The problem of finding an extension of a function given on the boundary of an open set into the interior of the set in a way which achieves the least possible value of some functional defined on such extensions is pervasive in mathematics and its applications. If the functional is the Dirichlet integral, one is led to the Dirichlet problem for the Laplace equation. If the functional is, instead, the supremum norm of the length of the gradient, uniqueness of minimal extensions is lost unless the problem is reinterpreted - the resulting notion is here called "absolutely minimizing". Owing to work of Aronsson in the 60's and Jensen in the 90's, one knows that, upon this proper reinterpretation, there is an associated Dirichlet problem which has a unique solution. However, the partial differential equation involved is now a highly degenerate quasilinear elliptic equation, the "infinity-Laplace" equation. Moreover, the infinity-Laplace operator must be understood in the generalized sense of viscosity solutions. In the original works, the "length of the gradient" was the Euclidean length. For some other notions of length, such as the natural maximum norm, there is only an existence theory and uniqueness is unsettled. This is one simple example of a basic question in this arena as yet unresolved. Others include, even in the case of the Euclidean length, uniqueness in unbounded domains, the famous problem of differentiability of infinity harmonic functions (recently settled in two space dimensions by Savin), and the possibility of other more refined results - perhaps generic - on the structure of solutions. These issues will be studied in this project.
The infinity-Laplace equation is an archetypal object in the subject called "calculus of variations in L-infinity" by its founders. Roughly speaking, it corresponds to designing for the worst case, clearly an important issue with potential relevance to many fields. One well established area of application of absolutely minimizing functions is in image processing. While this project is primarily mathematical, the questions asked are basic and aimed at archetypes (as opposed to technically complex generalizations). Success will illuminate the general theory and thereby its current and future applications.
