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The aim of this paper is to develop existence and regularity results for free boundary problems by presenting a functional $$J(\Omega)$$ whose argument is a domain $$\Omega$$ and proving that (i) J has a local minimum and (ii) if $$\Omega$$ is a stationary point of J then $$\Gamma =\partial \Omega$$ is the free boundary. The author concentrates upon two problems from plasma physics. The first is: In a fixed domain D $$-\Delta u(x)={\bar \beta}(u)(x)(x)$$ in $$\Omega =\{u>0\}$$, and $$=0$$ otherwise; $$u=-1$$ on $$\partial D$$, where $${\bar \beta}(u)(x)=meas\quad \{y\in D:u(y)\leq u(x)\}$$ and all level sets of u must have measure zero. The second problem is $$-\Delta u-(d^ 2u^*/ds^ 2)\bullet \beta(u)=f$$ in $$\Omega u=\partial u/\partial n=0$$ on $\Gamma =\partial \Omega du^*(0)/ds=du^*(| \Omega |)/ds=0,$ where $$u^*$$ is the inverse of $$s=\nu(u)(t)= meas\{x\in \Omega: u(x)<t\}$$ and $$\beta(u)(x)= meas\{y\in \Omega: u(y)<u(x)\}$$.