A relaxation result for micromagnetics in SBV
The paper deals with minimization problems for functionals of the form \[ E(m) = \int_{\Omega} f (x, m(x), \triangledown m(x)) \,dx+ \int_{\Omega \cap S(m)} \big|[m](x)\big| \,dH^{N -1} +E_1 (m), \] where \(E_1\) is a continuous functional in the \(L^2\) norm, \([m]\) is the jump of \(m, S(m)\) is the set of discontinuities of \(m\), and \(m\) satisfies the pointwise constraint \(|m(x)| = 1\). Problems of this kind arise in some continuum models in micromagnetics. The correct framework to set the problem is the one of SBV spaces, and an important issue consists in characterizing the lower semicontinuous envelope of \(E\): \[ \inf \Big\{\lim_n \inf E(m_n ): |m_n|=1,\;m_n\rightarrow m\text{ in }L^1\Big\} . \] In the paper this characterization is provided under mild and natural assumptions on the integrand \(f\).
scientific article; zbMATH DE number 5601459