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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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##### References:
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