Title and Abstract Winkert

A new class of double phase variable exponent problems

Abstract: In this talk we discuss a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We present certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S+)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. We also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded and we discuss optimal critical embeddings. Finally we present some recent existence results for equations involving this new type of operator.