F. John 稳定性条件与 A. Carasso 的 SECB 约束在向后抛物问题中的比较
F. John's stability conditions vs. A. Carasso's SECB constraint for backward parabolic problems
AI总结：
本文证明在向后抛物问题中，F. John 的先验有界条件可被 A. Carasso 的 SECB 约束替代，从而获得改进的稳定性估计，并引入了一类新的最优稳定正则化解。
摘要：
为了解决向后抛物问题，F. John [{\it Comm. Pure. Appl. Math.} (1960)] 引入了两个约束：“$\|u(T)\| \le M$”和$\|u(0) - g \| \le δ$，其中$u(t)$满足向后热方程，$t\in(0,T)$，初始数据为$u(0)$。A. Carasso 在 [{\it SIAM J. Numer. Anal.} (1994)] 中引入了{\it 从延拓边界缓慢演化} (SECB) 约束，以实现向后抛物问题在延拓边界$t=T$处对数据的连续依赖性。额外的“SECB 约束”保证了直到$t=T$的稳定性显著改善。本文证明，仅使用这三个约束中的两个即可获得相同类型的稳定性。更准确地说，我们表明先验有界条件$\|u(T)\| \le M$是多余的。这意味着 Carasso 的 SECB 条件可用于替代 F. John 的先验有界条件，并具有改进的稳定性估计。此外，针对具有 SECB 约束的向后抛物问题，引入了一类新的正则化解。新的正则化解是最优稳定的，我们还提供了一种构造性方案进行计算。最后给出了数值算例。
英文摘要：
In order to solve backward parabolic problems F. John [{\it Comm. Pure. Appl. Math.} (1960)] introduced the two constraints "$\|u(T)\|\le M$" and $\|u(0) - g \| \le δ$ where $u(t)$ satisfies the backward heat equation for $t\in(0,T)$ with the initial data $u(0).$ The {\it slow-evolution-from-the-continuation-boundary} (SECB) constraint has been introduced by A. Carasso in [{\it SIAM J. Numer. Anal.} (1994)] to attain continuous dependence on data for backward parabolic problems even at the continuation boundary $t=T$. The additional "SECB constraint" guarantees a significant improvement in stability up to $t=T.$ In this paper we prove that the same type of stability can be obtained by using only two constraints among the three. More precisely, we show that the a priori boundedness condition $\|u(T)\|\le M$ is redundant. This implies that the Carasso's SECB condition can be used to replace the a priori boundedness condition of F. John with an improved stability estimate. Also a new class of regularized solutions is introduced for backward parabolic problems with an SECB constraint. The new regularized solutions are optimally stable and we also provide a constructive scheme to compute. Finally numerical examples are provided.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
