Regularity of solutions to a distributed and boundary optimal control problem governed by semilinear elliptic equations (Q1995758)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) (\(N=2,3\)), with boundary \(\Gamma\) of class \(C^{1,1}\), let \( L,f: \Omega\times \mathbb{R}\rightarrow \mathbb{R}\), \(l:\Gamma\times \mathbb{R}\rightarrow \mathbb{R}\) be Carathéodory functions, and let \( g_1: \Omega\times \mathbb{R}\rightarrow \mathbb{R}\), \(g_2:\Gamma\times \mathbb{R}\rightarrow \mathbb{R}\) be continuous functions. The authors study the existence of local minimizers \((y,u,v)\) of the cost function \(I:W^{1,4}(\Omega)\times L^4(\Omega)\times L^4(\Gamma)\rightarrow \mathbb{R}\) defined by \[ \begin{array}{lll} &I(y,u,v)=&\int_\Omega \left(L(x,y(x))+\frac{\lambda_1}{2}|u(x)|^2+\frac{\lambda_2}{4}|u(x)|^4\right)dx+\\ & &\int_\Gamma \left(l(x,y(x))+\frac{\mu_1}{2}|v(x)|^2+\frac{\mu_2}{4}|v(x)|^4\right)ds \end{array} \] subjected to the following conditions \[ \begin{array}{l} \left\{\begin{array}{ll} -\sum_{i,j=1}^{N}D_j(a_{ij}(x)D_iy)+a_0(x)y+f(x,y)=u \ \ \ &\text{in} \ \ \Omega;\\ \sum_{i,j=1}^{N}a_{ij}(x)\nu_j(x)D_iy=v \ \ \ &\text{on} \ \ \Gamma, \end{array}\right.\\ g_1(x,y(x)) + \epsilon_1u(x) + \gamma_1u(x)^3\leq 0 \ \ \ \ \ \ \text{for a.e.} \ \ x\in\Omega\\ g_2(x',y(x')) + \epsilon_2v(x') + \gamma_2v(x')^3\leq 0 \ \ \ \ \text{for a.e.} \ \ x'\in\Gamma\end{array} \] where \(\lambda_i,\mu_i,\epsilon_i,\gamma_i\) are positive numbers, \(a_{ij}=a_{ji}\in C^{0,1}(\overline{\Omega})\), \((i,j=1,2)\), \(a_0\in C(\overline{\Omega})\), and \[\inf_{\xi \in \mathbb{R}^N\setminus\{0\}}\|\xi\|^{-2}\sum_{i,j=1}^{N}a_{ij}\xi_i\xi_j>0.\] The main result of this paper ensures the existence of such a local minimizer in the space \(C^{1,\alpha}(\overline{\Omega})^2\times C^{1,\alpha}(\Gamma)\), with \(\alpha\in (0,1)\).