Nonlinear Partial Differential Equations in Fractional Sobolev Spaces: Existence, Regularity, and Stability

This paper explores the theory and application of fractional Sobolev spaces Ws,p(Ω) in the analysis of nonlinear partial differential equations (PDEs). Specifically, we examine the existence, uniqueness, regularity, and blow-up phenomena of solutions to fractional PDEs, such as the fractional Laplace equation. The paper presents several original theorems related to the embedding properties of fractional Sobolev spaces, energy minimization problems, and the maximum principle for fractional operators. We also investigate the stability of solutions under perturbations and establish the fractional Poincare inequality and trace theorem. Our results contribute to the understanding of the interplay between fractional Sobolev norms and nonlinear variational problems, offering insights into energy minimization, regularity results, and nonlocal effects in PDEs. These findings have significant implications for the study of nonlocal problems, fractional variational calculus, and fractional diffusion equations in applied mathematics and physics.