arXiv subjects

Sagun Chanillo

Publications and source records attributed to Sagun Chanillo.

At least 19 recordsLinked to original sources

arXiv2025

The Singular CR Yamabe Problem and Hausdorff Dimension

We consider a compact pseudo-hermitian manifold (M,\theta, J), that is a manifold equipped with a contact form \theta and CR structure J. We consider a conformal deformation of the contact form to obtain a complete, singular contact form and a corresponding Yamabe problem. We estimate then the Hausdorff dimension of the singular set. The conformal geometry analog of this result is due to R. Schoen and S. -T. Yau. Results of this type have their origin in work by Huber for Riemann surfaces. In the second part of our paper we investigate the CR developing map for three dimensional CR manifolds. We establish the injectivity of the developing map essentially using the same strategy as Schoen and Yau for the conformal case which is based on the positive mass theorem. Higher dimensional analogs of Huber's theorem in the conformal case for Q curvature are due to Alice Chang, Jie Qing and P. Yang.

math.DG↗

arXiv2024

Small Divisor problems and $A_p$ weights with an application

We establish a link between Muckenhoupt $A_p$ weights and a means to address small divisor problems. We use this link to obtain a quantitative version of the Ehrenpreis-Malgrange theorem of local solvability for constant coefficient PDE. We give an example as to how our theorem applies. In our quantitative version of the Ehrenpreis-Malgrange theorem, the loss of derivatives in the solvability estimate is measured in the scale of Sobolev spaces via the use of Muckenhoupt A_p weights. A part of our results are global in nature.

math.AP↗

arXiv2022

Resonances and Eigenvalues for the Constant Mean Curvature Equation

In this paper we study resonances and eigenvalues for the nonlinear constant mean curvature eqn. linearized around the bubbles found by Brezis-Coron. This nonlinear eqn. is also called a H-system eqn. For degree one bubbles(the degree relates to a certain winding number) we only find resonances. For higher degree we prove eigenvalues do occur. Our goal is to eventually obtain dispersive estimates for the wave eqn. associated to the H-systems eqn. in its linear and non-linear form, a study of which was initiated by Chanillo-Yung.

math.AP↗

arXiv2020

Sharp bounds on the Nusselt number in Rayleigh-B\'enard convection and a bilinear estimate via Carleson measures

We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit. Due to a maximum principle property for the temperature exploited by Constantin-Doering and Otto-Seis, this amounts to proving a-priori bounds for horizontally-periodic solutions of a fourth-order equation in a strip of large width. Such bounds are obtained here using Fourier analysis, integral representations, and a bilinear estimate due to Coifman and Meyer which uses the Carleson measure characterization of BMO functions by Fefferman.

math.AP↗

arXiv2019

Estimates of the amplitude of holonomies by the curvature of a connection on a bundle

We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.

math.DG↗

arXiv2018

Almost sure boundedness of iterates for derivative nonlinear wave equations

We study nonlinear wave equations on $\mathbb R^{2+1}$ with quadratic derivative nonlinearities, which include in particular nonlinearities exhibiting a null form structure, with random initial data in $H_x^1\times L^2_x$. In contrast to the counterexamples of Zhou \cite{Zhou} and Foschi-Klainerman \cite{FK}, we obtain a uniform time interval $I$ on which the Picard iterates of all orders are almost surely bounded in $C_t(I ; \dot H_x^1)$.

math.AP↗

arXiv2017

Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side data is manufactured from divergence free vector fields which are in L^1. Such a result was proved earlier by Jean Bourgain and Haim Brezis on Euclidean space.

math.AP↗

arXiv2017

Norm Constants in cases of the Caffarelli-Kohn-Nirenberg inequality

By methods based on elementary Linear Algebra we obtain sharp constants in cases of the Caffarelli-Kohn-Nirenberg inequality via quasi-conformal changes of variables. Some of our results were obtained earlier by Lam and Lu. Our proofs are radical simplifications of earlier proofs. In the case $α>0$ we establish that we have symmetry breaking and optimizers to the CKN inequalities do not exist. This case was not treated by Lam and Lu. Our results thus are in the full parameter range of α.

math.AP↗

arXiv2016

Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem

We offer a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on $\mathbb{R}^n$. We use this idea to extend the result to real hyperbolic spaces $\mathbb{H}^n$.

math.CA↗

arXiv2015

The CR Paneitz Operator and the Stability of CR Pluriharmonic Functions

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in $\mathbb{C}^2$ are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.

math.DG↗

arXiv2015

Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities.

math.AP↗

arXiv2015

A Remark on the Kernel of the CR Paneitz operator

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kernel of the CR Paneitz operator contains a supplementary space to the CR pluriharmonic functions.

math.DG↗

arXiv2015

Absence of Self-Similar Blow-up and Local Well-posedness for the Constant Mean-Curvature Wave Equation

In this note, we consider the constant-mean-curvature wave equation in $(1+2)$-dimensions. We show that it does not admit any self-similar blow-up. We also remark that the equation is locally well-posed for initial data in $\dot{H}^{3/2}$.

math.AP↗

arXiv2012

Conformal Geometry and The Composite Membrane Problem

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and various qualitative properties of the metric are easily obtained from the solution of the Composite Membrane problem. The problem of minimizing eigenvalues over conformal classes has a higher dimensional analog for the critical GJMS operator and leads to new classes and questions for higher order unstable free boundary problems. In particular in dimension 4 we are lead to free boundary problems involving the Paneitz operator and in odd dimensions to fractional free boundary problems of unstable type.

math.AP↗

arXiv2012

Embedded Three Dimensional CR Manifolds and the Non-Negativity of Paneitz Operators

Let $Ω$ be a bounded strictly pseudoconvex domain in $C^2$ with a smooth, connected and compact boundary M and having a CR structure $J_0$ induced from $C^2$. Assume this CR structure has zero Webster torsion. Then if we deform the CR structure through real-analytic dependence on the deformation parameter and such that each deformed structure along the deformation path is smooth and embeddable in $C^2$, we show that for small deformations of the CR structure $J$ from $J_0$, the associated CR Paneitz operator for $J$ is non-negative. We also show that the Webster curvature for any ellipsoid in $C^2$ is positive. The results in this paper complement and provide partial converses to our earlier paper, (to appear Duke Math. J.) arxiv: 1007.5020.

math.CV↗

arXiv2011

A Remark on the Geometry of Uniformly Rotating Stars

In this paper we classify the free boundary associated to equilibrium configurations of compressible, self-gravitating fluid masses, rotating with constant angular velocity. The equilibrium configurations are all critical points of an associated functional and not necessarily minimizers. Our methods also apply to alternative models in the literature where the angular momentum per unit mass is prescribed. The typical physical model our results apply to is that of uniformly rotating white dwarf stars.

math.AP↗

arXiv2011

Wave equations associated to Liouville systems and constant mean curvature equations

We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial data.

math.AP↗

arXiv2010

An improved Strichartz estimate for systems with divergence free data

Using the div-curl inequalities of Bourgain-Brezis [?MR2057026] and van Schaftingen [?MR2078071], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow $L^1_x$ norms of the inhomogeneity in the right hand side of the estimate.

math.AP↗