JournalsjemsVol. 9 , No. 1DOI 10.4171/jems/70

Upper bounds for singular perturbation problems involving gradient fields

  • Arkady Poliakovsky

    Technion - Israel Institute of Technology, Haifa, Israel
Upper bounds for singular perturbation problems involving gradient fields cover

Abstract

We prove an upper bound for the Aviles-Giga problem, which involves the minimization of the energy E\e(v)=\eΩ2v2dx+1\eΩ(1v2)2dxE_\e(v)=\e\int_\Omega\big|\nabla^2v\big|^2dx+\frac{1}{\e}\int_\Omega\big(1-|\nabla v|^2\big)^2dx over vH2(Ω)v\in H^2(\Omega), where \e>0\e>0 is a small parameter. Given vW1,(Ω)v\in W^{1,\infty}(\Omega) such that vBV\nabla v\in BV and v=1|\nabla v| =1 a.e., we construct a family {v\e}\{v_\e\} satisfying: v\evv_\e\to v in W1,p(Ω)W^{1,p}(\Omega) and E\e(v\e)13Jv+vv3dHN1E_\e(v_\e)\to\frac{1}{3}\int_{J_{\nabla v}}|\nabla^+v-\nabla^-v|^3\,d{\mathcal H}^{N-1}, as \e\e goes to 00.