PDE Geometric Analysis seminar: Difference between revisions
No edit summary |
No edit summary |
||
| Line 29: | Line 29: | ||
|Russell Schwab (Carnegie Mellon) | |Russell Schwab (Carnegie Mellon) | ||
|[[#Russell Schwab (Carnegie Mellon)| | |[[#Russell Schwab (Carnegie Mellon)| | ||
'' | ''On Aleksandrov-Bakelman-Pucci type estimates for | ||
integro-differential equations (comparison theorems with measurable | |||
ingredients)'']] | |||
|Armstrong | |Armstrong | ||
|- | |- | ||
| Line 69: | Line 71: | ||
===Russell Schwab (Carnegie Mellon)=== | ===Russell Schwab (Carnegie Mellon)=== | ||
''On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)'' | |||
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP]) | |||
===Valentin Ovsienko ()=== | ===Valentin Ovsienko ()=== | ||
Revision as of 16:09, 3 October 2011
PDE and Geometric Analysis Seminar
The seminar will be held in room B115 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
Seminar Schedule Fall 2011
| date | speaker | title | host(s) |
|---|---|---|---|
| Oct 3 | Takis Souganidis (Chicago) |
Stochastic homogenization of the G-equation |
Armstrong |
| Oct 10 | Scott Armstrong (UW-Madison) |
Partial regularity for fully nonlinear elliptic equations |
Local speaker |
| Oct 17 | Russell Schwab (Carnegie Mellon) |
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients) |
Armstrong |
| Oct 24 | Valentin Ovsienko () |
TBA |
Marí Beffa |
| Oct 31 | Adrian Tudorascu (West Virginia University) |
TBA |
Feldman |
| Nov 7 | James Nolen (Duke) |
TBA |
Armstrong |
| Dec 5 | Charles Smart (MIT) |
TBA |
Armstrong |
Abstracts
Takis Souganidis (Chicago)
Stochastic homogenization of the G-equation
The G-equation is a Hamilton-Jacobi equation, of level-set-type, which is used as a model in turbulent combustion. In the lecture I will present recent joint work with Pierre Cardaliaguet about the homogenization of the G-equation set in random media, when the problem is not coercive and, hence, falls outside the scope of the theory of stochastic homogenization.
Scott Armstrong (UW-Madison)
Partial regularity for fully nonlinear elliptic equations
I will present some regularity results for (nonconvex) fully nonlinear equations. Such equations do not possess smooth solutions, but in joint work with Silvestre and Smart we show that the Hausdorff dimension of the singular set is less than the ambient dimension. Using an argument with a similar flavor, we prove (jointly with Silvestre) a unique continuation result for such equations.
Russell Schwab (Carnegie Mellon)
On Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations (comparison theorems with measurable ingredients)
Despite much recent (and not so recent) attention to solutions of integro-differential equations of elliptic type, it is surprising that a fundamental result such as a comparison theorem which can deal with only measure theoretic norms of the right hand side of the equation (L-n and L-infinity) has gone unexplored. For the case of second order equations this result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back to circa 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f, the supremum of u is controlled by the L-n norm of f (n being the underlying dimension of the domain). We discuss extensions of this estimate to fully nonlinear integro-differential equations and present a recent result in this direction. (Joint with Nestor Guillen, available at arXiv:1101.0279v3 [math.AP])
Valentin Ovsienko ()
To be posted.
Adrian Tudorascu (West Virginia University)
To be posted.
James Nolen (Duke)
To be posted.
Charles Smart (MIT)
To be posted.