arXiv subjects
Publications and source records attributed to Nam Q. Le.
We show that for an $L^2$ drift $b$ in two dimensions, if the Hardy norm of $\text{div }b$ is small, then the weak solutions to $Δu+b\cdot\nabla u=0$ have the same optimal Hölder regularity as in the case of divergence-free drift, that is, $u\in C^α_{\text{loc}}$ for all $α\in (0, 1)$.
We present the results of classical molecular dynamics simulations to assess the relative contributions to interfacial thermal conductance from inelastic phonon processes at the interface and in the adjacent bulk materials. The simulated system is the prototypical interface between argon and "heavy argon" crystals, which enables comparison with many past computational studies. We run simulations interchanging the Lennard-Jones potential with its harmonic approximation to test the effect of anharmonicity on conductance. The results confirm that the presence of anharmonicity is correlated with increasing thermal conductance with temperature, which supports conclusions from prior experimental and theoretical work. However, in the model Ar/heavy-Ar system, anharmonic effects at the interface itself contribute a surprisingly small part of the total thermal conductance. The larger fraction of the thermal conductance at high temperatures arises from anharmonic effects away from the interface. These observations are supported by comparisons of the spectral energy density, which suggest that bulk anharmonic processes increase interfacial conductance by thermalizing energy from modes with low transmission to modes with high transmission.
We obtain $C^{2,β}$ estimates up to the boundary for solutions to degenerate Monge-Ampère equations of the type $$ \det D^2 u = f~~\text{in}~Ω, \quad \quad ~f\sim \text{dist}^α(\cdot, \partialΩ)~\text{near}~\partialΩ,~α>0. $$ As a consequence we obtain global $C^\infty$ estimates up to the boundary for the eigenfunctions of the Monge-Ampère operator $(\det D^2 u)^{1/n}$ on smooth, bounded, uniformly convex domains in $R^n$.
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.
In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Ampère operator.
In this paper, we investigate regularity for solutions to the linearized Monge-Ampère equations when the nonhomogeneous term has low integrability. We establish global $W^{1,p}$ estimates for all $p<\frac{nq}{n-q}$ for solutions to the equations with right hand side in $L^q$ where $n/2<q\leq n$. These estimates hold under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{1,p}$ estimates of N. Winter for fully nonlinear, uniformly elliptic equations.
In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Ampère measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman-Stampacchia-Weinberger for uniformly elliptic operators in divergence form. We also obtain the $L^{p}$ integrability for the gradient of the Green's function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge-Ampère equation.
We argue that the relative thermal conductance between interfaces with different morphologies is controlled by crystal structure through $M_{min}/M_c > 1$, the ratio between the {\it minimum mode} count on either side $M_{min}$, and the {\it conserving modes} $M_c$ that preserve phonon momentum transverse to the interface. Junctions with an added homogenous layer, "uniform", and "abrupt" junctions are limited to $M_c$ while junctions with interfacial disorder, "mixed", exploit the expansion of mode spectrum to $M_{min}$. In our studies with cubic crystals, the largest enhancement of conductance from "abrupt" to "mixed" interfaces seems to be correlated with the emergence of voids in the conserving modes, where $M_c = 0$. Such voids typically arise when the interlayer coupling is weakly dispersive, making the bands shift rigidly with momentum. Interfacial mixing also increases alloy scattering, which reduces conductance in opposition with the mode spectrum expansion. Thus the conductance across a "mixed' junction does not always increase relative to that at a "uniform" interface.
The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem in $W^{4,p}$ under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain $W^{4,p}$ solution to the second boundary value problem when the affine mean curvature belongs to $L^p$ with $p$ greater than the dimension. Our self-contained analysis also covers the case of Abreu's equation.
In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of $p$-Laplace Allen-Cahn energies and the second inner variation of their $Γ$-limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper \cite{Le} in the case $p=2$. The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen-Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg-Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg \cite{KS} by giving a relation between the limit of second variations of the Allen-Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincaré inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun \cite{SZ2}.
In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,γ}$ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded right hand side and $C^{1, 1}$ boundary data.
In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Ampère equations.
In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Ampère equations: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type $p-p$ estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge-Ampère setting forms a space of homogeneous type.
In this paper we prove the global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature equation where the affine mean curvature is only assumed to be in $L^{p}$. Our result extends previous result by Trudinger and Wang in the case of globally bounded affine mean curvature and also covers Abreu's equation.
In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow. We also establish a gap theorem for self-shrinkers.
In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the single-multiplicity property of the limiting energy, that the Morse indices of critical points of the Allen-Cahn functional are bounded from below by the Morse index of the limiting minimal hypersurface.
In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, using Perelman's $\mathcal{W}$-functional, we show that suitable blow-ups of our evolving metrics converge in the pointed Cheeger-Gromov sense to a Gaussian shrinker. If the flow has uniformly bounded scalar curvature and develops Type II singularities at $T$, we show that suitable scalings of the potential functions in Perelman's entropy functional converge to a positive constant on a complete, Ricci flat manifold. We also show that if the scalar curvature is uniformly bounded along the flow in certain integral sense then the flow either develops a type II singularity at $T$ or it can be smoothly extended past time $T$.
In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions $N\leq 3$. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are $(i)$ well-preparedness of the initial data and $(ii)$ smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.