We address homogenization problems for variational inequalities issue from unilateral constraints for the \n p\n-Laplacian posed in perforated domains of \n {\mathbb{R}}^{n}\n, with \n n\hspace{0.17em}\ge \hspace{0.17em}3\n and \n p\in [2,n].\epsilon \n is a small parameter which measures the periodicity of the structure while \n {a}_{\epsilon}\u2a7d\epsilon \n measures the size of the perforations. We impose constraints for solutions and their fluxes (associated with the \n p\n-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter \n {\beta}_{\epsilon}\n which may be very large, namely, \n {\beta}_{\epsilon}\to \infty \n as \n \epsilon \to 0\n. We first consider the case where \n p\hspace{0.17em}<\hspace{0.17em}n\n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the different parameters of the problem: \n p,n,\epsilon ,{a}_{\epsilon}\n and \n {\beta}_{\epsilon}\n. Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for \n p\hspace{0.17em}=\hspace{0.17em}n\n and the case of non periodically distributed isoperimetric perforations. We make it clear that in the averaged constants of the problem, the perimeter of the perforations appears for any shape.

Keywords: Nonlinear homogenization, perforated media, variational inequalities, critical relations for parameter

^{1}; Lobo, Miguel

^{1}; Pérez, Eugenia

^{1}; Podolskii, Alexander V.

^{1}; Shaposhnikova, Tatiana A.

^{1}

@article{COCV_2018__24_3_921_0, author = {G\'omez, Delfina and Lobo, Miguel and P\'erez, Eugenia and Podolskii, Alexander V. and Shaposhnikova, Tatiana A.}, title = {Unilateral problems for the {p-Laplace} operator in perforated media involving large parameters}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {921--964}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017026}, mrnumber = {3877188}, zbl = {1409.35023}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017026/} }

TY - JOUR AU - Gómez, Delfina AU - Lobo, Miguel AU - Pérez, Eugenia AU - Podolskii, Alexander V. AU - Shaposhnikova, Tatiana A. TI - Unilateral problems for the p-Laplace operator in perforated media involving large parameters JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 921 EP - 964 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017026/ DO - 10.1051/cocv/2017026 LA - en ID - COCV_2018__24_3_921_0 ER -

%0 Journal Article %A Gómez, Delfina %A Lobo, Miguel %A Pérez, Eugenia %A Podolskii, Alexander V. %A Shaposhnikova, Tatiana A. %T Unilateral problems for the p-Laplace operator in perforated media involving large parameters %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 921-964 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017026/ %R 10.1051/cocv/2017026 %G en %F COCV_2018__24_3_921_0

Gómez, Delfina; Lobo, Miguel; Pérez, Eugenia; Podolskii, Alexander V.; Shaposhnikova, Tatiana A. Unilateral problems for the p-Laplace operator in perforated media involving large parameters. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 921-964. doi : 10.1051/cocv/2017026. http://archive.numdam.org/articles/10.1051/cocv/2017026/

An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18 (1992) 481–496 | DOI | MR | Zbl

, , and ,Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113 (1990) 209–259 | DOI | MR | Zbl

,Homogenization of p-Laplacian in perforated domain. Ann. Inst. H. Poincar Anal. Non Linaire 26 (2009) 2457–2479 | DOI | Numdam | MR | Zbl

, , and ,Variational inequalities with varying obstacles: the general form of the limit problem. J. Funct. Anal. 50 (1983) 329–386 | DOI | MR

and ,Boundary homogenization in perforated domains for adsorption problems with an advection term. Appl. Anal. 95 (2016) 1517–1533 | DOI | MR | Zbl

, , , and ,Sharp constants for some inequalities connected to the p-Laplace operator. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005) 56 | MR | Zbl

,Homogenization results for elliptic problems in periodically perforated domains with mixed-type boundary conditions. Asymptot. Anal. 80 (2012) 45–56 | MR | Zbl

, and ,Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains. Rend. Mat. Appl. 16 (1996) 387–413 | MR | Zbl

,A strange term coming from nowhere, in Topics in the Mathematical Modelling of Composite Materials, edited by and Birkäuser. Progr. Nonlinear Differ. Equ. Appl. 31 (1997) 45–93 | MR | Zbl

and ,Nonhomogeneous Neumann problems in domains with small holes. RAIRO Modél. Math. Anal. Numér. 22 (1988) 561–607 | DOI | Numdam | MR | Zbl

and ,A generalized strange term in Signorini’s type problems. ESAIM: M2AN 37 (2003) 773–805 | DOI | Numdam | MR | Zbl

, and ,Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24 (1997) 239–290 | Numdam | MR | Zbl

and ,Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations. Res. Notes Math. 106, Pitman (1985) | MR | Zbl

,On the homogenization of some nonlinear problems in perforated domains, Rend. Sem. Mat. Univ. Padova 84 (1991) 91–108 | Numdam | MR | Zbl

and ,Homogenization for the p-Laplace operator and nonlinear Robin boundary conditions in perforated media along manifolds. Dokl. Math. 89 (2014) 11–15 | DOI | Zbl

, , , and ,Homogenization of a variational inequality for the p-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: p equal to the dimension of the space. Dokl. Math. 93 (2016) 140–144 | DOI | MR | Zbl

, , , and ,Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds. Appl. Anal. 92 (2013) 218–237 | DOI | MR | Zbl

, , and ,On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems. Math. Methods Appl. Sci. 38 (2015) 2606–2629 | DOI | MR | Zbl

, , , and ,Homogenization for the p-Laplace operator in perforated media with nonlinear restrictions on the boundary of the perforations: a critical case. Dokl. Math. 92 (2015) 433–438 | DOI | MR | Zbl

, , and ,On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems. Asymptot. Anal. 80 (2012) 289–322 | MR | Zbl

, and ,The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries, in Homogenization and Applications to Material Sciences. GAKUTO Internat. Ser. Math. Sci. Appl. Gakkotosho 9 (1995) 203–213 | MR | Zbl

,Homogenization and porous media. Springer Velag (1997) | DOI | MR | Zbl

,Homogenization of a variational inequality for the Laplace operator with nonlinear restriction for the flux on the interior boundary of a perforated domain. Nonlinear Anal. Real World Appl. 15 (2014) 367–380 | DOI | MR | Zbl

, and ,The Poisson equation with semilinear boundary conditions in domains with many tiny holes. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989) 43–86 | MR | Zbl

,The Poisson equation with nonautonomous semilinear boundary conditions in domains with many time holes. SIAM J. Math. Anal. 22 (1991) 1222–1245 | DOI | MR | Zbl

,Application of uniform distribution to homogenization of a thin obstacle problem with p-Laplacian. Commun. Partial Differ. Equ. 39 (2014) 1870–1897 | DOI | MR | Zbl

and ,An Introduction to Variational Inequalities and their Applications. Moscow: Mir (1983) | MR | Zbl

and ,Homogenization of a nonlinear Dirichlet problem in a periodically perforated domain, in Recent Advances in Nonlinear Elliptic and Parabolic Problems, Pitman Res. Notes Math. Ser. 208, Longman Sci. Tech. (1989) | MR | Zbl

and ,Notes on the p-Laplace equation. Report. University of Jyväskylä Department of Mathematics and Statistics, Jyväskylä: University of Jyväskylä 102 (2006) | MR | Zbl

,Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod (1969) | MR | Zbl

,Boundary Value Problems in Domains with a Fine-grained Boundary. Izdat. Naukova Dumka, in Russian (1974) | MR | Zbl

and ,Error estimates for the homogenization of a quasilinear Dirichlet problem in a periodically perfored domain, in Progress in Partial Differential Equations: the Metz Surveys, 2, Pitman Res. Notes Math. Ser. 296, Longman Sci. Tech. (1993) | MR | Zbl

and ,On homogenization problem for the Laplace operator in partially perforated domains with Neumann’s condition on the boundary of cavities. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 6 (1995) 133–142 | MR | Zbl

and ,On homogenization of the Poisson equation in partially perforated domains with arbitrary density of cavities and mixed type conditions on their boundary. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. Ser. 7 (1996) 129–146 | MR | Zbl

and ,Homogenization of a class of elliptic problems with nonlinear boundary conditions in domains with small holes. Carpathian J. Math. 31 (2015) 77–88 | DOI | MR | Zbl

and ,Homogenization of a mixed problem with Signorini condition for an elliptic operator in a perforated domain. Sb. Math. 192 (2001) 245–260 | DOI | MR | Zbl

,A homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin type boundary conditions. Dokl. Math. 90 (2014) 489–494 | DOI | MR | Zbl

, and ,Solution continuation and homogenization of a boundary value problem for the p-Laplacian in a perforated domain with a nonlinear third boundary condition on the boundary of holes. Dokl. Math. 91 (2015) 30–34 | DOI | Zbl

,Homogenization for the p-Laplacian in a n-dimensional domain perforated by fine cavities, with nonlinear restrictions on their booundary, when p = n. Dokl. Math. 92 (2015) 464–470 | DOI

and ,Homogenization limit for the boundary value problem with the p-Laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain. Funct. Differ. Equ. 19 (2012) 351–370 | MR

and ,Random homogenization of p-Laplacian with obstacles in perforated domain. Commun. Partial Differ. Equ. 37 (2012) 538–559 | DOI | MR | Zbl

,On regularity of the solutions of variational inequalities. Zap. Nauchn. Semin. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27(1972) 211–219 | MR

,Decay rate estimates for the quasi-linear Timoshenko system with nonlinear control and damping terms. J. Math. Phys. 52 (2011) 18pp | MR | Zbl

and ,Homogenization of boundary value problems in perforated domains with the third boundary condition and the resulting change in the character of the nonlinearity in the problem. Differ. Equ. 47 (2011) 78–90 | DOI | MR | Zbl

and ,*Cited by Sources: *