Courses

1. Andrea Cianchi (University of Florence, Italy)
   Orlicz-Sobolev spaces and applications to elliptic PDEs
   After introducing the Orlicz-Sobolev spaces and presenting their basic definitions and properties, optimal embeddings of diverse kinds will be presented. Embeddings into Orlicz spaces, into Orlicz-Lorentz spaces and into spaces of continuous functions will be described. Trace embeddings will also be considered. The second part of my lectures will be devoted to a few applications of Orlicz-Sobolev embeddings to the regularity theory of nonlinear elliptic partial differential equations and of the calculus of variations under growth conditions of non-polynomial type. In particular, the classical question of the boundedness of solutions to boundary value problems for equations of this sort will be discussed.
    
2. Lars Diening (University of Bielefeld, Germany)

   Maximal Regularity of Degenerate Elliptic Equations 
   We study the maximal regularity of the weighted Poisson and p-Poisson system (p-Laplacian). In particular, we examine how the regularity of the data transfers to the one the gradient and the stress of the solution.  We are interested in the local as well as global results up to the boundary. The weights fall into the class of Muckenhoupt weights with a certain notion of small oscillations. This includes some degenerate weights.  We put a strong focus in obtaining estimates that are optimal with respect to the smallness parameters of the weights and the boundary. We will support this with suitable examples.
   
   This series of talks combines joint results with Balci, Breit, Byun, Cianchi, Lee, Maz'ya and Weimar.
   
   
   

3. Piotr Gwiazda (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland)
   PDEs in Anisotropic Musielak-Orlicz Spaces

   This lecture 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 Musielak–Orlicz 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 will be  established in the first part of the lecture, 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.
   
   The lecture is based mainly on the recent monograph:
   Iwona Chlebicka, Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska-Kamińska, Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces, Springer Monographs in Mathematics. Springer, Cham, 2021.

   
    
4. Juha Kinnunen (Aalto University, Helsinki, Finland)

   Parabolic BMO with applications to PDEs
   

   We discuss parabolic bounded mean oscillation (BMO) and Muckenhoupt weights and related to a doubly nonlinear parabolic partial differential equation (PDE). In the natural geometry of the PDE, the time variable scales to the power in the structural conditions for the PDE. Consequently, the Euclidean balls and cubes are replaced by parabolic rectangles respecting this scaling in all estimates. The main challenge is that the definition of parabolic BMO consists of two conditions on the mean oscillation of a function, one in the past and the other one in the future with a time lag between the estimates.
   
   The main results include a parabolic John-Nirenberg inequality, which gives exponential decay estimates for the oscillation of a function, and a characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO through parabolic Muckenhoupt weights and maximal functions. 
   
   We also discuss connections and applications of the results to the regularity theory of nonlinear parabolic partial differential equations. Harnack’s inequality gives a scale and location invariant pointwise bound for a positive solution at a given time in terms of its values at later times. The argument is based on a successive application of Sobolev’s inequality and energy estimates together with the fact that a logarithm of a positive weak solution belongs to the parabolic BMO and the parabolic John–Nirenberg lemma.
   
   
   

5. Paolo Marcellini (University of Florence, Italy) 
   Regularity for elliptic equations under p,q-growth conditions

   In the main part of the course we propose a full self-contained proof of the interior regularity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form, allowing the general case with x and u explicit dependence too. We focus on the growth conditions, since we deal with natural growth conditions, where the model example is the p-Laplacian equation, as well as with general growth conditions and the p,q-growth conditions, which will be considered in detail in this course.

   In the first part we will discuss about different steps to get interior regularity of weak solutions: a first step starts from a weak solution, which a-priori exists in some Sobolev or Orlicz spaces, to its local Lipschitz continuity; i.e., to a local L^\infinity gradient bound. Then, a second step, with different assumptions, is oriented to further interior regularity; i.e. C^1,\alpha or full regularity. We show the different nature of the assumptions, in particular we emphasize why the growth conditions essentially are not needed in this second step, while they are characteristic and necessary in the first step.