July  2010, 28(3): 1207-1235. doi: 10.3934/dcds.2010.28.1207

Vanishing viscosity for fractal sets

1. 

Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, United States

2. 

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma "La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy

Received  April 2010 Published  April 2010

We imbed an array of thin highly conductive fibers in a surrounding two-dimensional medium with small viscosity. The resulting composite medium is described by a second order elliptic operator in divergence form with discontinuous singular coefficients on an open domain of the plane. We study the asymptotic spectral behavior of the operator when, simultaneously, the viscosity vanishes and the fibers develop fractal geometry. We prove that the spectral measure of the operator converges to the spectral measure of a self-adjoint operator associated with the lower-dimensional fractal limit of the thin fibers. The limit fiber is a compact set that disconnects the initial domain into infinitely many non-empty open components. Our approach is of variational nature and relies on Hilbert space convergence of quadratic energy forms.
Citation: Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207
[1]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1

[2]

Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure & Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9

[3]

R. Estrada. Boundary layers and spectral content asymptotics. Conference Publications, 1998, 1998 (Special) : 242-252. doi: 10.3934/proc.1998.1998.242

[4]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711

[5]

José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002

[6]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

[7]

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239

[8]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677

[9]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753

[10]

Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155

[11]

Sarah Constantin, Robert S. Strichartz, Miles Wheeler. Analysis of the Laplacian and spectral operators on the Vicsek set. Communications on Pure & Applied Analysis, 2011, 10 (1) : 1-44. doi: 10.3934/cpaa.2011.10.1

[12]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[13]

Paul Loya and Jinsung Park. On gluing formulas for the spectral invariants of Dirac type operators. Electronic Research Announcements, 2005, 11: 1-11.

[14]

Andrea Braides, Valeria Chiadò Piat. Non convex homogenization problems for singular structures. Networks & Heterogeneous Media, 2008, 3 (3) : 489-508. doi: 10.3934/nhm.2008.3.489

[15]

Maria Rosaria Lancia, Valerio Regis Durante, Paola Vernole. Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1493-1520. doi: 10.3934/dcdss.2016060

[16]

Mario Ahues, Filomena D. d'Almeida, Alain Largillier, Paulo B. Vasconcelos. Defect correction for spectral computations for a singular integral operator. Communications on Pure & Applied Analysis, 2006, 5 (2) : 241-250. doi: 10.3934/cpaa.2006.5.241

[17]

Natalia O. Babych, Ilia V. Kamotski, Valery P. Smyshlyaev. Homogenization of spectral problems in bounded domains with doubly high contrasts. Networks & Heterogeneous Media, 2008, 3 (3) : 413-436. doi: 10.3934/nhm.2008.3.413

[18]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787

[19]

Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004

[20]

Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]