Nonuniformly elliptic problems

Modern theory of PDEs is centered around nonlinear problems. We concentrate on a wide class of singular and degenerate elliptic and parabolic equations with nonstandard structure. They are employed in modelling the properties of anisotropic and inhomogeneous materials, diffusion-type processes in media having nonstandard properties (including the motions of multiphased fluids in strongly inhomogeneous porous media), and in models ranging from modern material science to meteorology and image restoration. Nonuniformly elliptic problems require plenty of modern, sophisticated tools to build a suitable functional analytical setting. We want to discuss various aspects of nonstandard growth PDE, such as density of smooth functions in unconventional function spaces and optimal regularity conditions for solutions or minimizers to variational functionals and the Lavrentiev gap phenomenon. Our aim is to share the newest developments in the field.