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Connor Mooney

Publications and source records attributed to Connor Mooney.

44 records · Page 3Linked to original sources
arXiv2016

The sharp quantitative Euclidean concentration inequality

The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean isoperimetric inequality from \cite{fuscomaggipratelli}.

math.AP↗
arXiv2016

Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

We demonstrate that $C^{2,α}$ estimates for the Monge-Ampère equation depend in a highly nonlinear way both on the $C^α$ norm of the right-hand side and $1/α$. First, we show that if a solution is strictly convex, then the $C^{2,α}$ norm of the solution depends polynomially on the $C^α$ norm of the right-hand side. Second, we show that the $C^{2,α}$ norm of the solution is controlled by $\exp((C/α)\log(1/α))$ as $α\to 0$. Finally, we construct a family of solutions in two dimensions to show the sharpness of our results.

math.AP↗
arXiv2015

Some counterexamples to Sobolev regularity for degenerate Monge-Ampère equations

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial Ω} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.

math.AP↗
arXiv2013

Partial regularity for singular solutions to the Monge-Ampere equation

We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.

math.AP↗