# 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)

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