Partial Differential Equations and Variational Problems

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The principal investigator would like to continue his work on partialdifferential equations and variational problems. The first theme ofthis project, and also of the research of the PI over the past severalyears, is to study special sets associated to solutions to differentialequations and variational problems. The problems that the PI wouldcontinue to work on include the geometric structure of level sets, inparticular the nodal sets and the singular sets. The study is partlymotivated by the desire to understand to what extent the solutions canbe described quantitatively by polynomials or by homogeneous solutions.These problems have a close connection with other fields in mathematics,including several complex variables and algebraic geometry. A newresearch trend is the study of the relation between the growth of nodalsets and the growth of solutions themselves. It is expected that theyare closely related to each other. The second theme of this project isto study the isometric embedding of two-dimensional metrics in the threedimensional Euclidean space, both locally and globally. In general, theisometric embedding is based on the complicated Nash-Moser iteration,which requires detailed discussions of the linearized equations. Moreover,the global isometric embedding is expected to meet some topologicalobstacles. Difficulties arise when the Gaussian curvature changes itssign. Singular sets may appear even if the embedding exists.The problems of the singular sets originate from the material scienceand the control theory. Singular sets, as the name suggests, are thosesets where singularities occur. Precise definitions vary according toproblems where they arise. In reality it is impossible to eliminatesingular sets, the so-called "bad sets". An example is provided bycracks in the building material. Hence one of the central tasks is toinvestigate the conditions under which the singular sets can becontrolled and hence can be made small. Another application involvesthe high-performance computing, in particular the image processing. Oneproblem is to recover the image from a distorted copy and the differenceis measured exactly by some singular sets. It is highly expected thatsuch sets should be small enough to be neglected. The problems statedin the project are simplified mathematical models. It is the hope bythe investigator that the discussion of these mathematical problemswould improve the methods to control the singular sets in variousapplications.