Aleksandr A. Kovalevskij Livres

This monograph explores trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations, focusing on the existence and properties of weak and entropy solutions for elliptic second-order equations and certain fourth-order equations with L1-data. It addresses the removability of singularities in solutions to elliptic and parabolic second-order equations in divergence form. The text examines both localized and nonlocalized singularly peaking boundary regimes for various classes of quasilinear parabolic second- and higher-order equations in divergence form. The work is intended for researchers and post-graduate students specializing in partial differential equations and nonlinear analysis. The contents include: - Nonlinear elliptic equations with L^1-data, covering second-order equations with L^1-data and fourth-order equations with strengthened coercivity. - Removability of singularities in quasilinear elliptic and parabolic equations, detailing solutions for both types and including quasilinear elliptic equations with coefficients from the Kato class. - Boundary regimes with peaking for quasilinear parabolic equations, discussing energy methods for localized regimes and functional inequalities in peaking regimes for higher-order equations, as well as nonlocalized regimes with singular peaking. The appendix provides formulations and proofs of auxiliary results, complemented by a bibliography.