Recovering a variable exponent (Q2034392)

The paper deals with the inverse problem of recovering the exponent \(p(x)\) in the one-dimensional variable exponent \(p(x)\)-Laplace equation \[ -\left( \gamma(x) |u'(x)|^{p(x) - 2} u'(x) \right)' = 0 \] from the Dirichlet-to-Neumann map. The function \(\gamma(x)\) is supposed to be fixed. The authors prove the uniqueness theorem for the case of a constant positive \(\gamma\) and obtain a reconstruction procedure for \(\gamma \equiv 1\). Moreover, for the case of an arbitrary \(\gamma \in L_+^{\infty}([a,b])\), it is proved that some information on \(p(x)\) and \(\gamma(x)\) can be recovered from the Dirichlet-to-Neumann map. The proof technique is based on the reduction of the inverse problem to determining a function from its \(L^p\)-norms and on using the properties of a moment problem.