This chapter deals with some simple problems for P.D.E. by deriving a first order differential inequality for an appropriate (non-negative) measure of the solution, and by deducing therefrom an inequality for the measure itself, in terms of the data for the problem. For extensive surveys of the “volume integral method” - and of related matters in elasticity, together with many references - the reader is referred to the review article by Horgan and Knowles and to an update thereto by Horgan. The chapter shows how decay estimates of Phragmèn-Lindelöf type may be obtained for semi-infinite rectangular region. Roughly speaking, this means: a measure of the solution decays exponentially with distance as one moves towards infinity, if it is known that the measure does not increase asymptotically any faster than an exponential function of given exponent. Attention is confined to a three-dimensional region, and it is left to the reader to make the obvious adaptation to a two-dimensional context.