zbMATH — the first resource for mathematics

The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. (English) Zbl 0798.35123
This paper is devoted to the study of the nonlinear Cahn-Hilliard equation \[ u_ t= \Delta(-\varepsilon^ 2\Delta u+ F'(u)),\quad x\in \Omega,\;\partial u/\partial n= \partial(-\varepsilon^ 2 \Delta u+ f'(u))/\partial n=0,\quad x\in \partial\Omega. \] \(\Omega\) is taken to be a bounded smooth domain in \(\mathbb{R}^ 2\), \(u(x,t)\) stands for the concentration at \(x\) at time \(t\), \(F\) is the bulk free energy and is assumed to have two equal nondegenerate minima at \(\pm 1\).
The authors consider a family \(u^ \varepsilon\), \(\varepsilon\in ]0,\varepsilon_ 0]\) which, as \(\varepsilon\to 0\), approaches a step function with values \(-1\), \(+1\) which is discontinuous along a smooth curve \(\Gamma\subset\Omega\) and give “semi-classical estimates” for the principal eigenvalue of the linearized Cahn-Hilliard operator about \(u= u^ \varepsilon\), \[ \Delta(- \varepsilon^ 2\Delta\phi+ F''(u^ \varepsilon)\phi)= -\lambda\phi,\quad x\in \Omega,\;\partial\phi/\partial n= \partial\Delta\phi/\partial n=0,\quad x\in \partial\Omega. \] The function \(u^ \varepsilon\) is chosen to have a very specific behavior in the direction orthogonal to \(\Gamma\) as \(\varepsilon\to 0\).
Reviewer: B.Helffer (Paris)

35P15 Estimates of eigenvalues in context of PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35K55 Nonlinear parabolic equations
Full Text: DOI