Existence of solutions for quasilinear problem with Neumann boundary conditions

My abstract is:This paper is devoted to the study of the existence of weak solutionsfor quasilinear systems of a partial dierential equations which are the combinationof the Perona-Malik equation and the Heat equation. The proof of the main resultsare based on the compactness method and the motonocity arguments.

Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

Monte Carlo simulations of two-component coulomb gases applied in surface electrodes

In this work, we study the gapped Surface Electrode (SE), a planar system composed of two-conductor flat regions at different potentials with a gap G between both sheets. The computation of the electric field and the surface charge density requires solving Laplace’s equation subjected to Dirichlet conditions (on the electrodes) and Neumann Boundary Conditions over the gap. In this document, the GSE is modeled as a Two-Dimensional Classical Coulomb Gas having punctual charges +q and −q on the inner and outer electrodes, respectively, interacting with an inverse power law 1~r-potential. The coupling parameter Γ between particles inversely depends on temperature and is proportional to q2. Precisely, the density charge arises from the equilibrium states via Monte Carlo (MC) simulations. We focus on the coupling and the gap geometry effect. Mainly on the distribution of particles in the circular and the harmonically-deformed gapped SE. MC simulations differ from electrostatics in the strong coupling regime. The electrostatic approximation and the MC simulations agree in the weak coupling regime where the system behaves as two interacting ionic fluids. That means that temperature is crucial in finite-size versions of the gapped SE where the density charge cannot be assumed fully continuous as the coupling among particles increases. Numerical comparisons are addressed against analytical descriptions based on an electric vector potential approach, finding good agreement.

Existence and multiplicity of solutions for a p(x)-biharmonic problem with Neumann boundary conditions

In this paper, we study the p(x)-biharmonique problem with Neumannboundary conditions. Using the three critical point Theorem, we establish the existence of at least threesolutions of this problem.

Casimir effect with one large extra dimension

In this work, I study the Casimir effect of a massive complex scalar field in the presence of one large compactified extra dimension. I investigate the case of a scalar field confined between two parallel plates in the macroscopic three dimensions, and examine the cases of Dirichlet and mixed (Dirichlet–Neumann) boundary conditions on the plates. The case of Neumann boundary conditions is uninteresting, since it yields the same result as the case of Dirichlet boundary conditions. The scalar field also permeates a fourth compactified dimension of a size that could be comparable to the distance between the plates. This investigation is carried out using the [Formula: see text]-function regularization technique that allows me to obtain exact expressions for the Casimir energy and pressure. I discover that when the compactified length of the extra dimension is similar to the plate distance, or slightly larger, the Casimir energy and pressure become significantly different than their standard three-dimensional values, for either Dirichlet or mixed boundary conditions. Therefore, the Casimir effect of a quantum field that permeates a compactified fourth dimension could be used as an effective tool to explore the existence of large compactified extra dimensions.

Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal

The chemotaxis-Stokes system [Formula: see text] is considered subject to the boundary condition [Formula: see text] with [Formula: see text] and a given nonnegative function [Formula: see text]. In contrast to the well-studied case when the second requirement herein is replaced by a homogeneous Neumann boundary condition for [Formula: see text], the Dirichlet condition imposed here seems to destroy a natural energy-like property that has formed a core ingredient in the literature by providing comprehensive regularity features of the latter problem. This paper attempts to suitably cope with accordingly poor regularity information in order to nevertheless derive a statement on global existence within a generalized framework of solvability which involves appropriately mild requirements on regularity, but which maintains mass conservation in the first component as a key solution property.

Nonlinear Neumann boundary value problem for semilinear heat equations with critical power nonlinearities

We study the nonlinear Neumann boundary value problem for semilinear heat equation ∂ t u − Δ u = λ | u | p , t > 0 , x ∈ R + n , u ( 0 , x ) = ε u 0 ( x ) , x ∈ R + n , − ∂ x u ( t , x ′ , 0 ) = γ | u | q ( t , x ′ , 0 ) , t > 0 , x ′ ∈ R n − 1 where p = 1 + 2 n , q = 1 + 1 n and ε > 0 is small enough. We investigate the life span of solutions for λ , γ > 0. Also we study the global in time existence and large time asymptotic behavior of solutions in the case of λ , γ < 0 and ∫ R + n u 0 ( x ) d x > 0.