Recent Developments in Nonlocal Theory (Partial Differential Equations and Measure Theory)

Oversized hardcover, still sealed in publisher's original shrinkwrap. Gentle rubbing to tips of board corners. -- 464 pages. This multi-author volume consolidates recent progress in the analysis of nonlocal partial differential equations, particularly those involving the fractional Laplacian and related integro-differential operators. It offers both theoretical advancements and practical methodologies applicable across several branches of mathematics, physics, and engineering. The work opens with Claudia Bucur's accessible overview of fractional Sobolev spaces and the fractional Laplacian, providing foundational context for non-experts while still serving as a rigorous reference for advanced readers. Each subsequent section explores a specific subject, showcasing technical results, methodological innovations and open problems. Chen and Zhang's chapter on heat kernels for non-symmetric nonlocal operators is directly relevant to stochastic processes and anomalous diffusion, offering sharp estimates and stability results essential for probabilistic modeling. Da Lio's study of fractional harmonic maps connects nonlocal operators to geometric variational problems, highlighting regularity results critical for understanding geometric flows and manifold-valued functions. Danielli and Salsa's extensive treatment of obstacle problems involving the fractional Laplacian addresses both static and time-dependent formulations, which are pertinent to mathematical finance, fluid dynamics, and elasticity theory, and features optimal regularity results and detailed boundary behavior analysis. Dipierro and Valdinoci's work on nonlocal minimal surfaces emphasizes quantitative estimates and phenomena such as boundary stickiness, which are crucial in materials science and image processing. Rupert Frank provides a meticulous review of eigenvalue bounds for fractional Schrödinger operators, including sharp inequalities and asymptotics, which are central in quantum mechanics and spectral theory. González's chapter on conformal fractional Laplacians bridges nonlocal theory with conformal geometry, offering analytic formulations for the fractional Yamabe problem and connecting scattering theory with geometric PDEs. Kassmann's contribution elucidates the probabilistic interpretation of nonlocal operators through jump processes, integrating analytic and stochastic techniques and offering regularity estimates in Hölder spaces that are applicable to both pure and applied problems. The chapter by Kuusi, Mingione & Sire on the fractional p-Laplacian delivers refined existence and regularity results using modern potential-theoretic tools, addressing operators with measurable coefficients that are highly relevant to nonlinear analysis. Ros-Oton advances the discussion with results on boundary regularity and Pohozaev-type identities, demonstrating how fine estimates enable nonexistence and uniqueness theorems in critical domains. Molica Bisci's use of variational and topological methods to establish existence and multiplicity of solutions for periodic nonlocal problems is particularly strong in applications to nonlinear dynamics and bifurcation theory, using techniques such as Morse theory and pseudo-index arguments. Patrizi closes the volume with a detailed analysis of crystal dislocation dynamics through nonlocal models, highlighting multiscale modeling techniques that link microscopic and macroscopic behaviors via homogenization and viscosity solution theory. Academically, the book excels in unifying diverse subfields under the framework of nonlocality, illustrating how nonlocal models arise naturally in geometric analysis, stochastic processes, variational calculus, and physical modeling. Overall, the volume offers a robust synthesis of theory, methods, and applications, serving as both a state-of-the-art compendium and a practical toolkit for ongoing and future investigations in the field.…