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Non-linear problems involving integro-differential equations March 01, 2011 (10:00 AM PST - 12:00 PM PST)
Parent Program: --
Location: MSRI: Simons Auditorium
Description Location:  Simons Auditorium

Speakers: Erik Lindgren and Nestor Guillen

Title: Non-linear problems involving integro-differential equations  (Part I)

Elliptic or diffusion equations typically describe situations where a
field is or is trying to revert to some (usually infinitesimal )
average of itself. Thus, harmonic functions have the mean value
property and solutions to parabolic equations tend to equilibrium
configurations (e.g. harmonic functions). Elliptic
Integro-differential equations correspond to the same principle except
the averages considered are not just infinitesimal but instead take
into account all cales, a typical example being fractional powers of
the Laplacian. Many classical models (obstacle problems, phase
transitions, Hamilton-Jacobi equations, stochastic control problems)
have natural and well-known analogues in this setting, and their
analysis has presented mathematicians with significant new challenges.
In a series of talks we will present some of the mathematical and
physical problems that led to the study of such equations and explain
some recent results. In particular we will focus on the regularity
theory for non-linear integro-differential equations  and the
regularity of the free boundary in the obstacle problem for the
fractional Laplacian, open questions and possible research directions
will be presented.
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