Perona-Malik方程半离散格式解的极限的单调性性质
Monotonicity properties of limits of solutions to the semi-discrete scheme for the Perona-Malik equation
AI总结：
本文研究一维Perona-Malik方程半离散逼近解的极限，给出了严格收敛性不被保持的病态反例，并证明了所有广义解均满足最大值原理和全变差单调性。
摘要：
我们考虑一维Perona-Malik方程的广义解，其定义为半离散逼近解的所有可能极限，其中关于空间变量的导数被差商代替。我们的第一个结果是一个病态反例，其中初始数据作为有界变差函数严格收敛，但严格收敛并非对所有正时间都保持，特别地，许多基本量（如上确界或全变差）无法取到极限。然而，在我们的第二个结果中，我们证明了所有广义解均满足经典光滑解的某些性质，即最大值原理和全变差的单调性。反例的验证依赖于与合适的下/上解的比较结果。单调性结果是对一类更一般的演化曲线（我们称之为$uv$-演化）证明的。
英文摘要：
We consider generalized solutions of the Perona-Malik equation in dimension one, defined as all possible limits of solutions to the semi-discrete approximation in which derivatives with respect to the space variable are replaced by difference quotients. Our first result is a pathological example in which the initial data converge strictly as bounded variation functions, but strict convergence is not preserved for all positive times, and in particular many basic quantities, such as the supremum or the total variation, do not pass to the limit. Nevertheless, in our second result we show that all our generalized solutions satisfy some of the properties of classical smooth solutions, namely the maximum principle and the monotonicity of the total variation. The verification of the counterexample relies on a comparison result with suitable sub/supersolutions. The monotonicity results are proved for a more general class of evolution curves, that we call $uv$-evolutions.
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