The Lewy equation is an example of an inhomogenous linear partial differential equation that has no solutions. Note that we’re not imposing any boundary-value or initial-value conditions on the equation; the equation simply has no solutions. The proof that it has no solutions is a surprisingly simple application of complex analysis. (Also available in postscript.)

The paper Fifty years of local solvability surveys the development of the theory (known as local solvability) in the wake of Lewy’s discovery. Numerical linear algebra and solvability of partial differential equations describes an analogy between local solvability and numerically computing matrix eigenvalues.