Related papers: The insulated conductivity problem with $p$-Laplac…
This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems. We consider a simple model of two-dimensional steady-state heat conduction described by elliptic partial differential equations…
The conductance through a semi-infinite one-dimensional wire, partly embedded in a superconducting bulk electrode, is studied. When the electron-electron interactions within the wire are strongly repulsive, the wire effectively decouples…
In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-\Delta)^{p+{\frac{\alpha}{2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{R}^n,\\…
We study optimal convergence rates in the periodic homogenization of linear elliptic equations of the form $-A(x/\varepsilon):D^2 u^{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We show that the optimal rate for…
We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently…
An ideal conductor electrode in contact with a semi-infinite two-dimensional two-component plasma in an external potential is considered. The model is mapped onto an integrable sine-Gordon theory with Dirichlet boundary conditions. The…
We consider the conductivity problem in the presence of adjacent circular inclusions having arbitrary constant conductivity. When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the…
We discuss upper and lower bounds on the electrical conductivity of finite temperature strongly coupled quantum field theories, holographically dual to probe brane models, within linear response. In a probe limit where disorder is…
We solve the Neumann problem in the half space $\mathbb{R}^{n+1}_+$, for higher order elliptic differential equations with variable self-adjoint $t$-independent coefficients, and with boundary data in the negative smoothness space $\dot…
We analyze an optimization problem of the conductivity in a composite material arising in a heat conduction energy storage problem. The model is described by the heat equation that specifies the heat exchange between two types of materials…
Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in…
In this article, the problems to be studied are the following \leqnomode \begin{equation*} \label{p} \left\{\begin{array}{ll} (-\Delta )_p^s u \pm \dfrac{|u|^{p-2}u}{|x|^{sp}} = \lambda f(x,u) & \quad \mbox{in }\ \Omega\\[0.3cm] u= 0 &…
This paper is concerned with the elliptic equation $-\Delta u=\frac{\lambda }{(a-u)^p}$ in a connected, bounded $C^2$ domain $\Omega$ of $\mathbb{R}^N$ subject to zero Dirichlet boundary conditions, where $\lambda>0$, $N\geq 1$, $p>0$ and…
The dispersion law of plasmons running along thin wires with radius $a$ is known to be practically linear. We show that in a wire with a dielectric constant $\kappa$ much larger than that of its environment $\kappa_e$, such dispersion law…
We study decay rates of one-body reduced density matrices in insulators, described by a tight-binding model, where not only an external potential but also hoppings are spatially modulated. We determine analytically the power in the power…
We present a closed description of the charge carrier injection process from a conductor into an insulator. Common injection models are based on single electron descriptions, being problematic especially once the amount of charge-carriers…
We consider the problem of optimally insulating a given domain $\Omega$ of ${\mathbb{R}}^d$; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the…
We investigate pairing symmetry and a transition temperature in a quasi-one-dimensional repulsive Hubbard model. We solve the Eliashberg equation using the third-order perturbation expansion with respect to the on-site repulsion $U$. We…
We prove new optimal $C^{1,\alpha}$ regularity results for obstacle problems involving evolutionary $p$-Laplace type operators in the degenerate regime $p > 2$. Our main results include the optimal regularity improvement at free boundary…
We study temperature distribution in a heat conducting problem, for a system of p-Laplace equation, giving rise to a free boundary.