Boundary control problem for elliptic equations (Q5938977)

The problem of minimization of the functional \[ F[\Omega] = \int_{\Omega}f(x, u(x), \bigtriangledown u(x)) dx \tag{1} \] on solutions of a boundary value problem, for example, the Dirichlet problem, for an elliptic equation \[ -\text{div}(A(x)\bigtriangledown u(x))= \rho(x), \quad x \in \Omega, \] \[ u\mid_{\partial \Omega}= \varphi(x), \quad \tag{2} \] where \(\Omega\) is a bounded domain in the Euclidean space \(\mathbb{R}^n\), \(n\geq 2\), \(A(x)\) is the symmetric matrix with coefficients \(a_{ij} \in {L}_{\infty}(\mathbb{R}^n)\), is considered. It is supposed that a part \(S\) of the boundary \(\partial \Omega\) is fixed and another part \(\Gamma\) (the so-called free part) can be changed in a class of functions. Let \(u(x) \equiv\) for \(x \in \Gamma\): \[ u \mid_{S}= \varphi (x), \quad \tag{3} \] \[ u \mid_{\Gamma}= 0. \quad \tag{4} \] Then the stated problem (1)--(4) can be considered as a problem of minimization of the functional (1) by the control of the free part \(\Gamma\) of the boundary \(\partial \Omega\). A short review of approaches to the problem is given. Existence of a solution in a class of domains with boundary uniformly satisfying Lipschitz conditions is proved.