Mapping properties of weakly singular periodic volume potentials in Roumieu classes. (English) Zbl 1514.47085

Summary: The analysis of the dependence of integral operators on perturbations plays an important role in the study of inverse problems and of perturbed boundary value problems. In this paper, we focus on the mapping properties of the volume potentials with weakly singular periodic kernels. Our main result is to prove that the map which takes a density function and a periodic kernel to a (suitable restriction of the) volume potential is bilinear and continuous with values in a Roumieu class of analytic functions. This result extends to the periodic case of some previous results obtained by the authors for nonperiodic potentials, and it is motivated by the study of perturbation problems for the solutions of boundary value problems for elliptic differential equations in periodic domains.


47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
47H60 Multilinear and polynomial operators
Full Text: DOI arXiv Euclid


[1] G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146, Springer, 2002. Mathematical Reviews (MathSciNet): MR1859696
Zentralblatt MATH: 0990.35001
· Zbl 0990.35001
[2] H. Ammari and H. Kang, Polarization and moment tensors: with applications to inverse problems and effective medium theory, Applied Mathematical Sciences 162, Springer, 2007. Mathematical Reviews (MathSciNet): MR2327884
Zentralblatt MATH: 1220.35001
· Zbl 1220.35001
[3] H. Ammari, H. Kang, and H. Lee, Layer potential techniques in spectral analysis, Mathematical Surveys and Monographs 153, American Mathematical Society, Providence, RI, 2009. Mathematical Reviews (MathSciNet): MR2488135
Zentralblatt MATH: 1167.47001
· Zbl 1167.47001
[4] N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series) 36, Kluwer, Dordrecht, 1989. Mathematical Reviews (MathSciNet): MR1112788
Zentralblatt MATH: 0692.73012
· Zbl 0692.73012
[5] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications 5, North-Holland, Amsterdam, 1978. Mathematical Reviews (MathSciNet): MR503330
Zentralblatt MATH: 0404.35001
· Zbl 0404.35001
[6] A. Charalambopoulos, “On the Fréchet differentiability of boundary integral operators in the inverse elastic scattering problem”, Inverse Problems 11:6 (1995), 1137-1161. Mathematical Reviews (MathSciNet): MR1361766
Zentralblatt MATH: 0847.35144
Digital Object Identifier: doi:10.1088/0266-5611/11/6/002
· Zbl 0847.35144
[7] D. Cioranescu and P. Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications 17, The Clarendon Press, New York, 1999. Mathematical Reviews (MathSciNet): MR1765047
Zentralblatt MATH: 0939.35001
· Zbl 0939.35001
[8] M. Costabel and F. Le Louër, “Shape derivatives of boundary integral operators in electromagnetic scattering, Part I: Shape differentiability of pseudo-homogeneous boundary integral operators”, Integral Equations Operator Theory 72:4 (2012), 509-535. Mathematical Reviews (MathSciNet): MR2904609
· Zbl 1331.47045
[9] M. Costabel and F. Le Louër, “Shape derivatives of boundary integral operators in electromagnetic scattering, Part II: Application to scattering by a homogeneous dielectric obstacle”, Integral Equations Operator Theory 73:1 (2012), 17-48. Mathematical Reviews (MathSciNet): MR2913658
· Zbl 1263.78001
[10] M. Dalla Riva and M. Lanza de Cristoforis, “A perturbation result for the layer potentials of general second order differential operators with constant coefficients”, J. Appl. Funct. Anal. 5:1 (2010), 10-30. Mathematical Reviews (MathSciNet): MR2675022
Zentralblatt MATH: 1200.35111
· Zbl 1200.35111
[11] M. Dalla Riva and P. Musolino, “A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite”, SIAM J. Appl. Math. 73:1 (2013), 24-46. Mathematical Reviews (MathSciNet): MR3033138
Zentralblatt MATH: 1412.74064
Digital Object Identifier: doi:10.1137/120886637
· Zbl 1412.74064
[12] M. Dalla Riva, M. Lanza de Cristoforis, and P. Musolino, “Analytic dependence of volume potentials corresponding to parametric families of fundamental solutions”, Integral Equations Operator Theory 82:3 (2015), 371-393. Mathematical Reviews (MathSciNet): MR3355785
Zentralblatt MATH: 1332.31011
Digital Object Identifier: doi:10.1007/s00020-015-2236-3
· Zbl 1332.31011
[13] K. Deimling, Nonlinear functional analysis, Springer, 1985. Mathematical Reviews (MathSciNet): MR787404
Zentralblatt MATH: 0559.47040
· Zbl 0559.47040
[14] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften 224, Springer, 1983. Mathematical Reviews (MathSciNet): MR737190
Zentralblatt MATH: 0562.35001
· Zbl 0562.35001
[15] H. Haddar and R. Kress, “On the Fréchet derivative for obstacle scattering with an impedance boundary condition”, SIAM J. Appl. Math. 65:1 (2004), 194-208. Mathematical Reviews (MathSciNet): MR2112394
Zentralblatt MATH: 1084.35103
Digital Object Identifier: doi:10.1137/S0036139903435413
· Zbl 1084.35103
[16] F. Hettlich, “Fréchet derivatives in inverse obstacle scattering”, Inverse Problems 11:2 (1995), 371-382. Mathematical Reviews (MathSciNet): MR1324650
Zentralblatt MATH: 0821.35147
Digital Object Identifier: doi:10.1088/0266-5611/11/2/007
· Zbl 0821.35147
[17] F. John, Plane waves and spherical means applied to partial differential equations, Interscience, New York, 1955. Mathematical Reviews (MathSciNet): MR0075429
Zentralblatt MATH: 0067.32101
· Zbl 0067.32101
[18] F. John, Partial differential equations, 4th ed., Springer, 1982. Zentralblatt MATH: 0472.35001
· Zbl 0472.35001
[19] D. Kapanadze, G. Mishuris, and E. Pesetskaya, “Exact solution of a nonlinear heat conduction problem in a doubly periodic 2D composite material”, Arch. Mech. \((\) Arch. Mech. Stos.\() 67\):2 (2015), 157-178. Mathematical Reviews (MathSciNet): MR3381129
Zentralblatt MATH: 1329.80005
· Zbl 1329.80005
[20] D. Kapanadze, G. Mishuris, and E. Pesetskaya, “Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials”, Complex Var. Elliptic Equ. 60:1 (2015), 1-23. Mathematical Reviews (MathSciNet): MR3295084
Zentralblatt MATH: 1316.30041
Digital Object Identifier: doi:10.1080/17476933.2013.876418
· Zbl 1316.30041
[21] A. Kirsch, “The domain derivative and two applications in inverse scattering theory”, Inverse Problems 9:1 (1993), 81-96. Mathematical Reviews (MathSciNet): MR1203018
Zentralblatt MATH: 0773.35085
Digital Object Identifier: doi:10.1088/0266-5611/9/1/005
· Zbl 0773.35085
[22] R. Kress and L. Päivärinta, “On the far field in obstacle scattering”, SIAM J. Appl. Math. 59:4 (1999), 1413-1426. Mathematical Reviews (MathSciNet): MR1692655
Zentralblatt MATH: 0945.35069
Digital Object Identifier: doi:10.1137/S0036139997332257
· Zbl 0945.35069
[23] M. Lanza de Cristoforis, “A domain perturbation problem for the Poisson equation”, Complex Var. Theory Appl. 50:7-11 (2005), 851-867. Mathematical Reviews (MathSciNet): MR2155447
Zentralblatt MATH: 1130.35039
Digital Object Identifier: doi:10.1080/02781070500136993
· Zbl 1130.35039
[24] M. Lanza de Cristoforis and P. Musolino, “A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients”, Far East J. Math. Sci. \((\) FJMS \() 52\):1 (2011), 75-120. Mathematical Reviews (MathSciNet): MR2839280
Zentralblatt MATH: 1239.31003
· Zbl 1239.31003
[25] M. Lanza de Cristoforis and P. Musolino, “A quasi-linear heat transmission problem in a periodic two-phase dilute composite: a functional analytic approach”, Commun. Pure Appl. Anal. 13:6 (2014), 2509-2542. Mathematical Reviews (MathSciNet): MR3248402
Zentralblatt MATH: 1304.35686
Digital Object Identifier: doi:10.3934/cpaa.2014.13.2509
· Zbl 1304.35686
[26] M. Lanza de Cristoforis and L. Rossi, “Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density”, J. Integral Equations Appl. 16:2 (2004), 137-174. Mathematical Reviews (MathSciNet): MR2100384
Zentralblatt MATH: 1094.31001
Digital Object Identifier: doi:10.1216/jiea/1181075272
Project Euclid: euclid.jiea/1181075272
· Zbl 1094.31001
[27] M. Lanza de Cristoforis and L. Rossi, “Real analytic dependence of simple and double layer potentials for the Helmholtz equation upon perturbation of the support and of the density”, pp. 193-220 in Analytic methods of analysis and differential equations: AMADE 2006, edited by A. A. Kilbas and S. V. Rogosin, Cambridge Sci. Publ., 2008. Mathematical Reviews (MathSciNet): MR2562110
[28] F. Le Louër, “On the Fréchet derivative in elastic obstacle scattering”, SIAM J. Appl. Math. 72:5 (2012), 1493-1507. Mathematical Reviews (MathSciNet): MR3022273
· Zbl 1259.35154
[29] C.-S. Lin and C.-L. Wang, “Elliptic functions, Green functions and the mean field equations on tori”, Ann. of Math. \((2) 172\):2 (2010), 911-954. Mathematical Reviews (MathSciNet): MR2680484
Zentralblatt MATH: 1207.35011
Digital Object Identifier: doi:10.4007/annals.2010.172.911
· Zbl 1207.35011
[30] M. Mamode, “Fundamental solution of the Laplacian on flat tori and boundary value problems for the planar Poisson equation in rectangles”, Bound. Value Probl. (2014), art. id. 221. Mathematical Reviews (MathSciNet): MR3277912
Zentralblatt MATH: 1304.35238
Digital Object Identifier: doi:10.1186/s13661-014-0221-4
· Zbl 1304.35238
[31] G. W. Milton, The theory of composites, Cambridge Monographs on Applied and Computational Mathematics 6, Cambridge University Press, 2002. Mathematical Reviews (MathSciNet): MR1899805
Zentralblatt MATH: 0993.74002
· Zbl 0993.74002
[32] G. S. Mishuris and L. I. Slepyan, “Brittle fracture in a periodic structure with internal potential energy”, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470:2165 (2014), art. id. 20130821. Mathematical Reviews (MathSciNet): MR3177234
Zentralblatt MATH: 1371.74247
· Zbl 1371.74247
[33] V. Mityushev and P. M. Adler, “Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders, I: A single cylinder in the unit cell”, ZAMM Z. Angew. Math. Mech. 82:5 (2002), 335-345. Mathematical Reviews (MathSciNet): MR1902260
Zentralblatt MATH: 1005.76087
Digital Object Identifier: doi:10.1002/1521-4001(200205)82:5<335::AID-ZAMM335>3.0.CO;2-D
· Zbl 1005.76087
[34] V. V. Mityushev, E. Pesetskaya, and S. V. Rogosin, “Analytical methods for heat conduction in composites and porous media”, pp. 121-164 in Cellular and porous materials: thermal properties simulation and prediction, edited by A. Öchsner et al., Wiley, Weinheim, Germany, 2008. Zentralblatt MATH: 1150.74038
Digital Object Identifier: doi:10.3846/1392-6292.2008.13.67-78
· Zbl 1150.74038
[35] A. B. Movchan, N. V. Movchan, and C. G. Poulton, Asymptotic models of fields in dilute and densely packed composites, Imperial College Press, London, 2002. Mathematical Reviews (MathSciNet): MR1944927
Zentralblatt MATH: 1008.74002
· Zbl 1008.74002
[36] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967. Mathematical Reviews (MathSciNet): MR0227584
· Zbl 1225.35003
[37] R. Potthast, “Fréchet differentiability of boundary integral operators in inverse acoustic scattering”, Inverse Problems 10:2 (1994), 431-447. Mathematical Reviews (MathSciNet): MR1269018
Zentralblatt MATH: 0805.35157
Digital Object Identifier: doi:10.1088/0266-5611/10/2/016
· Zbl 0805.35157
[38] R. Potthast, “Domain derivatives in electromagnetic scattering”, Math. Methods Appl. Sci. 19:15 (1996), 1157-1175. Mathematical Reviews (MathSciNet): MR1410203
Zentralblatt MATH: 0866.35019
Digital Object Identifier: doi:10.1002/(SICI)1099-1476(199610)19:15<1157::AID-MMA814>3.0.CO;2-Y
· Zbl 0866.35019
[39] R. Potthast, “Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain”, J. Inverse Ill-Posed Probl. 4:1 (1996), 67-84. Mathematical Reviews (MathSciNet): MR1393355
Zentralblatt MATH: 0858.35139
Digital Object Identifier: doi:10.1515/jiip.1996.4.1.67
· Zbl 0858.35139
[40] L. Preciso, Perturbation analysis of the conformal sewing problem and related problems, Ph.D. thesis, University of Padova, Padova Digital University Archive, 1998.
[41] L. Preciso, “Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Roumieu type spaces”, Nat. Acad. Sci. Belarus, Proc. Inst. Math. 5 (2000), 99-104. Zentralblatt MATH: 0955.47039
· Zbl 0955.47039
[42] G. · Zbl 0655.35002
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