A free boundary problem with log-term singularity (Q1687827)

Summary: We study a minimum problem for a non-differentiable functional whose reaction term does not have scaling properties. Specifically we consider the functional \[ \mathcal J(v)=\int_\Omega\left(\frac{|\nabla v|^2}{2} -v^+(\text{log}\: v-1)\right)dx\rightarrow \text{min} \] which should be minimized in some natural admissible class of non-negative functions. Here, \(v^+=\max\{0,v\}.\) The Euler-Lagrange equation associated with \(\mathcal J\) is \[ -\Delta u= \chi_{\{u>0\}}\text{log}\: u, \] which becomes singular along the free boundary \(\partial\{u>0\}.\) Therefore, the regularity results do not follow from classical methods. Besides, the logarithmic forcing term does not have scaling properties, which are very important in the study of free boundary theory. Despite these difficulties, we obtain optimal regularity of a minimizer and show that, close to every free boundary point, they exhibit a super-characteristic growth like \[ r^2|\text{log}\: r|. \] This estimate is crucial in the study of analytic and geometric properties of the free boundary.