Sur l’unicité d’un problème inverse spectral bidimensionnel. (On the uniqueness of an inverse spectral bidimensional problem). (French) Zbl 0679.35083

El Badia, Abdellatif

C. R. Acad. Sci., Paris, Sér. I 308, No. 10, 273-276 (1989).

We show that the spectral characteristics \((\lambda_ n,\| \phi_ j(.,.,\lambda_ n)\|)_{n\geq 0}\); \(j=1,...,m_ n\), are sufficient to ensure the uniqueness of the associated inverse spectral bidimensional problem: \(\Delta \phi +(\lambda -q)\phi =0,\) in (0,1)\(\times (0,1)\), and \(q(x,y)=q(x).\)
Theorem. Let \(B(q,j_ 1,j_ 2,j_ 3,j_ 4)\) be the problem: \[ \Delta \phi +(\lambda -q)\phi =0\quad in\quad \Omega, \] \[ \partial_ n\phi +j_ 1\phi =0\quad on\quad]0,1[\times \{0\},\quad \partial_ n\phi +j_ 2\phi =0\quad on\quad]0,1[\times \{1\}, \] \[ \partial_ n\phi +j_ 3\phi =0\quad on\quad \{0\}\times]0,1[,\quad \partial_ n\phi +j_ 4\phi =0\quad on\quad \{1\}\times]0,1[. \] Let p and q in \(C^ 0(0,1):\int^{1}_{0}p(t)dt=\int^{1}_{0}q(t)dt.\)
Let \((\lambda_ n,\| \phi_ j(.,.,\lambda_ n)\|)_{n\geq 0}\), \((\mu_ n,\| \psi_ j(.,.,\mu_ n)\|)_{n\geq 0}\); \(j=1,...,m_ n\) be the spectral characteristics of the problems \(B(p,j_ 1,j_ 2,j_ 3,j_ 4)\) and \(B(q,j_ 1,j_ 2,j_ 3,j_ 4)\), respectively.
If \(\lambda_ n=\mu_ n\) and \(\| \phi_ j(.,.,\lambda_ n)\| =\| \psi_ j(.,.,\mu_ n)\|\) for all \(n\geq 0\) then \(p=q\) in (0,1).

Reviewer: A.El Badia

MSC:

35R30	Inverse problems for PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

inverse spectral problem; spectral characteristics