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PROJECT SUMMARY
The main themes of this project study the Loovariational problems and associated Euler
Lagrange equations, called the Aronsson equation. There are many problems in which the
appropriate cost functional must be the supermum (or infimum) of integrand functions. To
attack these problems, we plan to systematically study the variation problems in Loo and
its analysis by PDE methods in the following aspects: (1) complete characterization of absolute
minimizers of Loofunctionals by the Aronsson equations, i.e. finding generic sufficient
conditions on the Hamiltonian functions H such that absolute minimizers of H and viscosity
solutions to the Aronsson equation are equivalent, (2) Uniqueness problems for viscosity
solutions to the Aronsson equation and absolute minimizers, (3) Regularity issues of viscosity
solutions to the Aronsson equation in dimensions two or higher, (4) applications of
LCXJvariational problems to the study of weak KAM theory in the Hamiltonian dynamics,
and (5) LCXJvariational problems with constraints such as the principal eigenvalue problem
for the infinity Laplace operator.
The proposed problems belong to the field of nonlinear PDEs which provide basic laws
governing most natural phenomena and play crucial roles in studying problems arising from
calculus of variations, analysis, probality, and applied sciences including optimal control and
imagine processing. Variational problems in L= are very important from point of views of
both practical issues and mathematical theories. For example, find a chemotherapy regimen
for treatment of cancer seeking to minimize the maximum tumor load, recover partially
damaged or lost imagines by optimal extensions in appropriate sense of observed boundary
imagines, and seek strategies to maximize wining rate in a two person "tugofwar" games.
etc. On the other hand, LOGvariational problems usually yield extremely degenerate elliptic
PDEs, such as ooLaplace equation or the Aronsson equation, in which the existing theory
of partial differential equations usually reaches its limit and new ideas and techniques must
be developed in order to better understand the problems. Moreover, better understanding of
LCXJvariational problems will be important to understand the Hamiltonian dynamics in which
the Hamiltonian functions are farfromintegrable.
Over the past few years, the PIs have been actively involved and made several interesting
contributions to these problems, and we would like to continrme our studies in this direction
within the proposed project.
1
TPI6519672
Status  Finished 

Effective start/end date  7/1/06 → 6/30/11 
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Projects
 1 Finished

Collaborative Research: Linfinity variational problems and the Aronsson equation
Wang, C.
7/1/06 → 6/30/11
Project: Research project