Abstract
In this work we introduce volume constraint problems involving the nonlocal operator \((-\Delta )_{\delta }^{s}\), closely related to the fractional Laplacian \((-\Delta )^{s}\), and depending upon a parameter \(\delta >0\) called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when \(\delta \rightarrow 0^+\) and \(\delta \rightarrow +\infty \). Through these limit processes on \((-\Delta )_{\delta }^{s}\) we derive spectral convergence to the local Laplacian and to the fractional Laplacian as \(\delta \rightarrow 0^+\) and \(\delta \rightarrow +\infty \) respectively, as well as we prove the convergence of solutions of these problems to solutions of a local Dirichlet problem involving \((-\Delta )\) as \(\delta \rightarrow 0^+\) or to solutions of a nonlocal fractional Dirichlet problem involving \((-\Delta )^s\) as \(\delta \rightarrow +\infty \).
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Notes
Since \(\{\mathcal {D}_{\delta }\}_{\delta \ge 0}\) is monotone increasing we have \(\liminf \nolimits _{\delta \rightarrow +\infty } \mathcal {D}_{\delta }=\bigcup \nolimits _{\delta \ge 0}\bigcap \nolimits _{\mu \ge \delta } \mathcal {D}_{\mu }=\bigcup \nolimits _{\delta \ge 0}\mathcal {D}_{\delta }\) as well as \(\limsup \nolimits _{\delta \rightarrow +\infty } \mathcal {D}_{\delta }=\bigcap \nolimits _{\delta \ge 0}\bigcup \nolimits _{\mu \ge \delta } \mathcal {D}_{\mu }=\bigcap \nolimits _{\delta \ge 0}\bigcup \nolimits _{\mu \ge 0} \mathcal {D}_{\mu }=\bigcup \nolimits _{\delta \ge 0}\mathcal {D}_{\delta }\). Then, \(\lim \nolimits _{\delta \rightarrow +\infty }\mathcal {D}_{\delta }= \bigcup \nolimits _{\delta \ge 0}\mathcal {D}_{\delta }=\mathcal {D}\).
References
Alali, B., Albin, N.: Fourier multipliers for nonlocal Laplace operators. Appl. Anal. 1, 1–21 (2019)
Alberti, G., Bellettini, G.: A non-local anisotropic model for phase transitions: asymptotic behaviour of rescaled energies. Eur. J. Appl. Math. 9, 261–284 (1998)
Andrés, F., Muñoz, J.: A type of nonlocal elliptic problem: existence and approximation through a Galerkin–Fourier method. SIAM J. Math. Anal. 47, 498–525 (2015)
Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo-Melero, J.J.: Nonlocal diffusion problems. Mathematical Surveys and Monographs, American Mathematical Society, Providence, vol. 165. RI; Real Sociedad Matemática Española, Madrid (2010)
Bellido, J.C., Mora-Corral, C.: Existence for nonlocal variational problems in peridynamics. SIAM J. Math. Anal. 46, 890–916 (2014)
Bellido, J.C., Mora-Corral, C., Pedregal, P.: Hyperelasticity as a \(\Gamma \)-limit of peridynamics when the horizon goes to zero. Calc. Var. Part. Differ. Equ. 54, 1643–1670 (2015)
Boulanger, J., Elbau, P., Pontow, C., Scherzer, O.: Non-local functionals for imaging. In: Fixed-point Algorithms for Inverse Problems in Science and Engineering, vol. 49 of Springer Optim. Appl. Springer, New York, pp. 131–154 (2011)
Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces, in Optimal control and partial differential equations, pp. 439–455. IOS, Amsterdam (2001)
Braides, A.: \(\Gamma \)-convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford (2002)
Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional \(p\)-Laplacian. Discret. Contin. Dyn. Syst. 36, 1813–1845 (2016)
Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext, Springer, New York (2011)
Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15, 657–699 (2016)
Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications. Lecture Notes of the Unione Matematica Italiana, Springer, [Cham], vol. 20. Unione Matematica Italiana, Bologna (2016)
D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66, 1245–1260 (2013)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Du, Q.: Nonlocal modeling, analysis, and computation. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol. 94., Philadelphia PA (2019)
Duo, S., Wang, H., Zhang, Y.: A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discret. Contin. Dyn. Syst. Ser. B 24, 231–256 (2019)
Evgrafov, A., Bellido, J.C.: From non-local Eringen’s model to fractional elasticity. Math. Mech. Solids 24, 1935–1953 (2019)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multisc. Model. Simul. 7, 1005–1028 (2008)
Kassmann, M., Mengesha, T., Scott, J.: Solvability of nonlocal systems related to peridynamics. Commun. Pure Appl. Anal. 18, 1303–1332 (2019)
Kindermann, S., Osher, S., Jones, P.W.: Deblurring and denoising of images by nonlocal functionals. Multiscale Model. Simul. 4, 1091–1115 (2005)
Leoni, G., Spector, D.: Characterization of Sobolev and \(BV\) spaces. J. Funct. Anal. 261, 2926–2958 (2011)
Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181 (1972)
Lischke, A., Pang, G., Gulian, M., et al.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020)
Mengesha, T.: Nonlocal Korn-type characterization of Sobolev vector fields. Commun. Contemp. Math. 14, 1250028 (2012)
Mengesha, T., Du, Q.: Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140, 82–111 (2016)
Mengesha, T., Spector, D.: Localization of nonlocal gradients in various topologies. Calc. Var. Part. Differ. Equ. 52, 253–279 (2015)
Ponce, A.C.: A new approach to Sobolev spaces and connections to \(\Gamma \)-convergence. Calc. Var. Part. Differ. Equ. 19, 229–255 (2004)
Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Mat. 60, 3–26 (2016)
Ros-Oton, X., Serra, J.: Fractional Laplacian: Pohozaev identity and nonexistence results. Compte. R. Mathematique 350, 505–508 (2012)
Servadei, R., Valdinoci, E.: A Brezis-Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12, 2445–2464 (2013)
Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)
Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discret. Contin. Dyn. Syst. 33, 2105–2137 (2013)
Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb.: Sect. Math. 144, 831–855 (2014)
Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Spector, D.: On a generalization of \(L^p\)-differentiability. Calc. Var. Part. Differ. Equ. 55, 21 (2016)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Comm. Part. Differ. Equ. 35, 2092–2122 (2010)
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Communicated by A. Malchiodi.
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Appendix A: Proof of secondary results
Appendix A: Proof of secondary results
Proof of Lemma 2
It is immediate to see that \(\langle \cdot ,\cdot \rangle _{\mathbb {H}_0^{\delta ,s}}\) is a scalar product on \(\mathbb {H}_0^{\delta ,s}(\Omega )\times \mathbb {H}_0^{\delta ,s}(\Omega )\). Let us see that \(\mathbb {H}_0^{\delta ,s}(\Omega )\) is complete with respect to the norm . Let \(\{v_n\}_{n\in \mathbb {N}}\) be a Cauchy sequence in \(\mathbb {H}_0^{\delta ,s}(\Omega )\). Hence, given \(varepsilon>0\), and keeping in mind Proposition 1 and Lemma 1, there exist \(c>0\) and \(n_0=n_0(\varepsilon )\in \mathbb {N}\) such that, for \(m,n\ge n_0\), we have
and consequently \(\{v_n\}_{n\in \mathbb {N}}\) is also a Cauchy sequence in \(L^2(\Omega )\), and since this is a complete space, there exists \(v\in L^2(\Omega )\) such that \(v_n\rightarrow v\) in \(L^2(\Omega )\) as \(n\rightarrow +\infty \). Arguing as in [32, Lemma 7], it is easily proved that \(v\in \mathbb {H}_0^{\delta ,s}(\Omega )\) and that \(\{v_n\}_{n\in \mathbb {N}}\) converges in \(\mathbb {H}_0^{\delta ,s}(\Omega )\). Hence, the space \(\mathbb {H}_0^{\delta ,s}(\Omega )\) is a complete space. \(\square \)
Proof of Lemma 3
Since the sequence \(\{v_j\}_{j\in \mathbb {N}}\) is bounded in \(\mathbb {H}_0^{\delta ,s}(\Omega )\), because of Proposition 1, it is bounded in \(H^s(\Omega _{\delta })\) so that it is also bounded in \(H^s(\Omega )\) and, hence, in \(L^2(\Omega )\). Observe that, although \(v=0\) on \(\partial _{\delta }\Omega \), we have \(\Vert \cdot \Vert _{H^s(\Omega )}\ne \Vert \cdot \Vert _{H^s(\Omega _{\delta })}\) since the latter norm takes into account the interaction between \(\Omega \) and \(\partial _{\delta }\Omega \). Then, by [15, Corollary 7.2], there exists \(v\in L^p(\Omega )\) such that, up to a subsequence,
As mentioned before, the proof of Proposition 2 can be done similarly as to the proof [33, Proposition 9] so that we sketch the main steps.
Let us consider the energy functional
and observe that \(\big \langle \mathcal {J}'(u)\big |v\big \rangle =\frac{c_{N,s}}{2}\langle u,v\rangle _{\mathbb {H}_0^{\delta ,s}}=\big \langle \mathcal {J}'(v)\big | u\big \rangle \) for all \(u,v\in \mathbb {H}_0^{\delta ,s}(\Omega )\).
Lemma 8
Let \(\mathcal {E}\subseteq \mathbb {H}_0^{\delta ,s}(\Omega )\) be a weakly closed subspace and \(\mathcal {E}_1=\{u\in \mathcal {E}:\Vert u\Vert _{L^2(\Omega )}=1\}\). Then, there exists \(u^{\star }\in \mathcal {E}_1\) such that
Moreover,
for all \(\varphi \in \mathcal {E}\) and
Proof
Let \(\{u_j\}\subset \mathcal {E}_1\) be a minimizing sequence for \(\mathcal {J}\), i.e.,
Hence, the sequence \(\mathcal {J}(u_j)\) is bounded in \(\mathbb {R}\) and by definition of \(\mathcal {J}\), the sequence is also bounded. Moreover, because of Lemma 2 the space \(\mathbb {H}_0^{\delta ,s}(\Omega )\) is a Hilbert space and, hence, it is also a reflexive space. Then, up to a subsequence (that we do nor relabel), \(u_j\) converges weakly in \(\mathbb {H}_0^{\delta ,s}(\Omega )\) to some \(u^{\star }\in \mathcal {E}\) since \(\mathcal {E}\) is weakly closed. Therefore, we conclude
On the other hand, since is bounded, because of Lemma 3, we have
so that \(\Vert u^{\star }\Vert _{L^2(\Omega )}=1\) and then \(u^{\star }\in \mathcal {E}_1\). Observe that the functional \(\mathcal {J}\) is continuous and convex in \(\mathbb {H}_0^{\delta ,s}(\Omega )\), so that \(\mathcal {J}\) is weakly lower semicontinuous in \(\mathbb {H}_0^{\delta ,s}(\Omega )\). Then,
and, by (A.3), we conclude \(\mathcal {J}(u^{\star })=\min \limits _{u\in \mathcal {E}_1}\mathcal {J}(u)\). Next, we prove that \(\lambda ^{\star }=\mathcal {J}(u^{\star })\). Let us set \(\varepsilon \in (-1,1)\), \(\varphi \in \mathcal {E}\) and \(u_{\varepsilon }=\frac{u^{\star }+\varepsilon \varphi }{\Vert u^{\star }+\varepsilon \varphi \Vert _{L^2(\Omega )}}\in \mathcal {E}_1\). Then,
Because of the minimality of \(u^{\star }\) we have \(\left. \frac{d}{d\varepsilon }\mathcal {J}(u_{\varepsilon })\right| _{\varepsilon =0}=0\), from where we conclude . \(\square \)
Proof of Proposition 2-(1)
Taking \(\mathcal {E}=\mathbb {H}_0^{\delta ,s}(\Omega )\) in Lemma 8 we find that minimum defining \(\lambda _1^{\delta ,s}\) exists and \(\lambda _1^{\delta ,s}\) is an eigenvalue of problem (\(EP_{\delta }^{s}\)), proving (2.7). Moreover, because of Lemma 8, the minimum defining \(\lambda _1^{\delta ,s}\) is attained at some function \(\varphi _1^{\delta ,s}\in \mathbb {H}_0^{\delta ,s}(\Omega )\) with \(\Vert \varphi _1^{\delta ,s}\Vert _{L^2(\Omega )}=1\). Then, by (A.2), we conclude
proving (2.8). The proof of the non-negativity of the first eigenfunction follows using the triangle inequality and the minimality of \(\varphi _1^{\delta ,s}\) as in [33, Proposition 9-b)]. To prove that \(\lambda _1^{\delta ,s}\) is simple we can argue exactly as in [33, Proposition 9-c)] so we omit the details. \(\square \)
Proof of Proposition 2-(2)
Let us define \(\lambda _k^{\delta ,s}\) by (2.12) and observe that, by Lemma 8 with
which is weakly closed by Lemma 2; the minimium in (2.12) exists, it is attained at some function \(\varphi _k^{\delta ,s}\in \mathbb {P}_{k}^{\delta }\) and
In order to show that \(\lambda _{k}^{\delta ,s}\) is an eigenvalue associated to the eigenfunction \(\varphi _{k}^{\delta ,s}\), it remains to prove that (A.4) holds for any \(\phi \in \mathbb {H}_0^{\delta ,s}(\Omega )\). This can be done inductively as in [33, Proposition 9-d)]. Note that the initial induction step has been proved in Proposition 2-(1). Let us write
and, given \(\phi \in \mathbb {H}_0^{\delta ,s}(\Omega )\), write \(\phi =\phi _1+\phi _2\) with \(\phi _1=\sum _{i=1}^{k}c_i\varphi _i^{\delta ,s}\) and \(\phi _2\in \mathbb {P}_{k+1}^{\delta }\).
Testing the equation for \(\varphi _{k+1}^{\delta ,s}\) against \(\phi _2=\phi -\phi _1\) and \(\varphi _i^{\delta ,s}\) for \(i=1,\ldots ,k\) it is easily proved that
so that (A.4) holds for any \(\phi \in \mathbb {H}_0^{\delta ,s}(\Omega )\).
We continue by proving (2.10). Since \(\displaystyle \mathbb {P}_{k+1}^{\delta }\subseteq \mathbb {P}_{k}^{\delta }\subseteq \mathbb {H}_0^{\delta ,s}(\Omega )\) we have \(0<\lambda _1^{\delta ,s}\le \lambda _2^{\delta ,s}\le \ldots \le \lambda _{k}^{\delta ,s}\le \ldots \). Moreover, since \(\varphi _2^{\delta ,s}\in \mathbb {P}_{2}^{\delta }\), by Proposition 2-(1) we get \(\lambda _1\ne \lambda _2\) and (2.10) follows. To prove (2.11) let us assume that \(\lambda _{k}^{\delta ,s}\rightarrow c<+\infty \) so that the sequence is bounded in \(\mathbb {H}_0^{\delta ,s}(\Omega )\). Hence, because of Lemma 3 we conclude that, up to a subsequence, \(\varphi _{k}^{\delta ,s}\rightarrow \overline{\varphi }\in \ L^2(\Omega )\) as \(k\rightarrow +\infty \) and \(\{\varphi _{k}^{\delta ,s}\}_{k\in \mathbb {N}}\) is a Cauchy sequence. On the other hand, letting \(k,h\in \mathbb {N}\), \(k>h\), since \(\varphi _{k}^{\delta ,s}\in \mathbb {P}_{k}^{\delta }\) we have that \(\langle \varphi _{k}^{\delta ,s},\varphi _{h}^{\delta ,s}\rangle _{\mathbb {H}_0^{\delta ,s}}=0\). Therefore \(\{\varphi _{k}^{\delta ,s}\}_{k\in \mathbb {N}}\) is an orthogonal sequence in \(\mathbb {H}_0^{\delta ,s}(\Omega )\). Moreover, taking in mind that \(\varphi _{k}^{\delta ,s}\) are eigenfunctions, it follows that
and we conclude that \(\{\varphi _{k}^{\delta ,s}\}_{k\in \mathbb {N}}\) is also an orthogonal sequence in both \(\mathbb {H}_0^{\delta ,s}(\Omega )\) and \(L^2(\Omega )\). Then, \(\Vert \varphi _k^{\delta ,s}-\varphi _j^{\delta ,s}\Vert _{L^2(\Omega )}^2=\Vert \varphi _k^{\delta ,s}\Vert _{L^2(\Omega )}^2+\Vert \varphi _j^{\delta ,s}\Vert _{L^2(\Omega )}^2=2\) and we get a contradiction with the fact that \(\{\varphi _{k}^{\delta ,s}\}_{k\in \mathbb {N}}\) is a Cauchy sequence. Therefore, \(\lambda _k\rightarrow +\infty \) as \(k\rightarrow +\infty \).
To complete the proof of Proposition 2-(2), it remains to prove that the sequence of eigenvalues given by (2.12) exhust all the eigenvalues of problem (\(EP_{\delta }^{s}\)). This can be done exactly as in [33, Proposition 9-d)] so we omit the details. \(\square \)
Proof of Proposition 2-(3)
The orthogonality of the set \(\{\varphi _k^{\delta ,s}\}_{k\in \mathbb {N}}\) in \(\mathbb {H}_0^{\delta ,s}(\Omega )\) and \(L^2(\Omega )\) follows by (A.5), so it remains to prove that it is a basis for both \(\mathbb {H}_0^{\delta ,s}(\Omega )\) and \(L^2(\Omega )\). This follows using a standard Fourier analysis technique exactly as in [33, Proposition 9-f)] so we omit the details for the sake of brevity. \(\square \)
Proof of Proposition 2-(4)
Let \(n\in \mathbb {N},\ n\ge 1\) and \(\lambda _k^{\delta ,s}>0\) be an eigenvalue satisfying (2.15), i.e.,
Because of Proposition 2-(2), equation (2.12), every function belonging to\(span\{\varphi _k^{\delta ,s},\ldots ,\varphi _{k+n}^{\delta ,s}\}\) is an eigenfunction corresponding to \(\lambda _k^{\delta ,s}=\ldots =\lambda _{k+n}^{\delta ,s}\). It remains to prove that each eigenfunction \(\psi \in \mathbb {H}_0^{\delta ,s}(\Omega )\), \(\psi \ne 0\), associated to \(\lambda _{k}^{\delta ,s}\) belongs to \(span\{\varphi _k^{\delta ,s},\ldots ,\varphi _{k+n}^{\delta ,s}\}\). Let us write
and, consequently, \(\psi =\psi _1+\psi _2\) with the functions \(\psi _1\in span\{\varphi _k^{\delta ,s},\ldots ,\varphi _{k+n}^{\delta ,s}\}\) and \(\psi _2\in \left( span\{\varphi _k^{\delta ,s},\ldots ,\varphi _{k+n}^{\delta ,s}\}\right) ^{\bot }\). Using the orthogonality of the eigenfunctions and arguing as in [33, Proposition 9-g)] it can be easily proven that \(\psi _2\equiv 0\) and, hence, \(\psi =\psi _1\in span\{\varphi _k,\ldots ,\varphi _{k+h}\}\). \(\square \)
Up to minor changes indicated below, the proof of Lemma 5 is similar to the proof of [31, Proposition 4] so that we sketch the main steps of the proof.
Proof of Lemma 5
Let \(\varphi \in \mathbb {H}_0^{\delta ,s}(\Omega )\), \(\varphi \ne 0\) and \(\lambda >0\) such that
Up to multiplication by a constant, we can assume that \(\Vert \varphi \Vert _{L^2(\Omega )}^2=\varepsilon \) with \(\varepsilon >0\) to be chosen. Next, for \(k\in \mathbb {N}\), let
where \(v^+:=\max \{v,0\}\). Notice that \(w_k\in \mathbb {H}_0^{\delta ,s}(\Omega )\), since \(\mathbb {H}_0^{\delta ,s}(\Omega )\) is a linear space and \(\varphi (x)-C_k=-C_k\le 0\) a.e. \(x\in \partial _{\delta }\Omega \) so that \(w_k=(-C_k)^+=0\) on \(\partial _{\delta }\Omega \). Moreover, \(0\le w_k\le |\varphi |+|C_k|\le |\varphi |+1\in L^2(\Omega )\) because \(\Omega \) is a bounded domain and \(\lim \nolimits _{k\rightarrow +\infty }w_k=(\varphi -1)^+\). Hence, by the Dominated Convergence Theorem,
Proceeding in a similar way as in [31, Proposition 4] we can prove that
In addition, since \(\{w_{k+1}>0\}\subseteq \{w_k>2^{-(k+1)}\}\) for any \(k\in \mathbb {N}\),
On the other hand, using Hölder inequality with exponents \(p=\frac{2_s^*}{2}\) and \(q=\frac{N}{2s}\), the fractional Sobolev inequality (cf.[15, Theorem 6.5]) and Proposition 1, we find,
for a positive constant \(c=c(N,s,\delta )\). As a consequence, by (A.7) and (A.8),
with \(\beta := 1+\frac{2s}{N}\) and \(\theta =\theta (N,s,\delta ,\lambda )>1\). Next, let us choose \(\varepsilon >0\) such that \(\varepsilon ^{\beta -1}<\frac{1}{\theta ^{1/(\beta -1)}}\) and fix \(\eta \in (\varepsilon ^{\beta -1},\frac{1}{\theta ^{1/(\beta -1)}})\). Let us note that, since \(\theta >1\) and \(\beta >1\), we have \(\eta \in (0,1)\). Arguing inductively, it is easily proved that, for any \(k\in \mathbb {N}\),
Then, by (A.6), we get \(\displaystyle \int _{\Omega }\left[ (\varphi -1)^+\right] ^2dx=\lim \limits _{k\rightarrow +\infty }U_k=0\) since \(\eta \in (0,1)\). Thus, \((\varphi -1)^+=0\) a.e. in \(\Omega \) so that \(\varphi \le 1\) a.e. in \(\Omega \). By replacing \(\varphi \) with \(-\varphi \) it follows that \(\Vert \varphi \Vert _{L^{\infty }(\Omega )}\le 1\). Taking in mind the scaling \(\Vert \varphi \Vert _{L^2(\Omega }^2=\varepsilon \), we conclude
for a constant \(C>0\) depending on \(N,s,\delta \) and \(\lambda \). \(\square \)
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Bellido, J.C., Ortega, A. A restricted nonlocal operator bridging together the Laplacian and the fractional Laplacian. Calc. Var. 60, 71 (2021). https://doi.org/10.1007/s00526-020-01896-1
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DOI: https://doi.org/10.1007/s00526-020-01896-1