Nonlinear elliptic Dirichlet boundary value problems on time scales

TL;DR

This work develops existence and uniqueness results for nonlinear elliptic Dirichlet problems on time-scale domains via a spectral theory for the $\nabla\Delta$-Laplacian and operator reformulations. Under a Lipschitz nonlinearity with constant $L$, uniqueness follows from a contraction mapping when $L<\lambda_1$, the first eigenvalue of the linear Dirichlet problem $-\Delta_{\,\mathbb{T}}u+\lambda u=0$. When only a one-sided growth condition with $\alpha<\lambda_1$ holds, existence is obtained through the Leray--Schauder fixed point theorem, aided by a priori bounds. The analysis constructs the one- and multi-dimensional eigenfunction frameworks, proving self-adjointness, positivity, and completeness of product eigenfunctions, and provides explicit eigenvalue bounds. Examples across discrete, hybrid, and higher-dimensional time scales illustrate sharpness and scope, including resonance and multiplicity phenomena. The results extend nonlinear elliptic theory to time scales, unifying continuous, discrete, and hybrid settings and suggesting avenues for general domains and multiplicity analysis.

Abstract

We establish existence and uniqueness results for nonlinear elliptic Dirichlet boundary value problems on n-dimensional time scale domains. Time scales provide a unified framework that encompasses continuous, discrete, and hybrid settings. Under a Lipschitz condition on the nonlinearity bounded by the first eigenvalue, we prove existence and uniqueness using the contraction mapping theorem. Under a weaker one-sided growth condition, we establish existence using the Leray--Schauder fixed point theorem. To apply these functional analytic methods, we reformulate the problem as an operator equation, which requires developing the spectral theory for the Dirichlet Laplacian with mixed nabla-delta derivatives. We establish self-adjointness, positivity, and completeness of eigenfunctions, and the product eigenfunctions form a complete orthonormal basis in the n-dimensional setting.

Figures (2)

• Figure 1: Boundary decomposition of a two-dimensional rectangular domain on time scales. The gray edges and points are the sets $A_1 \times \partial A_2$ and $\partial A_1 \times A_2$ and $\Omega$ consists of the black rectangles, line segments, and points.
• Figure 2: $G(t,t) = \dfrac{(t-a)(b-t)}{b-a}$ attains its maximum at the value $t = \dfrac{a+b}{2}$.

Theorems & Definitions (57)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Hilbert--Schmidt theorem
• Theorem 2.2: Contraction Mapping Theorem
• Theorem 2.3: Leray--Schauder fixed point theorem Kolmogorov-Fomin
• Remark 3.1
• Remark 3.2
• Theorem 3.3