[Paper Review] On a minimax theorem: an improvement, a new proof and an overview of its applications

Biagio Ricceri|arXiv (Cornell University)|Dec 24, 2016

Optimization and Variational Analysis22 citations

TL;DR

This paper presents a new inductive proof of a minimax theorem for functions on product spaces where the second variable lies in a convex set of a Hausdorff topological vector space. It improves upon prior results by extending Theorem 1.A to broader settings without relying on Mosconi's intermediate result, and establishes conditions under which either a minimax equality holds or multiple global minima exist, with applications in nonlinear PDEs and variational analysis.

ABSTRACT

Theorem 1 of [14], a minimax result for functions $f:X imes Y o {\bf R}$, where $Y$ is a real interval, was partially extended to the case where $Y$ is a convex set in a Hausdorff topological vector space ([15], Theorem 3.2). In doing that, a key tool was a partial extension of the same result to the case where $Y$ is a convex set in ${\bf R}^n$ ([7], Theorem 4.2). In the present paper, we first obtain a full extension of the result in [14] by means of a new proof fully based on the use of the result itself via an inductive argument. Then, we present an overview of the various and numerous applications of these results.

Motivation & Objective

• To extend Theorem 1.A from real intervals to convex sets in Hausdorff topological vector spaces.
• To provide a new proof of the extended minimax result using an inductive argument based on Theorem 1.A itself.
• To establish a minimax result for functions with concave dependence on the second variable, without relying on Mosconi's theorem.
• To present a comprehensive overview of applications in nonlinear analysis, PDEs, and optimization.
• To demonstrate the existence of multiple weak solutions in elliptic boundary value problems via the minimax framework.

Proposed method

• Use of an inductive proof technique on the dimension of the simplex $ S_n $, starting from the base case $ n=1 $.
• Construction of a continuous affine map $ \psi: S_k \times [0,1] \to S_{k+1} $ to relate higher-dimensional simplices to lower ones.
• Definition of a lifted function $ \tilde{f}(x, \lambda, \mu) = f(x, \psi(\lambda, \mu)) $ to transfer properties across dimensions.
• Application of the inductive hypothesis to $ \tilde{f} $, leveraging quasi-concavity and lower semicontinuity.
• Use of inf-compactness and lower semicontinuity to ensure attainment of infima and suprema.
• Application of the main minimax result to variational problems in Sobolev spaces, particularly $ H^1_0(\Omega) $, via functional minimization and perturbation arguments.

Experimental results

Research questions

• RQ1 Can Theorem 1.A be fully extended from real intervals to convex sets in Hausdorff topological vector spaces?
• RQ2 Does the minimax equality hold when the continuity condition on $ f(x, \cdot) $ is relaxed to upper semicontinuity?
• RQ3 Can the minimax result be proven without relying on Mosconi’s theorem for $ \mathbb{R}^n $?
• RQ4 What are the implications of the minimax result for the existence of multiple solutions in nonlinear elliptic PDEs?
• RQ5 Under what conditions does the functional associated with a PDE attain at least two global minima in $ H^1_0(\Omega) $?

Key findings

• The paper establishes a full extension of Theorem 1.A to convex sets in Hausdorff topological vector spaces, proving that either $ \sup_Y \inf_X f = \inf_X \sup_Y f $ or $ f(\cdot, \hat{y}) $ has at least two global minima.
• A new inductive proof is provided that relies solely on Theorem 1.A, avoiding external results like Mosconi’s.
• Theorem 1.2 generalizes Theorem 1.C by proving the minimax result for concave $ f(x, \cdot) $ without requiring upper semicontinuity.
• The minimax framework is applied to a class of Kirchhoff-type PDEs, showing existence of at least three weak solutions, two of which are global minima.
• For $ f \in \mathcal{A} $ with $ \sup_{\xi} \int_0^\xi f(x,t)dt > 0 $ on a set of positive measure and subquadratic growth, the problem $ -\Delta u = \lambda f(x,u) + \varphi $ has at least three weak solutions for large $ \lambda $ and some $ \varphi \in C $.
• The result is extended to $ H^{-1}(\Omega) $-valued perturbations, showing existence of multiple solutions for a broad class of nonlinearities with controlled growth.