Birkhoff-Kellogg type results in product spaces and their application to differential systems

We provide a new version of the well-known Birkhoff-Kellogg invariant-direction Theorem in product spaces. Our results concern operator systems and give the existence of component-wise eigenvalues, instead of scalar eigenvalues as in the classical case, that have corresponding eigenvectors with all components nontrivial and localized by their norm. We also show that, when applied to nonlinear eigenvalue problems for differential equations, this localization property of the eigenvectors provides, in turn, qualitative properties of the solutions. This is illustrated in two contexts of systems of PDEs and ODEs. We show the applicability of our theoretical results with two explicit examples.

💡 Research Summary

The paper presents a novel extension of the classical Birkhoff‑Kellogg invariant‑direction theorem to product spaces, addressing a gap in the literature concerning operator systems that act on a Cartesian product of two spaces. While the traditional theorem guarantees a scalar eigenvalue λ and a non‑trivial eigenvector on the boundary of a bounded open set, a direct application to product spaces may yield eigenvectors with one or more trivial components, which is undesirable for systems of differential equations. To overcome this, the authors introduce the notion of component‑wise eigenvalues (λ₁, λ₂) and prove that under suitable compactness and positivity assumptions each component of the eigenvector can be forced to have a prescribed norm, thereby ensuring that no component is zero.

The central result (Theorem 2.3) assumes two normed spaces X₁, X₂ (each either a cone Kᵢ or the whole space) and a compact map T = (T₁,T₂) defined on the product of “shells” C₁,r₁ × C₂,r₂, where Cᵢ,rᵢ denotes the intersection of a cone (or the whole space) with the sphere of radius rᵢ. The key hypothesis (2.1) requires that the norms of T₁ and T₂ are bounded away from zero on the boundary of this product set. By constructing retractions ρᵢ onto the boundary and an auxiliary map N that normalizes the images of T₁ and T₂, the authors apply Schauder’s fixed‑point theorem to obtain a point (x₀,y₀) on the product boundary satisfying x₀ = λ₁ T₁(x₀,y₀), y₀ = λ₂ T₂(x₀,y₀) with λᵢ = rᵢ/‖Tᵢ(x₀,y₀)‖ > 0. This yields a pair of positive component‑wise eigenvalues and eigenvectors whose components have exactly the prescribed norms r₁ and r₂.

Further extensions are provided: Theorem 2.5 treats the mixed case where C₁ is a cone and C₂ is a full infinite‑dimensional space, producing an additional eigenpair with λ₂ negative (by flipping the sign in the auxiliary map). Theorem 2.6 shows that when both factors are infinite‑dimensional spaces, four eigenpairs (with all sign combinations) can be obtained. These results broaden the spectrum of possible eigenvalues for operator systems, especially when one component’s kernel changes sign—a situation that frequently arises in differential systems.

The theoretical framework is then applied to two concrete classes of differential equations. First, a coupled elliptic system on the unit ball Ω⊂ℝⁿ with homogeneous Dirichlet boundary conditions is considered:  −Δu = λ₁ f(x,u,v), −Δv = λ₂ g(x,u,v), u=v=0 on ∂Ω. Using the Green’s function for the Laplacian on the ball, the PDEs are rewritten as a Hammerstein integral system u = λ₁ T₁(u,v), v = λ₂ T₂(u,v) where T₁,T₂ map the cone of non‑negative continuous functions into itself and are compact. By assuming lower bounds f(x)≤f(x,u,v) and g(x)≤g(x,u,v) on the rectangle