Calculus of Variations and Geometric Measure Theory

G. Bellettini - G. Fusco - N. Guglielmi

A concept of solution for forward-backward equations of the form $u_t = \frac{1}{2}(p'(u_x))_x$ and numerical experiments for the singular perturbation $u_t = - \eps^2 u_{xxxx} + \frac{1}{2} (\phi'(u_x))_x$

created by belletti on 18 Nov 2005
modified on 13 Dec 2006


Accepted Paper

Inserted: 18 nov 2005
Last Updated: 13 dec 2006

Journal: Discrete Cont. Dyn. Syst.
Volume: 16
Number: 4
Pages: 783-842
Year: 2006


We study the gradient dynamics associated with the functional $F_\phi(u) := \frac{1}{2}\int_{I} \phi(u_x) dx$ and with its singular perturbation $F_\phi^\epsilon(u):=\frac{1}{2}\int_I \left(\epsilon^2 ({u_{xx}})^2 + \phi(u_x)\right) dx$, where $\phi$ is {\it non convex}. We discuss, with the support of numerical simulations, various aspects of the global dynamics of solutions $u^\epsilon$ of the singularly perturbed equation $u_t = - \epsilon^2 u_{xxxx} + \frac{1}{2} \phi''(u_x)u_{xx}$ for small values of $\epsilon >0$. We develop a point of view that leads to a reinterpretation of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$, and to a well defined notion of solution which is conjectured to coincide with the limit of $u^\epsilon$ as $\epsilon \to 0^+$.