论文标题
半线性椭圆方程的逆源问题的唯一结果和量规破坏
Uniqueness results and gauge breaking for inverse source problems of semilinear elliptic equations
论文作者
论文摘要
我们研究与形式的半线性椭圆方程相关的反源问题 \ [ ΔU（x）+a（x，u）= f（x）， \] 在有限的域上$ω\ subset \ mathbb {r}^n $，$ n \ geq 2 $。我们表明，可以使用非线性打破一类非线性$ a（x，u）$的逆源问题的量规对称性。这与线性方程的反源问题相反，线性方程始终具有量规对称性。非线性类别包括某些多项式和指数非线性。对于这些非线性，我们确定$ a（x，u）$和$ f（x）$从关联的DN地图中唯一的。 此外，对于一般的非线性$ a（x，u）$，我们表明我们可以恢复衍生物$ \ partial_u^ka（x，u）$和源$ f（x）$升至量表。特别是，我们通过删除$ u \ equiv 0 $是解决方案的假设，将一般多项式非线性恢复到量规并概括[FO20，LLLS20]的结果。
We study inverse source problems associated to semilinear elliptic equations of the form \[ Δu(x)+a(x,u)=F(x), \] on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both $a(x,u)$ and $F(x)$ uniquely from the associated DN map. Moreover, for general nonlinearities $a(x,u)$, we show that we can recover the derivatives $\partial_u^ka(x,u)$ and the source $F(x)$ up to a gauge. Especially, we recover general polynomial nonlinearities up to a gauge and generalize results of [FO20,LLLS20] by removing the assumption that $u\equiv 0$ is a solution.
