A priori estimates for finite-energy sequences of Muller's functional with non-coercive two-well potential with symmetrically placed wells

Abstract

In this paper we obtain a priori estimates for finite-energy  sequences of M\"uller's functional  $$I^{\varepsilon}_a(v)=\int_{0}^{1}\Big({\varepsilon}^2v''^2(s)+W(v'(s))+a(s)v^2(s)\Big)ds\;,$$ where $v\in {\rm H}^{2}\oi{0}{1}$ and $W$ is non-coercive two-well potential with symmetrically placed zero-points. We prove  $\Gamma$-convergence of corresponding relaxed functionals accordingto the approach of G. Alberti and S. M\"uller  as  $\varepsilon\longrightarrow 0$ for $W$ which satisfies $\int_{-\infty}^{0}\sqrt{W}=\int_{0}^{+\infty}\sqrt{W}=+\infty$.