论文标题
非线性进化方程的扩展变分理论通过模块化空间
An extended variational theory for nonlinear evolution equations via modular spaces
论文作者
论文摘要
我们提出了进化方程的经典变分理论的扩展，该理论也可能是非反射性和不可分割的空间中的动力学。关键点是基于与给定凸功能相关的抽象模块空间建立新的变分结构。首先，我们表明新的变分三重适合构建进化，从某种意义上说，可以引入新颖的二元性折扣并具有广义的计算链规则。其次，我们在进化方程的扩展变化意义上证明了良好性，而无需依靠任何反思性假设和对非线性的任何多项式要求。最后，我们讨论了可以在此框架中解决的几个重要应用：这些覆盖物（但不限于Musielak-Orlicz-Sobolev空间中的方程），例如可变指数，Orlicz，加权Lebesgue和双相空间。
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 evolution, 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 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.
