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Nonlinear Analysis

Marta Lewicka -- The Monge-Ampere system and the isometric immersion system: convex integration in arbitrary dimension and codimension (unusual day/time)

Marta Lewicka (University of Pittsburg)

Location:  Hill 425
Date & time: Friday, 16 February 2024 at 2:00PM - 3:00PM

The Monge-Ampere equation det
abla^2 v =f posed on a d=2 dimensional domain omega and in which we are seeking a scalar (i.e. dimension k=1) field v, has a natural weak formulation that appears as the constraint condition in the Gamma-limit of the dimensionally reduced non-Euclidean elastic energies. This formulation reads: frac{1}{2} abla votimes  abla v) + sym abla w= - (curl curl)^{-1}f   and it allows, via the Nash-Kuiper scheme of convex integration, for constructing multiple solutions that are dense in C^0(omega), at the regularity C^{1,alpha} for any alpha<1/3, no matter the sign of the right hand side function f. Does a similar result hold in higher dimensions d>2 and codimensions k>1? Indeed it does, but one has to replace the Monge-Ampere equation by the Monge-Ampere system, by altering curl curl to the corresponding operator that arises from the prescribed Riemann curvature problem, similarly to how the prescribed Gaussian curvature problem leads to the Monge-Ampere equation in 2d.

Our main result is a proof of flexibility of the Monge-Ampere system at C^{1,alpha} for  alpha<1/(1+d(d+1)/k). This finding extends our previous result where d=2, k=1, and stays in agreement with the known flexibility thresholds for the isometric immersion problem: the Conti-Delellis-Szekelyhidi result alpha<1/(1+d(d+1)) when k=1, as well as the Kallen result where alphato 1 as ktoinfty. For d=2, the flexibility exponent may be even improved to alpha<1/(1+2/k), using the conformal invariance of 2d metrics to the flat metric. 

We will also discuss other possible improvements and parallel results and techniques valid for the isometric immersion system.

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