论文标题
分数方程的反问题，测量数量最少
Inverse problems for fractional equations with a minimal number of measurements
论文作者
论文摘要
在本文中，我们研究了与以下形式的分数微分方程相关的几个反问题： \ [ （-Δ） \]在有界域$ω\ subset \ mathbb {r}^n $，$ n \ geq 1 $中给出。对于任何有限的$ n $，我们表明$ a^{（k）}（x）$，$ k = 0,1，\ ldots，n $，可以由$ n+1 $ 1 $不同的cauchy数据在$ω_e中唯一确定：如果$ n = \ infty $，则通过使用无限多对cauchy数据来确定唯一性结果。结果非常有趣，因为它在本地情况下通常不正确，即$ s = 1 $，即使对于最简单的情况时，$ n = 0 $，fortiori $ n \ geq 1 $。非局部性在建立唯一性结果中起着关键作用。我们还通过使用最少数量的测量结果来建立其他几个独特的确定结果。此外，在此过程中，我们得出了非线性分数微分方程作为重要副产品的新型比较原理。
In this paper, we study several inverse problems associated with a fractional differential equation of the following form: \[ (-Δ)^s u(x)+\sum_{k=0}^N a^{(k)}(x) [u(x)]^k=0,\ \ 0<s<1,\ N\in\mathbb{N}\cup\{0\}\cup\{\infty\}, \] which is given in a bounded domain $Ω\subset\mathbb{R}^n$, $n\geq 1$. For any finite $N$, we show that $a^{(k)}(x)$, $k=0,1,\ldots, N$, can be uniquely determined by $N+1$ different pairs of Cauchy data in $Ω_e:=\mathbb{R}^n\backslash\overlineΩ$. If $N=\infty$, the uniqueness result is established by using infinitely many pairs of Cauchy data. The results are highly intriguing in that it generally does not hold true in the local case, namely $s=1$, even for the simplest case when $N=0$, a fortiori $N\geq 1$. The nonlocality plays a key role in establishing the uniqueness result. We also establish several other unique determination results by making use of a minimal number of measurements. Moreover, in the process we derive a novel comparison principle for nonlinear fractional differential equations as a significant byproduct.
