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An inverse problem originating from magnetohydrodynamics. (English) Zbl 0732.76096
The authors obtain a rather interesting result for the inverse problem for \(\Delta u=-f(u)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). In particular, suppose \(\Omega\) is a bounded polygonal domain in \(R^ 2\) and that u and v are classical solutions of the Dirichlet problems for \(\Delta u=-f(u)\) and \(\Delta v=-g(v)\) where f and g are smooth and nonnegative. Under a mild condition on the boundary angles, the authors show that, if \(\partial u/\partial n\) and \(\partial v/\partial n\) agree on a portion of \(\partial \Omega\), then in fact, there must be a level of agreement between f and g in the sense that \(f^{(k)}(0)=g^{(k)}(0)\) for \(k=0,1,..\).

MSC:
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
35R30 Inverse problems for PDEs
35J60 Nonlinear elliptic equations
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