Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces

This book provides a detailed study of nonlinear partial differential equations satisfying certain nonstandard growth conditions which simultaneously extend polynomial, inhomogeneous and fully anisotropic growth. The common property of the many different kinds of equations considered is that the growth conditions of the highest order operators lead to a formulation of the equations in MusielakOrlicz spaces. This high level of generality, understood as full anisotropy and inhomogeneity, requires new proof concepts and a generalization of the formalism, calling for an extended functional analytic framework. This theory is established in the first part of the book, which serves as an introduction to the subject, but is also an important ingredient of the whole story. The second part uses these theoretical tools for various types of PDEs, including abstract and parabolic equations but also PDEs arising from fluid and solid mechanics. For connoisseurs, there is a short chapter on homogenization of elliptic PDEs.

The book will be of interest to researchers working in PDEs and in functional analysis.

Iwona Chlebicka is an assistant professor at the University of Warsaw. Her interests focus on nonlinear PDEs with Orlicz and Musielak¿Orlicz growth, including their anisotropic variants, and the functional analysis of the underlying function spaces. Investigating elliptic and parabolic problems with data below duality she studies well-posedness of various notions of very weak solutions, as well as their regularity.

Piotr Gwiazda is a professor at the Institute of Mathematics of the Polish Academy of Sciences. His wide spectrum of research includes the topics of weak, renormalized and measure-valued solutions to nonlinear PDEs. A substantial part of his contributions concerns elliptic and parabolic equations in Musielak¿Orlicz spaces. He is also interested in PDEs arising from fluid and solid mechanics as well as mathematical biology.

Agnieszka ¿wierczewska-Gwiazda is a professor at the University of Warsaw. Her diverse scientific interests include nonlinear PDEs in Musielak¿Orlicz spaces as well as hyperbolic conservation laws, entropy methods in fluid dynamics and mathematical biology, weak convergence methods and measure-valued solutions.

Aneta Wróblewska-Kami¿ska is an assistant professor at the Institute of Mathematics of the Polish Academy of Sciences. Her scientific interests include nonlinear PDEs, existence of weak solutions, and their qualitative properties, mathematical models of fluid mechanics, non-Newtonian fluids, Musielak¿Orlicz spaces, singular limits, and Navier¿Stokes and Euler type equations.

Part I Overture: 1. Introduction.- 2.N-Functions.- 3. MusielakOrlicz Spaces.- Part II PDEs: 4. Weak Solutions.- 5. Renormalized Solutions.- 6. Homogenization of Elliptic Boundary Value Problems.- 7. Non-Newtonian Fluids.- Part III Auxiliaries: 8. Basics.- 9. Functional Inequalities.- References.- List of Symbols.- Index.