On a generalized maximum principle for a transport-diffusion model with log⁡\log-modulated fractional dissipation

We consider a transport-diffusion equation of the form \partial_t \theta +v \cdot \nabla \theta + \nu \A \theta =0, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator \A represents log-modulated fractional dissipation: \A=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)} and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator \A in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $|\theta(t)|_\infty \le e^{Ct} |\theta_0|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $|\theta(t)|_p$ with $1\le p \le \infty$. At the cost of an exponential factor, this extends a recent result of Hmidi (2011) to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.Comment: 14 page

C2C^2 estimate for oblique derivative problem with mean Dini coefficients

We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a $C^1$ function whose first derivatives are Dini continuous. This improves a recent result in [6]. An extension to fully nonlinear elliptic equations is also presented.Comment: 21 pages, submitte

On the Wp2W^2_p Estimate for Oblique Derivative Problem in Lipschitz Domains

The aim of this paper is to establish $W^2_p$ estimate for non-divergence form second-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [14, 15], which work for $C^{1,\alpha}$ domains with $\alpha > 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.Comment: 23 page

Optimal estimates for the conductivity problem by Green's function method

We study a class of second-order elliptic equations of divergence form, with discontinuous coefficients and data, which models the conductivity problem in composite materials. We establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions. The novelty of these estimates is that they unify the known results in the literature and answer open problem (b) proposed by Li-Vogelius (2000) for the isotropic conductivity problem. We also obtain more interesting higher-order derivative estimates, which answers open problem (c) of Li-Vogelius (2000). It is worth pointing out that the equations under consideration in this paper are nonhomogeneous.Comment: 23 pages, submitte

On novel elastic structures inducing plasmonic resonances with finite frequencies and cloaking due to anomalous localized resonances

This paper is concerned with the theoretical study of plasmonic resonances for linear elasticity governed by the Lam\'e system in $\mathbb{R}^3$, and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce plasmonic resonances. It is shown that if either one of the two convexity conditions on the Lam\'e parameters is broken, then we can construct certain plasmon structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexity conditions is required. Indeed, the existing plasmonic structures are a particular case of the general structures constructed in our study. Furthermore, we consider the plasmonic resonances within the finite frequency regime, and rigorously verify the quasi-static approximation for diametrically small plasmonic inclusions. Finally, as an application of the newly found structures, we construct a plasmonic device of the core-shell-matrix form that can induce cloaking due to anomalous localized resonance in the quasi-static regime, which also includes the existing study as a special case.Comment: 26 pages, comments are welcom

On quasi-static cloaking due to anomalous localized resonance in R3\mathbb{R}^3

This work concerns the cloaking due to anomalous localized resonance (CALR) in the quasi-static regime. We extend the related two-dimensional studies in [2,10] to the three-dimensional setting. CALR is shown not to take place for the plasmonic configuration considered in [2,10] in the three-dimensional case. We give two different constructions which ensure the occurrence of CALR. There may be no core or an arbitrary shape core for the cloaking device. If there is a core, then the dielectric distribution inside it could be arbitrary

Global H˙1∩H˙−1\dot H^1 \cap \dot H^{-1} solutions to a logarithmically regularized 2D Euler equation

We construct global $\dot H^1\cap \dot H^{-1}$ solutions to a logarithmically modified 2D Euler vorticity equation. Our main tool is a new logarithm interpolation inequality which exploits the $L^{\infty-}$-conservation of the vorticity.Comment: 8 page

On Anomalous Localized Resonance for the Elastostatic System

We consider the anomalous localized resonance due to a plasmonic structure for the elastostatic system in R^2. The plasmonic structure takes a general core-shell-matrix form with the metamaterial located in the shell. If there is no core, we show that resonance occurs for a very broad class of sources. If the core is nonempty and of an arbitrary shape, we show that there exists a critical radius such that anomalous localized resonance (ALR) occurs. Our argument is based on a variational technique by making use of the primal and dual variational principles for the elastostatic system, along with the construction of suitable test functions.Comment: 21 pages, no figur

On anomalous localized resonance and plasmonic cloaking beyond the quasistatic limit

In this paper, we give the mathematical construction of novel core-shell plasmonic structures that can induce anomalous localized resonance and invisibility cloaking at certain finite frequencies beyond the quasistatic limit. The crucial ingredient in our study is that the plasmon constant and the loss parameter are constructed in a delicate way that are correlated and depend on the source and the size of the plasmonic structure. As a significant byproduct of this study, we also derive the complete spectrum of the Neumann-Poinc\'are operator associated to the Helmholtz equation with finite frequencies in the radial geometry. The spectral result is the first one in its type and is of significant mathematical interest for its own sake

On three-dimensional plasmon resonance in elastostatics

We consider the plasmon resonance for the elastostatic system in $\mathbb{R}^3$ associated with a very broad class of sources. The plasmonic device takes a general core-shell-matrix form with the metamaterial located in the shell. It is shown that the plasmonic device in the literature which induces resonance in $\mathbb{R}^2$ does not induce resonance in $\mathbb{R}^3$. We then construct two novel plasmonic devices with suitable plasmon constants, varying according to the source term or the loss parameter, which can induce resonances. If there is no core, we show that resonance always occurs. If there is a core of an arbitrary shape, we show that the resonance strongly depends on the location of the source. In fact, there exists a critical radius such that resonance occurs for sources lying within the critical radius, whereas resonance does not occur for source lying outside the critical radius. Our argument is based on the variational technique by making use of the primal and dual variational principles for the elastostatic system, along with the highly technical construction of the associated perfect plasmon elastic waves.Comment: 25 pages, comments welcome. arXiv admin note: text overlap with arXiv:1601.0774