Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem

A. I. Parfenov

Matematičeskie trudy, Tome 20 (2017) no. 1, pp. 158-200.

Résumé

In a special Lipschitz domain treated as a perturbation of the upper half-space, we construct a perturbation theory series for a positive harmonic function with zero trace. The terms of the series are harmonic extensions to the half-space from its boundary of distributions defined by a recurrent formula and passage to the limit. The approximation error by a segment of the series is estimated via a power of the seminorm of the perturbation in the homogeneous Slobodestkiĭ space $b_N^{1-1/N}$. The series converges if the Lipschitz constant of the perturbation is small.

Bibliographie

[1] Belov S. A., “Differentsiruemost reshenii kraevykh zadach pri variatsii oblasti i priblizhennye resheniya”, Differents. uravneniya, 38:1 (2002), 63–69 | Zbl

[2] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975

[3] Gorbachuk V. I., Knyazyuk A. V., “Granichnye znacheniya reshenii differentsialno-operatornykh uravnenii”, UMN, 44:3(267) (1989), 55–91 | Zbl

[4] Kantorovich L. V., Krylov V. I., Priblizhennye metody vysshego analiza, GITTL, M.–L., 1950

[5] Kozlov V. A., Mazya V. G., “Otsenki $L_p$-srednikh i asimptotika reshenii ellipticheskikh kraevykh zadach v konuse. II. Operatory s peremennymi koeffitsientami”, Math. Nachr., 137:1 (1988), 113–139 | DOI | MR | Zbl

[6] Kolyada V. I., “Neravenstva tipa Galyardo–Nirenberga i otsenki modulei nepreryvnosti”, UMN, 60:6(366) (2005), 139–156 | DOI | Zbl

[7] Kufner A., “O zavisimosti resheniya zadachi Dirikhle ot izmeneniya oblasti opredeleniya”, Apl. Mat., 6:4 (1961), 263–273 | MR

[8] Lavrentev M. A., Konformnye otobrazheniya s prilozheniyami k nekotorym voprosam mekhaniki, GITTL, M.–L., 1946

[9] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971

[10] Parfënov A. I., “Vesovaya apriornaya otsenka v raspryamlyaemykh oblastyakh lokalnogo tipa Lyapunova–Dini”, Sib. elektron. matem. izv., 9 (2012), 65–150 | Zbl

[11] Parfënov A. I., “Otsenka pogreshnosti obobschennoi formuly M. A. Lavrenteva normoi drobnogo prostranstva Soboleva”, Sib. elektron. matem. izv., 10 (2013), 335–377 | Zbl

[12] Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, GIFML, M., 1962

[13] Simonenko I. B., “Novyi obschii metod issledovaniya lineinykh operatornykh uravnenii tipa singulyarnykh integralnykh uravnenii. I”, Izv. AN SSSR. Ser. matem., 29:3 (1965), 567–586 | Zbl

[14] Strett Dzh. V. (lord Relei), Teoriya zvuka, v. 1, GITTL, M.–L., 1940

[15] Khërmander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 1, Teoriya raspredelenii i analiz Fure, Mir, M., 1986

[16] Babuška I., Nistor V., “Boundary value problems in spaces of distributions on smooth and polygonal domains”, J. Comput. Appl. Math., 218:1 (2008), 137–148 | DOI | MR | Zbl

[17] Costabel M., Le Louër F., “Shape derivatives of boundary integral operators in electromagnetic scattering. I: Shape differentiability of pseudo-homogeneous boundary integral operators”, Integral Equations Oper. Theory, 72:4 (2012), 509–535 | DOI | MR | Zbl

[18] Earle C. J., Mitra S., “Variation of moduli under holomorphic motions”, In the Tradition of Ahlfors and Bers, Proc. of the First Ahlfors–Bers Colloquium (Stony Brook, 1998), Contemporary Mathematics, 256, eds. I. Kra, B. Maskit, Amer. Math. Soc., Providence, 2000, 39–67 | DOI | MR | Zbl

[19] Garabedian P. R., Schiffer M., “Convexity of domain functionals”, J. Anal. Math., 2 (1953), 281–368 | DOI | MR | Zbl

[20] Groemer H., Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia of Mathematics and Its Applications, 61, Cambridge Univ. Press, Cambridge, 1996 | MR | Zbl

[21] Henrot A., Pierre M., Variation et optimisation de formes. Une analyse géométrique, Mathématiques et Applications, 48, Springer, Berlin, 2005 | DOI | MR | Zbl

[22] Henry D., Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations, London Math. Soc. Lecture Note Ser., 318, Cambridge Univ. Press, Cambridge, 2005 | MR | Zbl

[23] Hu B., Nicholls D. P., “Analyticity of Dirichlet–Neumann operators on Hölder and Lipschitz domains”, SIAM J. Math. Anal., 37:1 (2005), 302–320 | DOI | MR | Zbl

[24] Hu B., Nicholls D. P., “The domain of analyticity of Dirichlet–Neumann operators”, Proc. R. Soc. Edinb., Sect. A, Math., 140:2 (2010), 367–389 | DOI | MR | Zbl

[25] Jerison D. S., Kenig C. E., “Boundary behavior of harmonic functions in non-tangentially accessible domains”, Adv. Math., 46:1 (1982), 80–147 | DOI | MR | Zbl

[26] Kautský J., “Approximation of solutions of Dirichlet's problem on nearly circular domains and their application in numerical methods”, Apl. Mat., 7:3 (1962), 186–200 | MR | Zbl

[27] Komatsu H., “An elementary theory of hyperfunctions and microfunctions”, Partial Differential Equations, v. 1, Banach Center Publications, 27, Polish Academy of Sciences, Warszawa, 1992, 233–256 | MR | Zbl

[28] Lanza de Cristoforis M., “A functional decomposition theorem for the conformal representation”, J. Math. Soc. Japan, 49:4 (1997), 759–780 | DOI | MR | Zbl

[29] Lanza de Cristoforis M., “A domain perturbation problem for the Poisson equation”, Complex Variables Theory Appl., 50:7–11 (2005), 851–867 | DOI | MR | Zbl

[30] Lanza de Cristoforis M., Rossi L., “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 | DOI | MR | Zbl

[31] Laugesen R. S., Siudeja B. A., “Magnetic spectral bounds on starlike plane domains”, ESAIM Control Optim. Calc. Var., 21:3 (2015), 670–689 | DOI | MR | Zbl

[32] Lions J. L., Magenes E., Non-Homogeneous Boundary Value Problems and Applications, v. III, Die Grundlehren der mathematischen Wissenschaften, 183, Springer-Verlag, Berlin etc., 1973 | MR | Zbl

[33] Meyer Y., Coifman R., Wavelets. Calderón–Zygmund and Multilinear Operators, Cambridge Studies in Advanced Mathematics, 48, Cambridge Univ. Press, Cambridge, 1997 | MR

[34] Potthast R., Stratis I. G., “On the domain derivative for scattering by impenetrable obstacles in chiral media”, IMA J. Appl. Math., 68:6 (2003), 621–635 | DOI | MR | Zbl

[35] Quittner P., Souplet Ph., Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser, Basel etc., 2007 | MR | Zbl

[36] Szegö G., “The virtual mass of nearly spherical solids”, Duke Math. J., 16:2 (1949), 209–223 | DOI | MR | Zbl

[37] Verbitsky I. E., “Superlinear equations, potential theory and weighted norm inequalities”, Nonlinear Analysis, Function Spaces and Applications, Proc. of the Spring School (Prague, Czech Repub., 1998), v. 6, eds. M. Krbec et al., Acad. Sci. Czech Repub., Prague, 1999, 223–269 | MR | Zbl

[38] Whitney H., “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36:1 (1934), 63–89 | DOI | MR