Date:
Wed, 28/12/202214:00-15:00
TITLE: The Monge-Ampere system: convex integration in arbitrary dimension and codimension
ABSTRACT: In this talk, we will be concerned with the flexibility of the weak solutions to the Monge-Ampere system via convex integration. This system is a natural extension of the Monge-Ampere equation \det
abla^2 v=f, in relation to: (i) the problem of isometric immersion and (ii) the context of nonlinear elasticity.
The main technical ingredient consists of the ``stage'' construction, in which we achieve the Holder regularity \mathcal{C}^{1,\alpha} of the approximating fields, for all \alpha<\frac{1}{1+ d(d+1)/k} where d is an arbitrary dimension and k\geq 1 is an arbitrary codimension. When d=2 and k=1, we recover the previous result by Lewicka-Pakzad for the Monge-Ampere equation. Our construction can be translated to the isometric immersion problem, where for k=1 we recover the result by Conti-Delellis-Szekelyhidi, and for large k we quantify a version of the result by Kallen.
ABSTRACT: In this talk, we will be concerned with the flexibility of the weak solutions to the Monge-Ampere system via convex integration. This system is a natural extension of the Monge-Ampere equation \det
abla^2 v=f, in relation to: (i) the problem of isometric immersion and (ii) the context of nonlinear elasticity.
The main technical ingredient consists of the ``stage'' construction, in which we achieve the Holder regularity \mathcal{C}^{1,\alpha} of the approximating fields, for all \alpha<\frac{1}{1+ d(d+1)/k} where d is an arbitrary dimension and k\geq 1 is an arbitrary codimension. When d=2 and k=1, we recover the previous result by Lewicka-Pakzad for the Monge-Ampere equation. Our construction can be translated to the isometric immersion problem, where for k=1 we recover the result by Conti-Delellis-Szekelyhidi, and for large k we quantify a version of the result by Kallen.