Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem. (English) Zbl 1217.35019

Mel’nyk, T. A.; Nakvasiuk, Iu. A.; Wendland, W. L.

Math. Methods Appl. Sci. 34, No. 7, 758-775 (2011).

The authors consider the elliptic problem \(-\Delta _{x}u(x)=f(x)\) in \(\Omega _{\varepsilon }\), with the homogeneous Dirichlet boundary condition \( u_{\varepsilon }(x)=0\) on \(\Gamma _{\varepsilon }\), with the Signorini-type conditions \(u_{\varepsilon }(x)\leq g(x)\), \(\partial _{\nu }u_{\varepsilon }(x)\leq \varepsilon d(x)\), \((u_{\varepsilon }(x)-g(x))(\partial _{\nu }u_{\varepsilon }(x)-\varepsilon d(x))=0\) on \(S_{\varepsilon }\), and with the Neumann boundary conditions \(\partial _{\nu }u_{\varepsilon }(x)=0\) on \( \partial \Omega _{\varepsilon }\backslash (S_{\varepsilon }\cup \Gamma _{\varepsilon })\). Here \(\Omega _{\varepsilon }\) is a thick junction of type 3:2:1 consisting of the body \(\Omega _{0}\) and of a large number of thin curvilinear and vertical cylinders posed on the top surface of \(\Omega _{0}\), contained in the plane \(\{x_{3}=0\}\). \(S_{\varepsilon }\) is the union of the lateral surfaces of the thin cylinders. \(\Gamma _{\varepsilon }\) is the union of the upper surfaces of the thin cylinders which are contained in the plane \(\{x_{3}=h\}\). \(f\), \(g\) and \(d\) are given functions and \(g\) belongs to some Sobolev space.
The authors propose a variational formulation of this problem introducing the convex set \(K_{\varepsilon }=\{\varphi \in H^{1}(\Omega _{\varepsilon };\Gamma _{\varepsilon }):\varphi |_{S\varepsilon }\leq g|_{S\varepsilon }\) a.e. on \(S_{\varepsilon }\}\). This leads to the definition of weak solutions of this problem. The main result of the paper describes the asymptotic behaviour of the weak solution of this problem. The authors first prove uniform estimates on this weak solution. They then define the notion of weak solution of the homogenized problem and prove the existence of a weak solution of this homogenized problem. Finally they introduce auxiliary functions in order to prove the convergence of the solution of the original problem to that of the limit one. The authors end their paper with the derivation of further properties of the homogenized solution first in the general case, then assuming some additional hypotheses on the data of the problem. In the general case, the authors prove that the equations in \(\Omega _{0}\) can also be treated by means of boundary integral equations.

Reviewer: Alain Brillard (Riedisheim)

MSC:

35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
35J20	Variational methods for second-order elliptic equations
47J05	Equations involving nonlinear operators (general)
74K30	Junctions
35J86	Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators

Keywords:

boundary integral equations; variational formulation

References:

[1]	Blanchard, Boundary homogenization and reduction of dimention in a Kirchhoff-Love plate, SIAM Journal on Mathematical Analysis 39 (6) pp 1764– (2008) · Zbl 1149.74037
[2]	Blanchard, Highly oscillating boundaries and reduction of dimension: the critical case, Analysis and Application 5 pp 137– (2007) · Zbl 1121.35012
[3]	Blanchard, Junction of a periodic family of elastic rods with 3d plate, Journal de Mathématiques Pures et Appliquées 88 (9) pp 1– (2007) · Zbl 1116.74038
[4]	Chechkin, Asymptotic analysis of boundary-value problems in thick cascade junctions, Reports of National Ukrainian Academy of Sciences 9 pp 16– (2008) · Zbl 1164.35316
[5]	D’Apice, Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2, Networks and Heterogeneous Media 2 pp 255– (2007) · Zbl 1127.35310
[6]	Mel’nyk, Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1, Mathematical Models and Methods in Applied Sciences 31 (9) pp 1005– (2008) · Zbl 1145.35327
[7]	Mel’nyk, Homogenization of a boundary-value problem with mixed type of boundary conditions in a thick junction, Differential Equations 43 (5) pp 696– (2007) · Zbl 1140.35340
[8]	Mel’nyk, Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of the type 3:2:1, Ukrainskii Matem. Zhurnal 52 pp 1524– (2000)
[9]	Mel’nyk, Asymptotic structure of the spectrum of the Neumann problem in a thin comb-like domain, Comptes Rendus de l’Académie des Sciences-Series I 319 pp 1343– (1994)
[10]	Nazarov, Junctions of singularly degenerating domains with different limit dimensions, Trudy Seminara imeni I.G. Petrovskogo 18 pp 3– (1995)
[11]	Mel’nyk, Asymptotics of the Neumann spectral problem solution in a domain of ’thick comb’ type, Trudy Seminara imeni I.G. Petrovskogo 19 pp 138– (1996)
[12]	Mel’nyk, Homogenization of the Poisson equation in a thick periodic junction, Zeitschrift für Analysis und ihre Anwendungen 18 pp 953– (1999) · Zbl 0938.35021
[13]	Mel’nyk, Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1, Mathematical Methods in the Applied Sciences 23 pp 321– (2000) · Zbl 0983.35044
[14]	Mel’nyk, Asymptotic analysis of the Neumann problem of the junction of a body and thin heavy rods, Algebra i Analiz 12 pp 188– (2000)
[15]	Fleury, Asymptotic and spectral properties of the acoustic vibrations of body, perforated by narrow channels, Bulletin des Sciences Mathématique 2 pp 149– (1986) · Zbl 0616.76094
[16]	Khruslov, On the resonance phenomenas in the one problem of diffraction, Teorija Funktsij, Funcionalnyj Analis i ix Prilozhenija (Izd-vo Kharkov University) 8 pp 113– (1968)
[17]	Kotliarov, On a limit boundary condition of some Neumann problem, Teorija Funktsij, Funcionalnyj Analis i ix Prilozhenija (Izd-vo Kharkov University) 10 pp 83– (1970)
[18]	Signorini, Questioni di elasticita non linearizzata o semilinearizzata, Rendiconti di Matematica e delle sue Applicazioni 18 pp 95– (1959)
[19]	Kinderlehrer, An Introduction to Variational Inequalities and their Applications (1980) · Zbl 0457.35001
[20]	Lavrentovich YI Knyzkova TV Pidlisnyuk VV The potential of application of new nanostructural materials for degradation of pesticides in water 167 169
[21]	Lions, Variational inequalities, Communications on Pure and Applied Mathematics 20 pp 493– (1976) · Zbl 0152.34601
[22]	Lions, Quelques méthodes De Résolution des Problémes Aux Limites Non Linéires (1969)
[23]	Hsiao, Applied Mathematical Sciences (2008)
[24]	Kazmerchuk, Homogenization of the Signorini boundary-value problem in a thick plane junction, Nonlinear Oscillations 12 (1) pp 4– (2009) · Zbl 1277.35035
[25]	Zhikov, On the homogenization of nonlinear variational problems in perforated domains, Russian Journal of Mathematical Physics 12 (3) pp 393– (1994) · Zbl 0914.35014
[26]	Pastukhova, Homogenization of a mixed problem with Signorini condition for an elliptic operator in a perforated domain, Sbornik Mathematics 192 pp 245– (2001) · Zbl 1097.35024
[27]	Vorobev, On homogenization of the nonuniform Signorini problem for the Poisson equation in a periodic perforated domain, Differential Equations 39 (3) pp 359– (2003)
[28]	Sandrakov, Homogenization of variational inequalities for problems with regular obstacles, Doklady Akademii Nauk 397 pp 170– (2004) · Zbl 1378.35025
[29]	Sandrakov, Homogenization of variational inequalities for non-linear diffusion problems in perforated domains, Izvestia: Mathematics 69 (5) pp 1035– (2005) · Zbl 1103.35016
[30]	Zeidler, Nonlinear Functional Analysis and its Applications 3 (1985) · Zbl 0583.47051
[31]	Glowinski, Numerical Methods for Nonlinear Variational Problems (1984/2008) · Zbl 0536.65054
[32]	Hsiao GC Wendland WL Boundary Element Methods: Foundation and Error Analysis. Encyclopedia of Computational Mechanics 2004 339 373
[33]	Riasanov, The Fast Solution of Boundary Integral Equations (2007)
[34]	Uspenskii, The traces of functions the Sobolev space Wl1, ..., lnp on smooth surfaces, Siberian Mathematical Journal 13 pp 298– (1972) · Zbl 0247.46050
[35]	Of, The fast multipole method for the symmetric boundary integral formulation, IMA Journal of Numerical Analysis 26 pp 272– (2006) · Zbl 1101.65114
[36]	Adams, Sobolev Spaces (1975)
[37]	Hellwig, Partial Differential Equations (1977)
[38]	Ciarlet, The Finite Element Method for Elliptic Problems (1978)
[39]	Hackbusch, Multi-grid Methods and Applications (2003)