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

Pilant, Michael; Rundell, William

SIAM J. Math. Anal. 18, 1801-1809 (1987).

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

Keywords:

recovered from the overposed boundary data; smoothness; existence; uniqueness