Abstract

We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem through the Caffarelli--Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and vertex-edge neighborhoods.

Keywords

  1. fractional Laplacian
  2. analytic regularity
  3. corner domains
  4. weighted Sobolev spaces

MSC codes

  1. 26A33
  2. 35A20
  3. 35B45
  4. 35J70
  5. 35R11

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References

1.
G. Acosta and J.P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), pp. 472--495.
2.
G. Albanese, A. Fiscella, and E. Valdinoci, Gevrey regularity for integro-differential operators, J. Math. Anal. Appl., 428 (2015), pp. 1225--1238.
3.
H. Abels and G. Grubb, Fractional-Order Operators on Nonsmooth Domains, https://arxiv.org/abs/2004.10134, 2020.
4.
N. Abatangelo and X. Ros-Oton, Obstacle problems for integro-differential operators: Higher regularity of free boundaries, Adv. Math., 360 (2020).
5.
A. Bonito, J.P. Borthagaray, R.H. Nochetto, E. Otárola, and A.J. Salgado, Numerical methods for fractional diffusion, Comput. Vis. Sci., 19 (2018), pp. 19--46.
6.
I. Babuška and B.Q. Guo, Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order, SIAM J. Math. Anal., 19 (1988), pp. 172--203.
7.
J.P. Borthagaray, W. Li, and R.H. Nochetto, Linear and nonlinear fractional elliptic problems, in 75 Years of Mathematics of Computation, Contemp. Math., 754 American Mathematical Society, Providence, RI, 2020, pp. 69--92.
8.
L. Banjai, J.M. Melenk, R.H. Nochetto, E. Otárola, A.J. Salgado, and Ch. Schwab, Tensor FEM for spectral fractional diffusion, Found. Comput. Math., 19 (2019), pp. 901--962.
9.
J.P. Borthagaray and R.H. Nochetto, Besov Regularity for the Dirichlet Integral Fractional Laplacian in Lipschitz Domains, https://arxiv.org/abs/2110.02801, 2021.
10.
U. Biccari, M. Warma, and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), pp. 387--409.
11.
M. Costabel, M. Dauge, and S. Nicaise, Analytic regularity for linear elliptic systems in polygons and polyhedra, Math. Models Methods Appl. Sci., 22 (2012).
12.
M. Cozzi, Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces, Ann. Mat. Pura Appl. (4), 196 (2017), pp. 555--578.
13.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), pp. 1245--1260.
14.
L.A. Caffarelli and P.R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), pp. 767--807.
15.
M. D'Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian, and Z. Zhou, Numerical methods for nonlocal and fractional models, Acta Numer., 29 (2020), pp. 1--124.
16.
A. Dall'Acqua, S. Fournais, T. Østergaard Sørensen, and E. Stockmeyer, Real analyticity away from the nucleus of pseudorelativistic Hartree-Fock orbitals, Anal. PDE, 5 (2012), pp. 657--691.
17.
A. Dall'Acqua, S. Fournais, T. Østergaard Sørensen, and E. Stockmeyer, Real analyticity of solutions to Schrödinger equations involving a fractional Laplacian and other Fourier multipliers, in Proceedings of the 17th International Congress on Mathematical Physics, 2013, pp. 600--609.
18.
J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble), 5 (1954), pp. 305--370.
19.
C. Ebmeyer, Mixed boundary value problems for nonlinear elliptic systems with p-structure in polyhedral domains, Math. Nachr., 236 (2002), pp. 91--108.
20.
C. Ebmeyer and J. Frehse, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math. Nachr., 203 (1999), pp. 47--74.
21.
L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
22.
M. Faustmann, M. Karkulik, and J.M. Melenk, Local Convergence of the FEM for the Integral Fractional Laplacian, SIAM J. Numer. Anal., 60 (2022), pp. 1055--1082.
23.
M. Faustmann, C. Marati, J.M. Melenk, and C. Schwab, Exponential Convergence of $hp$-FEM for the Integral Fractional Laplacian in 1D, https://arxiv.org/abs/2204.04113, 2022.
24.
M. Faustmann, C. Marcati, J.M. Melenk, and Ch. Schwab, Exponential Convergence of $hp$ FEM for the Integral Fractional Laplacian in polygons, https://arxiv.org/abs/2209.11468, 2022.
25.
M. Faustmann, J.M. Melenk, and D. Praetorius, Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian, Math. Comp., 90 (2021), pp. 1557--1587.
26.
B. Guo and I. Babuška, Regularity of the solutions for elliptic problems on nonsmooth domains in $\bold R^3$. I. Countably normed spaces on polyhedral domains, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), pp. 77--126.
27.
B. Guo and I. Babuška, Regularity of the solutions for elliptic problems on nonsmooth domains in $\bold R^3$. II. Regularity in neighbourhoods of edges, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), pp. 517--545.
28.
P. Grisvard, Elliptic problems in Nonsmooth Domains, Classics 69 Appl. Math., SIAM.
29.
G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Adv. Math., 268 (2015), pp. 478--528.
30.
H. Gimperlein, E.P. Stephan, and J. Štoček, Corner Singularities for the Fractional Laplacian and Finite Element Approximation, preprint, 2021; also available online from http://mat1.uibk.ac.at/heiko/corners.pdf.
31.
T. Horger, J.M. Melenk, and B. Wohlmuth, On optimal $L^2$- and surface flux convergence in FEM, Comput. Vis. Sci., 16 (2013), pp. 231--246.
32.
M. Karkulik and J.M. Melenk, $\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian, Adv. Comput. Math., 45 (2019), pp. 2893--2919.
33.
V.A. Kozlov, V.G. Maz'ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, Math. Surveys Monogr. 52, American Mathematical Society, Providence, RI, 1997.
34.
H. Koch, A. Rüland, and W. Shi, Higher regularity for the fractional thin obstacle problem, New York J. Math., 25 (2019), pp. 745--838.
35.
M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), pp. 7--51.
36.
A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, and G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020).
37.
C.B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss. 130, Springer New York, 1966.
38.
V. Maz'ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Math. Surveys Monogr. 162, American Mathematical Society, Providence, RI, 2010.
39.
J.M. Melenk and B. Wohlmuth, Quasi-optimal approximation of surface based Lagrange multipliers in finite element methods, SIAM J. Numer. Anal., 50 (2012), pp. 2064--2087.
40.
X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9), 101 (2014), pp. 275--302.
41.
G. Savaré, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal., 152 (1998), pp. 176--201.
42.
J. Štoček, Efficient Finite Element Methods for the Integral Fractional Laplacian and Applications, Ph.D. thesis, Heriot-Watt University, 2020.
43.
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007.

Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 6323 - 6357
ISSN (online): 1095-7154

History

Submitted: 15 December 2021
Accepted: 25 July 2022
Published online: 8 December 2022

Keywords

  1. fractional Laplacian
  2. analytic regularity
  3. corner domains
  4. weighted Sobolev spaces

MSC codes

  1. 26A33
  2. 35A20
  3. 35B45
  4. 35J70
  5. 35R11

Authors

Affiliations

Funding Information

ETH Zurich
Austrian Science Fund https://doi.org/10.13039/501100002428 : SFB F65

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