双非线性抛物型障碍问题的熵方法
An entropy approach to doubly nonlinear parabolic obstacle problems
AI总结：
本文针对双非线性抛物型障碍问题，引入重正化熵公式，结合有序惩罚程序等方法，证明了不同扩散情形下解的存在、唯一性与$L^1$稳定性，明确了极限反应的构成。
AI中文摘要：
本文研究双非线性抛物型障碍问题，我们引入了一种重正化熵公式，其中反应测度通过在障碍处计算熵乘子来编码。对于仅依赖梯度的扩散以及连续的与时间无关的障碍，我们证明了全局$L^1$比较估计、解的唯一性，以及无测度熵上解类中的极小性原理。对于同时依赖解及其梯度的扩散，我们结合边界控制分解，证明了连续时间相关障碍下的存在性。通过两种有序惩罚程序，我们得到了近似解及相应反应项的收敛性，极限反应由带有限密度的绝对连续部分和集中在规定紧空间区域的有限Radon测度组成。单侧时间正则化产生梯度的强收敛并确定非线性通量，综合这些结果，在共同假设下得到存在性、唯一性和$L^1$稳定性。
英文摘要：
This paper studies doubly nonlinear parabolic obstacle problems. We introduce a renormalized entropy formulation in which the reaction measure is encoded by evaluating the entropy multiplier at the obstacle. For diffusion depending only on the gradient and continuous time-independent obstacles, we prove a global $L^1$ comparison estimate, uniqueness of the solution, and a minimality principle among a measure-free entropy supersolution class. For diffusion depending on both the solution and its gradient, we prove existence for continuous time-dependent obstacles with a boundary-controlled decomposition. Using two ordered penalization procedures, we obtain convergence of the approximate solutions and of the corresponding reaction terms. The limiting reaction consists of an absolutely continuous part with bounded density and a finite Radon measure concentrated on a prescribed compact spatial region. A one-sided time regularization yields strong convergence of the gradients and identifies the nonlinear flux. Together, the results yield existence, uniqueness and $L^1$ stability under their common assumptions.
