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Domain variational formulation for free boundary problems. (English) Zbl 0537.35074
Optimization of distributed parameter structures, Vol. II, NATO Adv. Study Inst. Ser., E, Appl. Sci. 50, 1152-1194 (1981).
[For the entire collection see Zbl 0511.00034.]
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 \(\beta(u)(x)= meas\{y\in \Omega: u(y)<u(x)\}\).
Results for the first problem are stated in concise form, whereas for the second problem techniques are developed and partial results are given.
Reviewer: M.Brokate

35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
49J20 Existence theories for optimal control problems involving partial differential equations