Non Elliptic Theory: Spectral Properties and Regularity of the Fractal Laplacian

This paper systematically investigates the spectral properties and regularity ofthe fractal Laplacian, providing a mathematical foundation for the modeling andanalysis of nonlocal elliptic theory in complex fractal systems. Based on fractalSobolev spaces and self-similar measures, the domain of the fractal Laplacian isdefined and examined, with discussions on its spectral behavior and regularity un?der fractal geometric constraints. The spectral gap theorem and H¨older continuityare rigorously proven. Numerical analyses of the characteristic spectrum of theSierpinski triangle validate the theoretical accuracy. Furthermore, by incorporat?ing applications such as the quantum Schr¨odinger equation, fractal diffusion inecological systems, and mechanical properties of composite materials, the broadapplicability of the fractal Laplacian is demonstrated. The optimization of compu?tational frameworks and the ethical risks of fractal modeling are also proposed forfuture research.