Simple waveguide junctions whose straightforward modelling leads to unsolvable boundary value problems

Engineers and physicists are used to considering simplified models of the real physical phenomena. These models do not catch all features of the physical phenomena of interest but usually only those which are believed to be significant for the purpose of the model. Other features, which could be considered in more complicated models, are usually dropped. In these simplifying processes some features of the physical models of interest can be lost, but it is usually assumed that the resulting simplified models are still well posed, that is with a unique solution depending continuously on the data. In timeharmonic electromagnetics there is only one well known exception to this situation, which occurs when the forcing terms excite resonant modes of ideal cavity resonators. Quite surprisingly, it has been recently shown [1], [2] that for infinite many time harmonic electromagnetic boundary value problems any solution does not depend continuously on the source terms. In this contribution another more annoying aspect is pointed out. In particular, it is shown that the same models for which the continuous dependence is lost do not admit any solution for a very wide set of source terms. This fact is shown to be independent of any excitation of modes in ideal cavity resonators. Moreover, it is surprising that such a feature can arise in almost trivial waveguide discontinuity problems, which usually allow the calculation of the solution by the well known procedure based on modal expansions. The above considerations point out the importance of the a priori evaluation of the well posedness of the models of interest and give rise to additional doubts about the meaning of some results obtained by numerical simulators for problems involving one of the dielectric configurations defining ill posed problems.