We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly nonreflexive and nonseparable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. First, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality pairing can be introduced and a generalized computational chain rule holds. Second, we prove well-posedness in an extended variational sense for evolution 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.

An extended variational theory for nonlinear evolution equations via modular spaces

Scarpa L.
2021-01-01

Abstract

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly nonreflexive and nonseparable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. First, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality pairing can be introduced and a generalized computational chain rule holds. Second, we prove well-posedness in an extended variational sense for evolution 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.
2021
Modular spaces
Musielak-Orlicz-Sobolev spaces
Nonlinear evolution equations
Variational approach
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1195254
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