Connor Mooney
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View article: Semi-convex viscosity solutions of the special Lagrangian equation
Semi-convex viscosity solutions of the special Lagrangian equation Open
We prove smoothness and interior derivative estimates for viscosity solutions to the special Lagrangian equation with almost negative phases and small enough semi-convexity. We show by example that the range of phases we consider and the s…
View article: A half-space Bernstein theorem for anisotropic minimal graphs
A half-space Bernstein theorem for anisotropic minimal graphs Open
We prove that an anisotropic minimal graph over a half-space with flat boundary must itself be flat. This generalizes a result of Edelen–Wang to the anisotropic case. The proof uses only the maximum principle and ideas from fully nonlinear…
View article: Remarks on the quadratic Hessian equation
Remarks on the quadratic Hessian equation Open
We prove that viscosity solutions to the quadratic Hessian equation $$σ_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori interio…
View article: Solutions to the minimal surface system with large singular sets
Solutions to the minimal surface system with large singular sets Open
Lawson and Osserman proved that the Dirichlet problem for the minimal surface system is not always solvable in the class of Lipschitz maps. However, it is known that minimizing sequences (for area) of Lipschitz graphs converge to objects c…
View article: Bernstein theorems for nonlinear geometric PDEs
Bernstein theorems for nonlinear geometric PDEs Open
In this expository article we revisit the Bernstein problem for several geometric PDEs including the minimal surface, Monge-Ampère, and special Lagrangian equations. We also discuss the minimal surface system where appropriate. The article…
View article: Counterexamples to maximal regularity for operators in divergence form
Counterexamples to maximal regularity for operators in divergence form Open
In this paper, we present counterexamples to maximal $$L^p$$ -regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory tha…
View article: Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates
Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates Open
We establish global well-posedness and convergence of the score-based generative models (SGM) under minimal general assumptions of initial data for score estimation. For the smooth case, we start from a Lipschitz bound of the score functio…
View article: Book Review: Regularity Theory for Elliptic PDE
Book Review: Regularity Theory for Elliptic PDE Open
View article: Counterexamples to maximal regularity for operators in divergence form
Counterexamples to maximal regularity for operators in divergence form Open
In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such …
View article: Bernstein theorems for nonlinear geometric PDEs
Bernstein theorems for nonlinear geometric PDEs Open
View article: A half-space Bernstein theorem for anisotropic minimal graphs
A half-space Bernstein theorem for anisotropic minimal graphs Open
We prove that an anisotropic minimal graph over a half-space with flat boundary must itself be flat. This generalizes a result of Edelen-Wang to the anisotropic case. The proof uses only the maximum principle and ideas from fully nonlinear…
View article: The anisotropic Bernstein problem
The anisotropic Bernstein problem Open
We construct nonlinear entire anisotropic minimal graphs over $\mathbb{R}^{4}$ , completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes …
View article: On the Lawson-Osserman conjecture
On the Lawson-Osserman conjecture Open
We prove that if $u : B_1 \subset \mathbb{R}^2 \rightarrow \mathbb{R}^n$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. This solves a conjecture of Lawson and Osserman from 1977 i…
View article: A VERITAS/Breakthrough Listen Search for Optical Technosignatures
A VERITAS/Breakthrough Listen Search for Optical Technosignatures Open
The Breakthrough Listen Initiative is conducting a program using multiple telescopes around the world to search for "technosignatures": artificial transmitters of extraterrestrial origin from beyond our solar system. The VERITAS Collaborat…
View article: Sobolev regularity for optimal transport maps of non-convex planar domains
Sobolev regularity for optimal transport maps of non-convex planar domains Open
We prove a sharp global $W^{2,\,p}$ estimate for potentials of optimal transport maps that take a certain class of non-convex planar domains to convex ones.
View article: VERITAS discovery of very high energy gamma-ray emission from S3 1227+25 and multiwavelength observations
VERITAS discovery of very high energy gamma-ray emission from S3 1227+25 and multiwavelength observations Open
We report the detection of very high energy gamma-ray emission from the blazar S3 1227+25 (VER J1230+253) with the Very Energetic Radiation Imaging Telescope Array System (VERITAS). VERITAS observations of the source were triggered by the …
View article: Bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows
Bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows Open
The level-set curvature G-equation, a well-known model in turbulent combustion, has the following form $G_t + \left(1-d\, \mathrm{dvi}\left({\frac{DG}{|DG|}}\right)\right)_+|DG|+V(X)\cdot DG=0.$ Here the cutoff correction $()_+$ is imposed…
View article: Non $C^1$ solutions to the special Lagrangian equation
Non $C^1$ solutions to the special Lagrangian equation Open
We construct viscosity solutions to the special Lagrangian equation that are Lipschitz but not $C^1$.
View article: Homogeneous functions with nowhere-vanishing Hessian determinant
Homogeneous functions with nowhere-vanishing Hessian determinant Open
We prove that functions that are homogeneous of degree \alpha \in (0, 1) on \mathbb{R}^n and have nowhere-vanishing Hessian determinant cannot change sign.
View article: Singular structures in solutions to the Monge-Ampère equation with point masses
Singular structures in solutions to the Monge-Ampère equation with point masses Open
We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures …
View article: The anisotropic Bernstein problem
The anisotropic Bernstein problem Open
We construct nonlinear entire anisotropic minimal graphs over $\mathbb{R}^4$, completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to high…
View article: Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase
Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase Open
In this paper, we prove interior gradient estimates for the Lagrangian mean curvature equation, if the Lagrangian phase is critical and supercritical and $C^{2}$. Combined with the a priori interior Hessian estimates proved in [Bha21, Bha2…
View article: Singular structures in solutions to the Monge-Ampère equation with point masses
Singular structures in solutions to the Monge-Ampère equation with point masses Open
We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures …
View article: Homogeneous functions with nowhere vanishing Hessian determinant
Homogeneous functions with nowhere vanishing Hessian determinant Open
We prove that functions that are homogeneous of degree $\alpha \in (0,\,1)$ on $\mathbb{R}^n$ and have nowhere vanishing Hessian determinant cannot change sign.
View article: Entire solutions to equations of minimal surface type in six dimensions
Entire solutions to equations of minimal surface type in six dimensions Open
We construct nonlinear entire solutions in \mathbb{R}^6 to equations of minimal surface type that correspond to parametric elliptic functionals.
View article: Hilbert's $19^{\text{th}}$ problem revisited
Hilbert's $19^{\text{th}}$ problem revisited Open
In this survey article we revisit Hilbert's $19^{\text{th}}$ problem concerning the regularity of minimizers of variational integrals. We first discuss the classical theory (that is, the statement and resolution of Hilbert's problem in all…
View article: Hilbert's 19th problem revisited
Hilbert's 19th problem revisited Open
In these notes we revisit Hilbert's 19th problem concerning the regularity of minimizers of variational integrals. We first discuss the classical theory (that is, the statement and resolution of Hilbert's problem in all dimensions). We the…
View article: A proof by foliation that Lawson's cones are $A_Φ$-minimizing
A proof by foliation that Lawson's cones are $A_Φ$-minimizing Open
We give a proof by foliation that the cones over $\mathbb{S}^k \times \mathbb{S}^l$ minimize parametric elliptic functionals for each $k,\,l \geq 1$. We also analyze the behavior at infinity of the leaves in the foliations. This analysis m…
View article: Solutions to the Monge–Ampère Equation with Polyhedral and Y-Shaped Singularities
Solutions to the Monge–Ampère Equation with Polyhedral and Y-Shaped Singularities Open
View article: Strict $2$-convexity of convex solutions to the quadratic Hessian equation
Strict $2$-convexity of convex solutions to the quadratic Hessian equation Open
We prove that convex viscosity solutions to the quadratic Hessian inequality $σ_2(D^2u) \geq 1$ are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $…