# American Institute of Mathematical Sciences

September  2008, 3(3): 633-646. doi: 10.3934/nhm.2008.3.633

## On the conjugate of periodic piecewise harmonic functions

 1 Narvik University College, and Norut Narvik, P.O.B. 385 N-8505 Narvik, Norway 2 Narvik University College, P.O.B. 385 N-8505 Narvik, Norway 3 Department of Mathematics, Luleå University, SE-97187 Luleå, Sweden

Received  March 2007 Published  June 2008

The paper considers the conjugate of periodic functions which are piecewise harmonic. In particular, we consider the harmonic conjugate of the solution of the problem of stationary heat conduction through a periodic network of fibres and matrix of arbitrary shape. A numerical example is also presented.
Citation: Dag Lukkassen, Annette Meidell, Peter Wall. On the conjugate of periodic piecewise harmonic functions. Networks & Heterogeneous Media, 2008, 3 (3) : 633-646. doi: 10.3934/nhm.2008.3.633
 [1] 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 [2] 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 [3] 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 [4] Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61 [5] Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks & Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503 [6] Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361 [7] Francisco Crespo, Sebastián Ferrer. On the extended Euler system and the Jacobi and Weierstrass elliptic functions. Journal of Geometric Mechanics, 2015, 7 (2) : 151-168. doi: 10.3934/jgm.2015.7.151 [8] Shingo Takeuchi. The basis property of generalized Jacobian elliptic functions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2675-2692. doi: 10.3934/cpaa.2014.13.2675 [9] Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43 [10] 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 [11] 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 [12] Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91 [13] Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475 [14] Yuanxiao Li, Ming Mei, Kaijun Zhang. Existence of multiple nontrivial solutions for a $p$-Kirchhoff type elliptic problem involving sign-changing weight functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 883-908. doi: 10.3934/dcdsb.2016.21.883 [15] Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221 [16] Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451 [17] Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019041 [18] Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697 [19] Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064 [20] Fanghua Lin, Xiaodong Yan. A type of homogenization problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 1-30. doi: 10.3934/dcds.2003.9.1

2018 Impact Factor: 0.871