×

zbMATH — the first resource for mathematics

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