On optimal \begin{document}$ L^1 $\end{document} -control in coefficients for quasi-linear Dirichlet boundary value problems with \begin{document}$ BMO $\end{document} -anisotropic \begin{document}$ p $\end{document} -Laplacian
Umberto De Maio, Peter I. Kogut, Gabriella Zecca · Mathematical Control and Related Fields · 2020
We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principal part and \begin{document}$ L^1 $\end{document} -control in coefficient of the low-order term. We assume that the matrix of anisotropy belongs to BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we introduce a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces.