Allard has worked with a group under the direction of Professor Kenneth Cohen of the UNC Department of Opthalmology with the goal of measuring the effectiveness as well as suggesting modifications of the design of devices used for corneal topography. He has found the differential equations which govern this type of measurement. Graduate student James Rolf programmed the approximate solution of these equations as well as simulation tools for testing the validity and robustness of the approximate solution procedure. Allard studied the minimal surface equation for many years. This equation is satisfied by surfaces acted upon only by surface tension forces.
Soap films are examples of such surfaces.
Allard has proved a number of existence and regularity theorems for such surfaces. He has done a good deal of work on the elliptic problems which are a natural generalization of the minimal surface problems just mentioned. Allard has devoted a great deal of time to the study of certain problems in differential geometry, particularly those which involve partial differential equations.