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The Monge-Ampere system: convex integration in arbitrary dimension and codimension

Marta Lewicka (U of Pittsburgh)
Thursday, November 9, 2023 - 2:30pm to 3:30pm
PDL C-38
The Monge-Ampere equation \$ \det\nabla^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 on \(\omega\), 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: curl^2 (\nabla v\otimes \nabla v) = -2f 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<1/7, no matter the sign of the right hand side function f [Lewicka-Pakzad 2017].

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^2 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 [Lewicka 2022].

Our main result is a proof of flexibility of the Monge-Ampere system at C^{1,\alpha} for any \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 \alpha\to 1 as k\to\infty.
For d=2, the flexibility exponent may be even improved to \alpha<1/(1+4/k), using the conformal invariance of 2d metrics to the flat metric [Lewicka 2023]. We will also discuss other possible improvements when d>2.
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