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Global uniqueness for a semilinear elliptic inverse problem. (English) Zbl 0817.35126
Consider the Dirichlet problem \[ -\Delta u+ a(x,u)=0 \quad \text{in } \Omega, \qquad u=g \quad \text{on } \partial \Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) \((n\geq 3)\) with smooth boundary and \(a(x,s)\) satisfies \(a, a_ s, a_{ss}\in L^ \infty (\Omega\times [-S, S])\) for any \(S<\infty\), \(0\leq a_ s\) and \(g\) is sufficiently smooth. If \(a_ 1\), \(a_ 2\) satisfy the above conditions for \(a\) and if \(a_ 1 (x,0)= a_ 2 (x,0)=0\) and if in addition the corresponding Dirichlet to Neumann maps coincide and \(a_{1,s}\) and \(a_{2,s}\) are essentially bounded on \(\Omega\times \mathbb{R}\), then the authors prove that \(a_ 1= a_ 2\) on \(\Omega\times \mathbb{R}\).

MSC:
35R30 Inverse problems for PDEs
35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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