## An inverse problem for a nonlinear elliptic differential equation.(English)Zbl 0647.35081

Let $$-\Delta u=\gamma (x,y)+f(u)$$ in $$\Omega\subset R^ 2$$, where $$\Omega$$ is a bounded domain with a smooth boundary $$\Gamma =\Gamma_ 1\cup \Gamma_ 2$$. Assume that $$\gamma(x,y)$$ is known. The functions $$f(u)$$ and $$u(x,y)$$ are to be recovered from the overposed boundary data $$\alpha(x,y)U_ N+\beta (x,y)u=g_ 1(x,y)$$ on $$\Gamma_ 1$$, $$u_ N=g_ 2(x,y)$$ and $$u=\theta (x,y)$$ on $$\Gamma_ 2$$, where $$U_ N$$ is the normal derivative of u on $$\Gamma$$. Under some assumptions of smoothness on $$g_ 1$$, $$g_ 2$$, $$\theta$$ monotonicity on $$\theta$$ and a number of other assumptions the authors prove existence and uniqueness of the solution $$<f,u>$$ to the above problem.
Reviewer: A.G.Ramm

### MSC:

 35R30 Inverse problems for PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35J25 Boundary value problems for second-order elliptic equations
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