A minimization problem with free boundary related to a cooperative system.

I will talk about the minimum problem for the functional

$$\int_{\Omega}\left( \vert \nabla u \vert^{2} + Q^{2}\chi_{\{\vert u\vert>0\}}\right)dx$$

with the constraint $u_i\geq 0$ for $i=1,\ldots,m$ where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and $u=(u_1,\cdots,u_m)\in H^{1}(\Omega;\mathbb{R}^{m})$.

 

First we derive the Euler equation satisfied by each component. Then we show that the noncoincidence set $\{\vert u\vert>0\}$ is (locally) nontangentially accessible. Having this we are able to establish sufficient regularity of the force term appearing in the Euler equations and derive the regularity of the free boundary $\Omega\cap\partial\{\vert u\vert>0\}$. 

This is a joint work with Professor Luis A. Caffarelli from University of Texas at Austin and Professor Henrik Shahgholian from KTH Royal Institute of Technology.