通过测度空间和对偶性建立双重非线性方程的直接方法
A direct method for doubly nonlinear equations via convexification in spaces of measures and duality
AI总结：
本文提出了一种通过测度空间和对偶性直接求解双重非线性方程的方法，无需时间离散化，利用弱凸框架和Hamilton-Jacobi不等式解决能量耗散问题。
摘要：
在反射Banach空间中，双重非线性方程解的存在性通过一种全局时间变分方法得以建立，该方法受De Giorgi原理启发，将相关的流动描述为某个合适能量耗散函数在轨迹上的null极小解。与著名的最小化运动方案不同，所提出的方法不依赖任何时间离散化或迭代构造。相反，它提供了一种基于在测度空间中放松问题的直接方法，受连续方程约束：在此过程中，由于Ambrosio的超位置原理，没有引入间隙。在这一弱凸框架中，通过两个进一步步骤恢复null极小性质的正确性。首先，Von Neumann极小最大化定理的仔细应用将对偶问题识别为对光滑且有界的柱函数集的上确界，解决了一个Hamilton-Jacobi型不等式。其次，此类Hamilton-Jacobi系统解的适当
英文摘要：
Existence of solutions to doubly nonlinear equations in reflexive Banach spaces is established by resorting to a global-in-time variational approach inspired by De Giorgi's principle, which characterizes the associated flows as null-minimizers of a suitable energy-dissipation functional defined on trajectories. In contrast to the celebrated minimizing movements scheme, the proposed strategy does not rely on any time-discretization or iterative constructions. Instead, it provides a direct method based on the relaxation of the problem in spaces of measures, constrained by the continuity equation: in this procedure, no gap is introduced due to the Ambrosio's superposition principle. Within this weak convex framework, the validity of the null-minimization property is recovered through two further steps. First, a careful application of the Von Neumann minimax theorem yields an identification of the dual problem as a supremum over the set of smooth and bounded cylinder functions, solving an Hamilton-Jacobi-type inequality. Secondly, a suitable "backward boundedness" property of solutions to such Hamilton-Jacobi system gives a proper bound of the dual problem, ensuring that the minimum value of the original functional is actually zero. The proposed strategy naturally extends to non-autonomous equations, encompassing time- and space-dependent dissipation potentials and time-dependent potential energies.
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