Linear Elliptic Problems with non H^{-1} data and pathological solutions

In the theory of linear elliptic problems with data not belonging to H−1 two cases can be distinguished. When the right hand side in the equation is a summa- ble function we point out that the a priori estimates can be attained very quickly by symmetrization methods. On the other side, when the datum includes a distributional term, different and subtler tools have to be used. We deal with an equation in the plane, whose right hand side is a functional on a space of Hölder continuous functions with a suitable exponent. We obtain a priori bounds via duality arguments; these, in addition, show Serrin pathological solution (see Serrin in Ann. Scuola Norm. Sup. Pisa 18(3), 385–387, 1964 ) in its true light.