Memory effects in homogenisation: Linear second-order equations
The author studies homogenisation problems. The problems \(A_ \varepsilon\) employ linear differential equations, but the limit problem \(A_ 0\) is given by an integro-differential operator: The expression of the integral term is deduced under two different sets of assumptions: bounds in \(L^ 1\) or \(L^ 2\). The \(L^ 2\) setting uses the Fourier transform and natural assumptions of boundedness in \(L^ \infty\) and uniform ellipticity of the coefficients \(a^ \varepsilon\), \(b^ \varepsilon\) and \(c^ \varepsilon\). The \(L^ 1\) settings are obtained under additional assumptions. The description of the memory term is given for a problem on a bounded interval, by using the eigenfunction expansion and a representation theorem for Nevanlinna functions. An explicit study is given for one space dimensional problem. Generalization is given in several space dimensions where the operator \(-\partial^ 2_ x\) is replaced by the Laplace operator \(-\Delta\). Several examples are computed.
representation theorem for Nevanlinna functions
scientific article; zbMATH DE number 488201