On a Bernoulli-type overdetermined free boundary problem


  • Murat Akman University of Essex, Department of Mathematical Sciences
  • Agnid Banerjee Tata Institute of Fundamental Research, Center for Applicable Mathematics
  • Mariana Smit Vega Garcia Western Washington University, Department of Mathematics


Quasilinear elliptic equations and p-Laplacian, degenerate elliptic equations, free boundary problems, Bernoulli-type free boundary problems



In this article we study a Bernoulli-type free boundary problem and generalize a work of Henrot and Shahgholian in [25] to \(\mathcal{A}\)-harmonic PDEs. These are quasi-linear elliptic PDEs whose structure is modelled on the \(p\)-Laplace equation for a fixed \(1<p<\infty\). In particular, we show that if \(K\) is a bounded convex set satisfying the interior ball condition and \(c>0\) is a given constant, then there exists a unique convex domain \(\Omega\) with \(K\subset \Omega\) and a function \(u\) which is \(\mathcal{A}\)-harmonic in \(\Omega\setminus K\), has continuous boundary values 1 on \(\partial K\) and 0 on \(\partial\Omega\), such that \(|\nabla u|=c\) on \(\partial \Omega\). Moreover, \(\partial\Omega\) is \(C^{1,\gamma}\) for some \(\gamma>0\), and it is smooth provided \(\mathcal{A}\) is smooth in \(\mathbf{R}^n \setminus \{0\}\). We also show that the super level sets \(\{u>t\}\) are convex for \(t\in (0,1)\).



How to Cite

Akman, M., Banerjee, A., & Smit Vega Garcia, M. (2021). On a Bernoulli-type overdetermined free boundary problem. Annales Fennici Mathematici, 46(2), 601–618. Retrieved from https://afm.journal.fi/article/view/110563