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We consider the linear elliptic systems or equations in divergence form with periodically oscillating coefficients. We prove the large-scale boundary Lipschitz estimate for the weak solutions in domains satisfying the so-called…

Monte Carlo simulations are performed for hard ellipsoids for a number of values of its semi-axes in the range $c/a \in \{0.1,10\}$. The isotropic phase results are compared to the Vega equation of state [Mol. Phys. {\bf 92} (1997)…

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic…

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (\Delta -q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive…

We study a general nonlinear elliptic equation in the Orlicz setting with data not belonging to the dual of the energy space. We provide several Lorentz-type and Morrey-type estimates for the gradients of solutions under various conditions…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

We compute the phase transition curves for mesoscopic superconductors. Special emphasis is given to the limiting shape of the curve when the magnetic flux is large. We derive an asymptotic formula for the ground state of the Schr\"odinger…

We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either computationally expensive or hard to converge,…

We consider systems of semilinear wave equations in three space dimensions with quadratic nonlinear terms not satisfying the null condition. We prove small data global existence of the classical solution under a new structural condition…

In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fourier-type condition on internal micro-surfaces. The first contribution of this work is the…

Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of…

We investigate the systematic design of compliant morphing structures composed of materials reacting to an external stimulus. We add a perimeter penalty term to ensure existence of solutions. We propose a phase-field approximation of this…

The existence of entire solutions to quasilinear elliptic systems exhibiting both singular and convective reaction terms is discussed. An auxiliary problem, obtained by `freezing' the convection terms and `shifting' the singular ones, is…

We obtain Calder\'on-Zygmund type estimates for parabolic equations with Orlicz growth, where nonlinearities involved in the equations may be discontinuous for the space and time variables. In addition, we consider parabolic systems with…

We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…

We establish the Caccioppoli inequality, a reverse H\"older inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order $2m$ with certain lower order…

We consider a linear differential system of Mathieu equations with periodic coefficients over periodic closed orbits and we prove that, arbitrarily close to this system, there is a linear differential system of Hamiltonian damped Mathieu…

For a class of semi-linear elliptic equations with critical Sobolev exponents and boundary conditions, we prove point-wise estimates for blowup solutions and energy estimates. A special case of this class of equations is a locally defined…

Dynamical systems can display a plethora of ergodic and ergodicity breaking behaviors, ranging from simple periodicity to ergodicity and chaos. Here we report an unusual type of non-ergodic behavior in a many-body discrete-time dynamical…

We obtain some Poincar\'{e} type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*}…