Schedule for: 19w5130 - Convex Integration in PDEs, Geometry, and Variational Calculus

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the compressible Euler equations in multiple space dimensions. These equations describe the time evolution of a compressible inviscid fluid. We give an overview over some results showing existence of infinitely many solutions. All these results are based on the convex integration method, which was used by De Lellis and Szekelyhidi in the context of the incompressible Euler equations. After considering general initial data, we focus on a special type of two-dimensional initial data, namely those which are constant in each of the two half spaces (Riemann data).

We discuss a recent progress in an optimal Brownian stopping problem, called the optimal Skorokhod embedding problem, which is an active research area especially in relation to mathematical finance. Given two probability measures with appropriate order, the problem considers the stopping time under which the Brownian motion carries one probability measure to the other, while minimizing the transportation cost. We focus on the cost given by the distance between the initial and the final point. A strong duality result of this optimization problem is obtained, which enables us to prove that the optimal stopping time is given by the first hitting time to a barrier determined by the optimal dual solutions. The main part of this talk is based on joint work with Nassif Ghoussoub (UBC) and Aaron Palmer (UBC).

We study convex integration solutions in the context of the modelling of shape-memory alloys. In a first part we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. This is based on joint work with Angkana Rüland and Jamie Taylor.

We provide counterexamples to the chain rule for divergence-free vectorfields, based on a Gromov-type convex integration scheme. Joint work with G. Crippa, N. Gusev, and S. Spirito.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.