Summary: We study a linear elliptic partial differential equation of second order in a bounded domain \(\Omega\subset R^N\), with nonstandard boundary conditions on a part \(G\) of the boundary \(\partial\Omega\). Here, neither the solution nor its normal derivative are prescribed pointwise. Instead, the average of the solution over \(G\) is given and the normal derivative along \(G\) has to follow a prescribed shape function, apart from an additive (unknown) constant. We prove the well-posedness of the problem and provide a method for the recovery of the unknown boundary data.