On operator estimates for elliptic operators with mixed boundary conditions in two-dimensional domains with rapidly oscillating boundary

D. I. Borisov ; R. R. Suleimanov

Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 163-184.

Résumé

The paper considers a second-order elliptic operator with variable sufficiently smooth coefficients in an arbitrary two-dimensional domain with rapidly oscillating boundary under the assumption that the oscillation amplitude is small. The structure of the oscillations is fairly arbitrary in that no periodicity or local periodicity conditions are imposed. The oscillating boundary is divided into two components with the Dirichlet boundary condition posed on one of the components and the Neumann condition, on the other. Such mixed boundary conditions are preserved under homogenization; as a result, the functions in the domain of the homogenized operator have weak power-law singularities. Despite these singularities, we have been able to modify the technique in our previous papers appropriately so as to prove the uniform resolvent convergence of the perturbed operator to the homogenized operator and estimate the convergence rate.

Détail

Keywords: oscillating boundary, operator estimate, mixed boundary conditions.

Bibliographie

[1] E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., 127, Springer, New York, 1980 | MR

[2] O. A. Oleinik, G. A. Iosifyan, A. S. Shamaev, Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo MGU, M., 1990 | MR

[3] A. G. Belyaev, A. G. Mikheev, A. S. Shamaev, “Difraktsiya ploskoi volny na bystroostsilliruyuschei poverkhnosti”, Zh. vychisl. matem. i matem. fiz., 32:8 (1992), 1258–1272 | MR | Zbl

[4] V. V. Grushin, S. Yu. Dobrokhotov, “Osrednenie v zadache o dlinnykh volnakh na vode nad uchastkom dna s bystrymi ostsillyatsiyami”, Matem. zametki, 95:3 (2014), 359–375 | DOI | MR

[5] V. A. Kozlov, S. A. Nazarov, “Asimptotika spektra zadachi Dirikhle dlya bigarmonicheskogo operatora v oblasti s silno izrezannoi granitsei”, Algebra i analiz, 22:6 (2010), 127–184 | MR | Zbl

[6] S. A. Nazarov, “Asimptotika resheniya i modelirovanie zadachi Dirikhle v uglovoi oblasti s bystroostsilliruyuschei granitsei”, Algebra i analiz, 19:2 (2007), 183–225 | MR | Zbl

[7] S. E. Pastukhova, “Effekt ostsilliruyuschei granitsy pri usrednenii odnoi zadachi klimatizatsii”, Differents. uravneniya, 37:9 (2001), 1216–1222 | MR

[8] Y. Amirat, O. Bodart, G. A. Chechkin, A. L. Piatnitski, “Boundary homogenization in domains with randomly oscillating boundary”, Stochastic Process. Appl., 121:1 (2011), 1–23 | DOI | MR

[9] J. Arrieta, S. Brushi, “Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation”, Discrete Contin. Dyn. Syst. Ser. B, 14:2 (2010), 327–351 | MR

[10] G. R. Barrenechea, P. Le Tallec, F. Valentin, “New wall laws for the unsteady incompressible Navier–Stokes equations on rough domains”, M2AN Math. Model. Numer. Anal., 36:2 (2002), 177–203 | DOI | MR

[11] G. A. Chechkin, A. Friedman, A. L. Piatnitski, “The boundary-value problem in domains with very rapidly oscillating boundary”, J. Math. Anal. Appl., 231:1 (1999), 213–234 | DOI | MR

[12] E. N. Dancer, D. Daners, “Domain perturbation for elliptic equations subject to Robin boundary conditions”, J. Differential Equations, 138:1 (1997), 86–132 | DOI | MR

[13] M. K. Gobbert, C. A. Ringhofer, “An asymptotic analysis for a model of chemical vapor deposition on a microstructured surface”, SIAM J. Appl. Math., 58:3 (1998), 737–752 | DOI | MR

[14] W. Jäger, A. Mikelić, “Couette flows over a rough boundary and drag reduction”, Comm. Math. Phys., 232:3 (2003), 429–455 | DOI | MR

[15] Myong-Hwan Ri, Effective wall-laws for the Stokes equations over curved rough boundaries, arXiv: 1311.0977

[16] N. Neuss, M. Neuss-Radu, A. Mikelić, “Effective laws for the Poisson equation on domains with curved oscillating boundaries”, Appl. Anal., 85:5 (2006), 479–502 | DOI | MR

[17] D. Borisov, G. Cardone, L. Faella, C. Perugia, “Uniform resolvent convergence for strip with fast oscillating boundary.”, J. Differential Equations, 255:12 (2013), 4378–4402 | DOI | MR

[18] D. I. Borisov, “Ob operatornykh otsenkakh dlya ploskikh oblastei s neregulyarnym iskrivleniem granitsy: usloviya Dirikhle i Neimana”, Probl. mat. anal., 116 (2022), 69–84 | MR

[19] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR

[20] V. A. Kondratev, “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi ili uglovymi tochkami”, Tr. MMO, 16, Iz-vo Mosk. un-ta, M., 1967, 209–292 | MR | Zbl

[21] S. E. Pastukhova, “Ob otsenkakh usredneniya dlya singulyarno vozmuschennykh operatorov”, Probl. mat. anal., 106 (2020), 149–168 | MR

[22] G. Griso, “Interior error estimate for periodic homogenization”, Anal. Appl. (Singap.), 4:1 (2006), 61–79 | DOI | MR

[23] N. N. Senik, “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898 | DOI | MR

[24] T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic systems: $L_2$-operator error estimates”, Mathematika, 59:2 (2013), 463–476 | DOI | MR

[25] T. A. Suslina, “Homogenization of the Neumann problem for elliptic systems with periodic coefficients”, SIAM J. Math. Anal., 45:6 (2013), 3453–3493 | DOI | MR