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A. Menovschikov, A. Molchanova, L. Scarpa:
"An extended variational theory for nonlinear evolution equations via modular spaces";
in: "ASC Report 39/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 35.

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. Firstly, we show that the new variational triple is suited for framing the evolu-tion, in the sense that a novel duality paring can be introduced and a generalised computational chain rule holds. Secondly, we prove well-posedness in an extended variational sense for evolu-tion equations, without relying on any reflexivity assumption and any polynomial requirement on the nonlinearity. Finally, we discuss several important applications that can be addressed in this framework: these cover, but are not limited to, equations in Musielak-Orlicz-Sobolev spaces, such as variable exponent, Orlicz, weighted Lebesgue, and double-phase spaces.

Nonlinear evolution equations; variational approach; modular spaces; Musielak-Orlicz-Sobolev spaces.

http://www.asc.tuwien.ac.at/preprint/2020/asc39x2020.pdf