Spectral analysis, maximum principles and shape optimization for nonlinear superposition operators of mixed fractional order

TL;DR

This work develops a unified variational framework for nonlinear superposition operators of mixed fractional order, defined as $\mathcal{L}_{\mu,p} u=\int_{[0,1]}(-\Delta)_p^{s}u\,d\mu(s)$ with signed measure $\mu=\mu^{+}-\mu^{-}$, and studies Dirichlet eigenvalue problems, maximum principles, and shape optimization. Using a combination of Lusternik–Schnirelmann theory and nonlocal rearrangement techniques, it proves the existence, isolation, and sign properties of the principal eigenvalue $\lambda_{1,\mu}(\Omega)$ and characterizes a second eigenvalue via a mountain-pass variational scheme, along with a Faber–Krahn inequality for the principal frequency. The authors develop weak and strong maximum principles for the positive part operator $\mathcal{L}_{\mu,p}^{+}$, introduce a nonlocal tail and logarithmic estimates, and establish global boundedness of positive eigenfunctions. The results extend known local, fractional, and mixed operators, providing a robust variational toolkit for nonlinear nonlocal PDE models with potential applications in anomalous diffusion and population dynamics, and lay out a clear path for further regularity and Hong–Krahn–Szegö-type investigations. Overall, the paper delivers a comprehensive spectral and shape-optimization theory for nonlinear mixed-order operators in bounded Lipschitz domains.

Abstract

The main objective of this paper is to investigate the spectral properties, maximum principles, and shape optimization problems for a broad class of nonlinear ``superposition operators" defined as continuous superpositions of operators of mixed fractional order, modulated by a signed finite Borel measure on the unit interval. This framework encompasses, as particular cases, mixed local and nonlocal operators such as $-Δ_p+(-Δ_p)^s$, finite (possibly infinite) sums of fractional $p$-Laplacians with different orders, as well as operators involving fractional Laplacians with ``wrong" signs. The main findings, obtained through variational techniques, concern the spectral analysis of the Dirichlet eigenvalue problem associated with general superposition operators with special emphasis on various properties of the first eigenvalue and its corresponding eigenfunction. We establish weak and strong maximum principles for positive superposition operators by introducing an appropriate notion of the {\it nonlocal tail} for this class of superposition operators and deriving a logarithmic estimate, both of which are of independent interest. Utilizing these newly developed tools, we further investigate the spectral properties of such superposition operators and prove that the first eigenvalue is isolated. Moreover, we show that the eigenfunctions corresponding to positive eigenvalues are globally bounded and that they change sign when associated with higher eigenvalues. In addition, we demonstrate that the second eigenvalue is well-defined and provide the mountain pass characterization. Finally, we address shape optimization problems, in particular, the Faber--Krahn inequality associated with the principal frequency associated with the superposition operators.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• Lemma 2.2