Abstract. An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.

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The solution of a quasilinear elliptic state equation depends on the coefficient function belonging to an admissible set. The solution is evaluated by a cost functional the value of which is to be maximized over the admissible set, i.e., the reliable (safe) solution is searched for. Due to the nature of the equation, the Kirchhoff transformation can be applied to obtain both the existence of the true state solution and a cost sensitivity formula. In many cases, the latter makes it possible to determine the reliable solution immediately. The problem is approximated by means of the finite element method, and some convergence results are proven. Numerical examples illustrate the theory which can be directly generalized to spatial problems.

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