m\), and there is some constant \(C_j>0\) such that
For simplicity, we write j in place of k and fix
Theorem 5.1
Let \(f\in H^1(\Omega )\), \(\lambda \in \mathbb {R}\) and \(u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)\) satisfy in weak sense \(I_ju=f+\lambda u\) in \(\Omega \). Then for any \(\Omega '\subset \subset \Omega \) there is \(C=C(N,\Omega ,\Omega ',j,\lambda )>0\) such that
Proof
Let \(\Omega '\subset \subset \Omega \) and fix \(r\in (0,\frac{1}{8})\) small such that \(8r\le \text {dist}(\Omega ',\mathbb {R}^N{\setminus }\Omega )\). Moreover, fix \(x_0\in \Omega '\) and denote \(B_n:=B_{nr}(x_0)\). Note that by using assumption (4.1) with Lemma 2.2 we achieve, by making \(r>0\) small enough,
Let \(\eta \in C^{0,1}_c(B_4)\) with \(0\le \eta \le 1\), \(\eta \equiv 1\) on \(B_2\). Note that it holds
where we put as usual
Note that by choice we have \(\Vert \eta \Vert _{C^{0,1}(\mathbb {R}^N)}\le 1+\frac{1}{r}\le \frac{2}{r}\), so that for all \(x,y\in \mathbb {R}^N\)
Fix \(e\in \partial B_1(0)\) and \(h\in (0,r)\). Let
Let \(\psi =\eta ^2\delta _{h}u\in {\mathscr {D}}^j(B_4)\), where in the following \(\delta _hu:=\delta _{h,e}u\). Note that
Hence, we have
and using the translation invariance, we also have
In the following, for simplicity, we put \(v(x)=\eta (x)\delta _hu(x)\), \(x\in \mathbb {R}^N\). Note that by Definition, \(v\in {\mathscr {D}}^j(B_4)\). Then with the help of Young’s inequality for some \(\mu \in (0,1)\) such that
we find
By a rearrangement of the double integral with Young’s inequality for the same \(\mu \in (0,1)\) as above we have
Here, we indicate with \(\delta _{-h,y}\) (resp. \(\delta _{-h,z}\)) that \(\delta _{-h}\) acts on the y (resp. z) variable. Note that
Note here, that (5.6) satisfies
and (5.7) can be written as
For \(h\in (0,r)\), \(z\in \mathbb {R}^N{\setminus }\{0\}\) put
Then, by combining (5.4) and (5.5), we find
Next we show that we have \(\int _{\mathbb {R}^N}k_h(z)\ \textrm{d}z \le C h^{\alpha }\) for some \(C>0\). Clearly, we can bound
for some \(C_1=C_1(N,j)>0\), using that \(B_1(0)\cup B_1(he)\subset B_2(0)\) and the properties of j. In the following, by making \(C_j\) larger if necessary, we may also assume that assumption (2.1) reads
Then note that \(B_{2h}(he)\subset B_{3h}(0)\) and we have
While with \(b_{\sigma }(t)=\frac{1}{\sigma }t^{-\sigma }\) we have
Combining (5.11) with (5.12) and (5.13) and the choice \(\alpha =1-\sigma \in (0,1)\) we find \(C_2=C_2(N, j,\alpha )>0\) such that
Whence, from (5.10) with (5.14) we have
for a constant \(C_4=C_4(N,j,r,\alpha ,\lambda )>0\). By a standard covering argument, we then also find with a constant \(C_5=C_5(N,j,\Omega ,\Omega ',\alpha ,\lambda )>0\) and \(\Omega ''=\{x\in \Omega \;:\; \text {dist}(x,\mathbb {R}^N{\setminus }\Omega )>4r\}\)
The claim (5.1) then follows since \(f\in H^1(\Omega )\). \(\square \)
Remark 5.2
If additionally \(f\in L^{\infty }(\Omega )\), combining Theorem 5.1 with Corollary 4.2 it follows that we have in the situation of Theorem 5.1 for every \(\Omega '\subset \subset \Omega \)
Corollary 5.3
Assume \(m=1\). Let \(f\in C^2(\overline{\Omega })\), \(\lambda \in \mathbb {R}\), and let \(u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)\) satisfy in weak sense \(I_ju=\lambda u+ f\) in \(\Omega \). Then \(u\in H^{1}(\Omega ')\) and \(\partial _iu\in D^j(\Omega ')\) for any \(\Omega '\subset \subset \Omega \). More precisely, with \(\alpha \) as above there is for any \(\Omega '\subset \subset \Omega \) a constant \(C=C(N,\Omega ,\Omega ', j,\lambda )>0\) such that
so that \(u\in N^{2\alpha ,2}(\Omega ')\subset H^1(\Omega ')\), that is, there is also \(C'=C'(N, j,\Omega ,\Omega ',\alpha ,\lambda )>0\) such that
and, moreover,
Proof
Let \(\Omega _i\subset \subset \Omega \), \(i=1,\ldots ,7\) such that
Let \(\eta \in C^{\infty }_c(\Omega _{7})\) with \(\eta =1\) on \(\Omega _6\) and \(0\le \eta \le 1\). Fix \(e\in \partial B_1(0)\) and \(h\in (0,\frac{1}{2}r)\), where \(r=\min \{\text {dist}(\Omega _i,\Omega {\setminus } \Omega _{i+1})\;:\; i=1,\ldots ,6\}\). Then by Lemma 2.8 the function \(v=\eta \delta _h u\), where we write \(\delta _h\) instead of \(\delta _{h,e}\), satisfies \(I_jv=\lambda v+\tilde{f}\) in \(\Omega _5\), where \(\tilde{f}=\delta _hf+g_{\eta ,\delta _hu}\). Following the proof of Theorem 5.1 to (5.16) it follows with Theorem 4.1 that there is \(C=C(N, j, r,\alpha ,\lambda )>0\) (changing from line to line) such that
where we applied once more Theorem 5.1. Here, for \(x\in \Omega _4\) using the assumptions on the differentiability of j it follows that there is \(C=C(j)>0\) such that
Moreover, for \(x\in \Omega _1\) in a similar way, there is \(C=C(j)>0\) such that
Thus we have
The proof of the first part then is finished with Proposition 2.11 since \(2\alpha >1\). Next, write \(D_hp(x)=\frac{p(x+he)-p(x)}{h}\) for any function \(p:\mathbb {R}^N\rightarrow \mathbb {R}\), with \(e\in \partial B_1(0)\) fixed and \(h\in \mathbb {R}{\setminus }\{0\}\). Then with Lemma 2.8 for some \(\eta \in C^{\infty }_c(\Omega )\) such that \(0\le \eta \le 1\) and \(\eta \equiv 1\) on \(\Omega _2\subset \subset \Omega \) with \(\Omega '\subset \subset \Omega _1\subset \subset \Omega _2\) we have with \(v=\eta u\),
Next, let \(\mu \in C^{\infty }_c(\Omega _1)\) with \(0\le \mu \le 1\) and \(\mu \equiv 1\) on \(\Omega '\). Then with \(\phi =D_{-h}[\mu ^2D_hv]\in {\mathscr {D}}^j(\Omega _1)\) for h small enough we have for some \(C>0\) (which may change from line to line independently of h)
since
and
due to assumptions on the differentiability of j. Moreover, with a similar calculation as in the proof of Theorem 5.1 we have
where for some \(\Omega _2\subset \subset \Omega _3\subset \subset \Omega _4\subset \subset \Omega \) with h small enough
Combining this with (5.20) we find
Since also \(\mu D_hv\in {\mathscr {D}}^j(\Omega _2)\) for all \(h>0\) small enough (see Lemma 2.3) and since \(D^j(\Omega _2)\) is a Hilbert space, we conclude that \(\mu \partial _ev\in {\mathscr {D}}^j(\Omega _2)\) with
for \(h\rightarrow 0\). This finishes the proof. \(\square \)
Corollary 5.4
Let \(f\in C^{2m}(\overline{\Omega })\), \(\lambda \in \mathbb {R}\), and let \(u\in {\mathscr {V}}^j_{loc}(\Omega )\cap L^{\infty }(\mathbb {R}^N)\) satisfy in weak sense \(I_ju=\lambda u+f\) in \(\Omega \). Then \(u\in H^{m}(\Omega ')\) for any \(\Omega '\subset \subset \Omega \) and there is \(C=C(N, j,\Omega ,\Omega ',m)>0\) such that
In particular, if \(m=\infty \), then \(u\in C^{\infty }(\Omega )\).
Proof
By Corollary 5.3 the claim holds for \(m=1\) in particular with \(u|_{\Omega '}\in D^j(\Omega ')\) for all \(\Omega '\subset \subset \Omega \). Assume next, the claim holds for \(m-1\) with \(m\in \mathbb {N}\), \(m\ge 2\) in the following way: We have \(u\in H^{m-1}(\Omega ')\) and \(\partial ^{\beta }u|_{\Omega '}\in D^j(\Omega ')\) for any \(\Omega '\subset \subset \Omega \) and \(\beta \in \mathbb {N}_0^N\) with \(|\beta |\le m-1\), and there is \(C=C(N, j,\Omega ,\Omega ',m)>0\) such that
Fix \(\Omega '\subset \subset \Omega \) and let \(\Omega _i\subset \subset \Omega \), \(i=1,\ldots ,7\) and \(\eta \in C^{\infty }_c(\Omega _7)\) as in the proof of Corollary 5.3. Put \(v=\partial ^{\beta }(\eta u)\) for some \(\beta \in \mathbb {N}_0^N\), \(|\beta |=m-1\). Then \(Iv=\partial ^{\beta }f+\lambda v+\partial ^{\beta }g_{\eta ,u}\) in \(\Omega _5\) by Lemma 2.8 and direct computation using the assumptions on J. From here, proceeding as in the proof of Corollary 5.3 by applying Theorem 5.1 the claim follows. \(\square \)
Proof of Theorem 1.1
By assumption, it follows from [20] that \({\mathscr {D}}^j(\Omega )\) is compactly embedded into \(L^2(\Omega )\). This gives the existence of the sequence of eigenfunctions and corresponding eigenvalues. The fact that the first eigenfunction can be chosen to be positive follows from the fact that \(b_j(|u|)\le b_j(u)\), Proposition 1.2 and Proposition 1.5 (see also [19]). Now statement (1) follows from Theorem 4.3 (with \(h=f=0\)) and statement (2) follows directly from Corollary 5.4. \(\square \)
Proof of Theorem 1.6
The first part follows from the Poincaré inequality, i.e. under the assumptions it holds \(\Lambda _1(\Omega )>0\), and Theorem 4.3 with \(h=0=\lambda \). The last assertion follows from Corollary 5.4. \(\square \)
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This work is supported by DAAD and BMBF (Germany) within the project 57385104. The authors thank Mouhamed Moustapha Fall and Tobias Weth for helpful discussions. We also thank the anonymous referee for valuable suggestions.
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Appendix A: An inequality
Appendix A: An inequality
The following is a variant of [9, Lemma 10] (see also [20, Lemma 5.1]).
Lemma A.1
Let \(q\in L^1(\mathbb {R}^N)\) be a nonnegative even function with \(q=0\) on \(\mathbb {R}^N{\setminus } B_r(0)\) for some \(r>0\). Let \(\Omega \subset \mathbb {R}^N\) open and \(x_0\in \Omega \) such that \(B_{2r}(x_0)\subset \Omega \). Then for all measurable functions \(u:\Omega \rightarrow \mathbb {R}\) we have
Proof
In the following, we identify u with its trivial extension \(\tilde{u}:\mathbb {R}^N\rightarrow \mathbb {R}\), \(\tilde{u}(x)=u(x)\) for \(x\in \Omega \) and \(\tilde{u}(x)=0\) otherwise. Denote \(g(x,y)=(u(x)-u(y))^2\) for \(x,y\in \mathbb {R}^N\). Note that we have
By Fubini’s theorem we have
Note that since \(q=0\) on \(\mathbb {R}^N{\setminus } B_r(0)\), q is even, and \(B_r(x)\subset B_{2r}(x_0)\subset \Omega \) for any \(x\in B_r(x_0)\), we have
\(\square \)
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Feulefack, P.A., Jarohs, S. Nonlocal operators of small order. Annali di Matematica 202, 1501–1529 (2023). https://doi.org/10.1007/s10231-022-01290-y
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DOI: https://doi.org/10.1007/s10231-022-01290-y