The Bolzano mean-value theorem and partial differential equations

TL;DR

This paper extends the Bolzano mean-value theorem to infinite-dimensional Banach spaces, establishing existence results for solutions to abstract differential equations and inclusions with constraints, applicable to boundary value problems in PDEs.

Contribution

It introduces infinite-dimensional variants of Bolzano and Miranda theorems, providing new existence results for constrained solutions to differential equations and inclusions without structural assumptions.

Findings

Existence of solutions to abstract equations with constraints in Banach spaces.

Application to boundary value problems for elliptic PDEs with state constraints.

Development of set-valued solution existence results for differential inclusions.

Abstract

We study the existence of solutions to abstract equations of the form $0 = Au + F(u)$, $u\in K\subset E$, where A is an abstract differential operator acting in a Banach space $E$, $K$ is a closed convex set of constraints being invariant with respect to resolvents of A and perturbations are subject to different tangency condition. Such problems are closely related to the so-called Poicar\'e- Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results can and should be regarded as infinite-dimensional variants of Bolzano and Miranda-Poincar\'e theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction-diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.