Dirichlet Boundary Conditions Research Articles

Casimir effect of a rough membrane in 2 + 1 Hořava–Lifshitz theory

We investigate the Casimir effect of a rough membrane within the framework of the Hořava–Lifshitz theory in 2+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2+1$$\\end{document} dimensions. Quantum fluctuations are induced by an anisotropic scalar field subject to Dirichlet boundary conditions. We implement a coordinate transformation to render the membrane completely flat, treating the remaining terms associated with roughness as a potential. The spectrum is obtained through perturbation theory and regularized using the ζ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta $$\\end{document}-function method. We present an explicit example of a membrane with periodic border. Additionally, we consider the effect of temperature. Our findings reveal that the Casimir energy and force depend on roughness, the anisotropic scaling factor and temperature.

M-Voronoi and other random open and closed-cell elasto-plastic cellular materials: Geometry generation and numerical study at small and large strains

The present study deals with a numerical design strategy of a novel class of three-dimensional random Voronoi-type geometries, called M-Voronoi. These materials comprise random, non-quadratic convex void shapes and non-uniform intervoid ligament thicknesses, and can span high-to-low relative densities. The starting point for their generation is a random adsorption algorithm (RSA) construction with spherical voids embedded in an incompressible, nonlinear elastic matrix phase. The initial RSA geometry is subjected to large elastic volume changes by prescribing Dirichlet boundary conditions. Due to the incompressibility of the matrix phase, the externally imposed volume changes lead to significant void growth. The numerical growth process may be stopped at any desired porosity. The proposed M-Voronoi process is general and allows the formation of isotropic (or anisotropic) designs. As a byproduct of the developed approach, we also present a novel remeshing technique allowing to read arbitrary geometries of one or multiple phases. The elasto-plastic properties of the M-Voronoi porous materials are numerically investigated at small strains as well as large compressive and shear loads. Their response is assessed by comparison with other well-known random and periodic porous geometries such as polydisperse porous materials with spherical voids (RSA), classical TPMS Gyroid geometries and random Spinodoid topologies. The results show that M-Voronoi and RSA (with spherical voids) geometries exhibit the stiffest elastic and highest flow stress response compared to the other two geometries. This study shows unambiguously that randomness may or may not lead to enhanced mechanical response such as higher stiffness or flow stress.

Numerical Reconstruction of the Source in Dynamical Boundary Condition of Laplace’s Equation

In this work, we consider Cauchy-type problems for Laplace’s equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green’s second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations.

Existence and bifurcation of positive solutions to a class of predator-prey models with mutual interference among the predators

In this work, we study a class of diffusive predator-prey models with mutual interference among the predators while searching for food. Under Dirichlet boundary condition, by the fixed point index theory, we provide the necessary and sufficient conditions for the existence of coexistence states (i.e., stationary pattern). This is a strong contrast to the corresponding Neumann boundary problem, which exhibits bistability and admits no patterns.

Lazer-McKenna Conjecture for fractional problems involving critical growth

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of RN{Aαu=u+2α⁎−1+λu−s¯φ1,u>0, in Ω,u=0, on ∂Ω, where Aα is the spectral fractional operator, λ and s¯ are real numbers, Ω⊂RN is bounded, 2α⁎=2NN−2α is a critical exponent, 0<α<1, φ1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain.

Coexistence states in a cross-diffusion system of a competition model

The main goal of this paper is to study the stationary problem for a Lotka–Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by Kuto and Tsujikawa (2015) and Wang and Yan (2015), where the corresponding Neumann problem is analyzed.

Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

An application of Sobolev's inequality to one-dimensional Kirchhoff equations

We consider nonlocal differential equations with convolution coefficients of the form−M((a⁎(g∘|u′|))(1))u″(t)=λf(t,u(t)), t∈(0,1), in which the coefficient function M imparts a nonlocal structure to the problem. The function g satisfies p-q growth. A model case occurs when g(t):=tp, where p>1, and a(t)≡1 – i.e., the problem−M(‖u′‖Lpp)u″(t)=λf(t,u(t)), t∈(0,1). By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

A second-order [formula omitted]-[formula omitted] difference scheme for the nonlinear time–space fractional Schrödinger equation

In this paper, we mainly consider the numerical approximation for the nonlinear time–space fractional Schrödinger equation with the Dirichlet boundary condition. A linearized second-order finite difference scheme is presented by adopting the L2-1σ formula to approximate the Caputo fractional derivative and the second-order weighted and shifted Grünwald difference formula to approximate the Riesz fractional derivative. To our knowledge, obtaining numerical solutions in the numerical scheme with the nonlinear term is very expensive. In order to reduce the computational cost and memory, the nonlinear term is handled by the local extrapolation method, which is more efficient than implicit processing and without reduced the convergence accuracy. Meanwhile, the numerical scheme of the coupled nonlinear time–space fractional Schrödinger system is also presented. The optimal error estimate of the numerical solution with convergence order O(τ2+h2) is strictly proven. Finally, a series of numerical experiments are presented to confirm the accuracy and effectiveness of the proposed numerical schemes.

Relationship between two solidification problems in order to determine unknown thermal coefficients when the heat transfer coefficient is very large

The phase-change processes are found in a wide variety of dynamic systems, for example in the study of snow avalanches. When a thermal property of the material is unknown, we can add a boundary condition to formulate an Inverse Stefan Problem, and determine this property. In this paper we study a heat conduction phase-change problem with Robin and Neumann boundary condition at a fixed face. This overspecified condition allows to simultaneously determine two unknown thermal coefficients through a moving boundary problem or a free boundary problem. Formulae for different cases where obtained by Ceretani and Tarzia (2015) [6]. The formulation with these type of boundary conditions is a more realistic one than the heat conduction phase-change problems with Dirichlet and Neumann boundary condition at the fixed face, considered by Tarzia, (1982-1983). Therefore we propose to study the relationship between the problems with Robin-Neumann conditions, and the problems with Dirichlet-Neumann conditions. The main result of this work is the convergence analysis of these problems, when the heat transfer coefficient h of the Robin condition is very large. We present for each case of the free and moving boundary problems, an upper bound for the error of the two unknown parameters, obtaining in every case a bound of order o(1h). Finally we show a numerical example of the convergence, for a phase change material commonly used in heating or cooling processes.

Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

We generalize the one-dimensional population model of Anguige &amp; Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller &amp; Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

Non-classical two-phase Stefan problem with variable thermal coefficients

We study a one-dimensional two-phase Stefan problem governed by diffusion-convection equations with a Dirichlet boundary condition at the fixed face, variable thermal coefficients and particular heat sources. A similarity solution is obtained through the solution to a system of integral equations. Some particular cases are analyzed, providing also computational examples.

Quenching Reaction-Diffusion Systems in Bioengineering and Life Sciences

This research paper revolves around investigating the phenomenon of quenching in reaction-diffusion systems and highlighting its significance. The primary focus is on analyzing a specific type of parabolic singular reaction-diffusion model that incorporates positive Dirichlet boundary conditions. The objective is to establish the sufficiency of certain conditions for quenching to occur within a finite time frame and to demonstrate the global existence of solutions. The novelty of this paper lies in the simplicity of the conditions imposed on the nonlinearity. This simplicity allows us to choose it from a wide range of possibilities, thus facilitating the application of the model to numerous singular reaction-diffusion phenomena. To bolster our findings, we will present various real-world applications in the fields of bioengineering and life sciences, showcasing the practical relevance of quenching phenomena. Finally, the paper ends with a conclusion and some potential future perspectives for further research in this area.

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation

We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.

Mean field games of controls with Dirichlet boundary conditions

In this paper, we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field game of controls setting, that is in which the dynamics of each agent is affected not only by the average position of the rest of the agents but also by their average optimal choice. This setting allows the modeling of more realistic real-life scenarios in which agents not only will leave the domain at a certain point in time (like during the evacuation of pedestrians or in debt refinancing dynamics) but also act competitively to anticipate the strategies of the other agents. We shall establish the existence of Nash Equilibria for such class of mean-field game of controls systems under certain regularity assumptions on the dynamics and the Lagrangian cost. Much of the paper is devoted to establishing several a priori estimates which are needed to circumvent the fact that the mass is not conserved (as we are in a Dirichlet boundary condition setting). In the conclusive sections, we provide examples of systems falling into our framework as well as numerical implementations.

Asymptotic profiles of the steady state for a diffusive SIS epidemic model with Dirichlet boundary conditions

Asymptotic profiles of the steady state for a diffusive SIS epidemic model with Dirichlet boundary conditions

Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions

This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.

On a complete parametric Sturm-Liouville problem with sign changing coefficients

&lt;abstract&gt;&lt;p&gt;In this paper we study a complete second order differential equation of Sturm-Liouville type under Dirichlet boundary condition and where the variable coefficients are allowed to be sign changing. Through critical point theory, we obtain the existence of two nontrivial generalized solutions by requiring a specific growth on the nonlinearity. Moreover, the solutions turn out to be nonnegative and with opposite energy sign.&lt;/p&gt;&lt;/abstract&gt;