Wojciech Kryszewski: A generalized version of the Miranda-Poincaré theorem with applications to state-constrained elliptic PDE

The famous H. Poincar ́e conjecture dated 1893 stating that a continuous \(f : C = [−1, 1]^n\to \mathbb{R}^n\) such that for \(x, y \in C\) and \(i = 1, ..., n\) if \(x_i = −1\) and \(y_i = −1\), then \(f_i (x)f_i(y)\leq 0\), has a zero was proved in 1940 by C. Miranda. It easily appears that the Miranda-Poincar ́e theorem is equivalent to the Brouwer fixed point theorem. This result was generalized and applied many times, but most of these generalizations have rather technical character and do not reflect the real meaning of assumptions. In the talk I will give a brief survey of some recent developments of the Miranda theorem along with a description of their geometric meaning. We will discuss global and local aspects of results as well as topological tools allowing to detect zeros or fixed points of maps defined on subsets of Banach spaces. Finally the existence of stationary solutions to some Neumann and Dirichlet boundary value elliptic problems and periodic solutions of the corresponding parabolic problems under the presence of state constraints will be presented.