Dirichlet Boundary Conditions Research Articles

Separation property and asymptotic behavior for a transmission problem of the bulk-surface coupled Cahn–Hilliard system with singular potentials and its Robin approximation

We consider a class of bulk-surface coupled Cahn–Hilliard systems in a smooth, bounded domain [Formula: see text] [Formula: see text], where the bulk phase variable is connected to the surface phase variable via a Dirichlet boundary condition or its Robin approximation. For a general class of singular potentials (including the physically relevant logarithmic potential), we establish the regularity propagation of global weak solutions to the initial boundary value problem. In particular, when the spatial dimension is two, we prove the instantaneous strict separation property, which ensures that every global weak solution remains uniformly away from the pure states [Formula: see text] after any given positive time. In the three-dimensional case, we obtain the eventual strict separation property that holds for sufficiently large time. This strict separation property allows us to prove that every global weak solution converges to a single equilibrium as time goes to infinity using the Łojasiewicz–Simon approach. Finally, we study the double obstacle limit for the problem with logarithmic potentials in the bulk and on the boundary, showing that as the absolute temperature [Formula: see text] tends to zero, the corresponding weak solutions converge (for a suitable subsequence) to a weak solution of the problem with a double obstacle potential.

ASYMPTOTICS OF THE SOLUTION TO A BISINGULAR PROBLEM OF OPTIMAL DISTRIBUTED CONTROL IN A CONVEX DOMAIN WITH A SMALL PARAMETER IN ONE OF THE HIGHER DERIVATIVES

The paper considers the problem of optimal distributed control in a strictly convex planar domain with a smooth boundary and a small parameter in one of the higher derivatives of the elliptic operator. In this problem, a zero Dirichlet boundary condition is imposed, and the control enters additively into the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square-integrable functions. The solutions to the resulting boundary value problems are treated in the generalized sense as elements of a certain Hilbert space. The optimality criterion is the sum of the square of the norm of the state deviation from a given state and the square of the norm of the control, with a weighting coefficient. This structure of the optimality criterion allows either the first or the second term to be emphasized, depending on the need. In the first case, achieving the desired state is prioritized, while in the second case, minimizing resource costs becomes more important. The asymptotics of the problem are studied in detail, arising from a second-order differential operator with a small coefficient in one of the higher derivatives, to which a zero-order differential operator is added.

Dirichlet Boundary Conditions for FFT‐Based Micromechanics of Bicontinuous Stochastic Microstructures

ABSTRACTComputational homogenization may be used to evaluate effective properties of microstructured materials. As digital imaging techniques generally generate microstructure information on a regular grid, a common approach to compute the effective properties of the microstructured materials is to use fast Fourier transform (FFT)‐based methods. Traditionally, this approach involves periodic boundary conditions. However, for certain microstructure types, nonperiodic boundary conditions are of interest. In this work, we show how to efficiently impose Dirichlet boundary conditions for FFT‐based micromechanics on stochastic bicontinuous microstructures in combinations with a linear conjugate gradient (CG) solver. Consequently, we compute the effective properties of bicontinuous stochastic microstructures and compare the results to values reported in the literature computed by a finite element (FE) code.

Existence and uniqueness of solution for a non-local singular boundary value problem

The main goal of this work is the examination of a mixed singular problem, which is characterized by a singularity of order n at the origin. This problem incorporates a Dirichlet boundary condition alongside an integral with variable bounds. We start by outlining the functional framework for analyzing the posed problem. Initially, we establish a bilateral a priori estimate using energy inequalities. To demonstrate the existence of a solution, we illustrate the density of the image of the operator associated with the problem, relying on regularization operators for our analysis. This approach seeks to demonstrate the existence of a solution to the mixed singular problem, ensuring that the solution adheres to the Dirichlet boundary conditions and the integral with variable bounds. Central to this process are the regularization operators and energy inequalities, which help manage the singularities and establish the existence of a well-defined solution.

Eigenvalue variations of the Neumann Laplace operator due to perturbed boundary conditions

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold (M,g,∂M) under a singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions on a small portion of the boundary. We derive an asymptotic expansion of the perturbed eigenvalues as the Dirichlet part shrinks to a point x∗∈∂M in terms of the spectral parameters of the unperturbed system. This asymptotic expansion demonstrates the impact of the geometric properties of the manifold at a specific point x∗. Furthermore, it becomes evident that the shape of the Dirichlet region holds significance as it impacts the first terms of the asymptotic expansion. A crucial part of this work is the construction of the singularity structure of the restricted Neumann Green’s function which may be of independent interest. We employ a fusion of layer potential techniques and pseudo-differential operators during this work.

Global stability of reaction-diffusion equation with nonlocal delay

In this paper, we establish the global stability of the spatially nonhomogeneous steady state solution of a reaction diffusion equation with nonlocal delay under the Dirichlet boundary condition. To achieve this, we obtain the global existence and nonnegativity of solutions and give an extensive study on the properties of omega limit sets.

A high-order no image point sharp interface immersed boundary method for compressible flows

A high-order no-image point sharp interface immersed boundary method for compressible flow is presented. The method comprises a stable high-order compact scheme and a ghost point value determination method. By regulating dissipation, the stability of the compact scheme for either Dirichlet or Neumann boundary conditions is validated by the von Neumann method in one dimension. With regard to the use of ghost points, mirror points or Lagrange points are no longer employed. The boundary conditions at the intersection of arbitrary geometries and Cartesian grids are imposed on the basis function of Taylor polynomial interpolation, along with weighted least squares error minimization, in order to determine the values of the ghost points. Third-order accuracy is maintained for both subsonic and supersonic inviscid flow. Numerical simulations of several two-dimensional benchmark problems are carried out to provide evidence about the convergence order of the method.

A remark on an elliptic equation in an unbounded domain

We give in this work some remarks on the resolution of an elliptic equation with homogeneous Dirichlet boundary conditions posed on the unbounded domain Σ =(0,+∞ )×(0,1). Our aim is to use the famous Dore-Venni theorem, in UMD Banach spaces, to prove the existence, uniqueness and maximal regularity of the strict solution.

Exact solution to a class of problems for the Burgers’ equation on bounded intervals

In this study, we consider Burgers’ equation with fixed Dirichlet boundary conditions on generic bounded intervals. By employing the Hopf-Cole transformation and a recently established exact operational solution for linear reaction–diffusion equations, an exact solution in the time domain is derived through inverse Laplace transforms. In the event that analytic inverses do in fact exist, they can be obtained in closed form through the use of Mellin transforms. Nevertheless, highly efficient algorithms are available, and numerical inverses in the time domain are always feasible, regardless of the complexity of the Laplace domain expressions. Two illustrative tests demonstrate that the results align closely with those of classical exact solutions. In comparison to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent a novel alternative, offering new insights and perspectives. The exact solution via the inverse Laplace transform is shown to be more computationally efficient, providing a reference point for numerical and semi-analytical methods.

Quantum dissipative effects for a real scalar field coupled to a time-dependent Dirichlet surface in d+1 dimensions

We study the dynamical Casimir effect for a real scalar field φ in d+1 dimensions, in the presence of a mirror that imposes Dirichlet boundary conditions and undergoes time-dependent motion or deformation. Using a perturbative approach, we expand in powers of the deviation of the mirror’s surface Σ from a hyperplane, up to fourth order. General expressions for the probability of pair creation induced by motion are derived, and we analyze the impact of space-time dimensionality as well as of the nonlinear effects introduced by the fourth-order terms. Published by the American Physical Society 2024

Conforming embedded isogeometric analysis for B-Rep CAD models with strong imposition of Dirichlet boundary conditions using trivariate B++ splines

Strong imposition of Dirichlet boundary conditions for immersed finite element methods or immersed isogeometric methods remains a challenge. To address this issue, this paper presents a 3D conforming embedded isogeometric method for Boundary-Represented (B-Rep) solid CAD models by generalizing our bivariate B++ splines to trivariate B++ Splines. The proposed method can convert a B-Rep model into a trivariate B++ spline solid patch with body-fitted boundary representation while retaining key features of B-rep models, such as sharp points, sharp edges, and holes. The basis functions of the trivariate B++ spline solid patch satisfy the Kronecker delta property, which implies that we can strongly impose Dirichlet boundary conditions on B-Rep models without the necessity of Nitsche's method. The presented method can be viewed as a parameterization method that inherits the advantages of volumetric parameterization methods in that the basis functions of a reconstructed geometry satisfy the Galerkin method. In addition, compared with T-splines, the proposed method does not generate the extraordinary point and can achieve optimal convergence rate. Several numerical examples are used to demonstrate the reliability of the presented method.

A strong unique continuation property for weakly coupled elliptic systems

We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system-structure of the problem and Carleman estimates. Then, we use our unique continuation theorems to show two nonexistence results. The first one states the nonexistence of nontrivial solutions to a weakly coupled elliptic system with a critical nonlinearity and Dirichlet boundary condition on starshaped domains, whereas the second one yields nonexistence of symmetric least energy solutions for a critical system in more general domains.

The least gradient problem with Dirichlet and Neumann boundary conditions

Abstract In this paper, we consider the BV least gradient problem with Dirichlet condition on a part Γ ⊂ ∂ ⁡ Ω {\Gamma\subset\partial\Omega} and Neumann boundary condition on its complementary part ∂ ⁡ Ω \ Γ {\partial\Omega\backslash\Gamma} . We will show that in the plane this problem is equivalent to an optimal transport problem with import/export taxes on ∂ ⁡ Ω \ Γ {\partial\Omega\backslash\Gamma} . Thanks to this equivalence, we will be able to show existence and uniqueness of a solution to this mixed least gradient problem and we will also prove some Sobolev regularity on this solution. We note that these results generalize those in [S. Dweik, W 1 , p W^{1,p} regularity on the solution of the BV least gradient problem with Dirichlet condition on a part of the boundary, Nonlinear Anal. 223 2022, Article ID 113012], where we studied the pure Dirichlet version of this problem.

Heat transfer from a cylindrical heater to a medium with variable thermophysical characteristics and heat source power

The formulation and solution of the problem of non-stationary thermal conductivity in a composite material with variable thermophysical characteristics are considered. Density, heat capacity, thermal conductivity, as well as the power of the heat source due to the hydration reactions of the binder components change during the concrete hardening process. The heat transfer problem is formulated for the general case when there are no calculation formulas for thermophysical transfer coefficients. The “microprocess method” was used to calculate the dynamics of the temperature field. According to this method, the space from the outer surface of the heater is modeled by a system of successively located “rings”. When moving from the previous “ring” to the next one, the charge in the transfer coefficients and the power of the volumetric heat source were taken into account. At the same time, the initial and boundary conditions were corrected. The boundary value problems are formulated in the form of a differential equation of non-stationary heat transfer with an arbitrary initial distribution of transfer potentials, Dirichlet boundary conditions, and a heat source in the form Po=f(Fo).The obtained solutions are analyzed for some particular cases. Prospects for further theoretical and experimental research are determined.

A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions

A semilinear diffusion PDE with variable order time-fractional Caputo derivative subject to homogeneous Dirichlet boundary conditions

Zaremba problem with degenerate weights

Abstract We present new regularity result in the study of elliptic equations with mixed boundary conditions. We obtain small higher integrability of the gradient (Meyers property). The result is new for both linear and nonlinear equations with degenerate coefficients with a new sharp Maz’ya-type capacity condition on the part with Dirichlet boundary condition, while prior research was limited to uniformly elliptic weights.

Existence of weak solutions for nonlocal Dirichlet problems via Young measure theory

This article investigates the existence of weak solutions for a class of nonlocal problems with Dirichlet boundary conditions. The proof of the existence result relies on Galerkin's approximation and Young's measure theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/73/abstr.html

Current Density Induced by a Cosmic String in de Sitter Spacetime in the Presence of Two Flat Boundaries

In this paper, we investigate the vacuum bosonic current density induced by a carrying-magnetic-flux cosmic string in a (D+1)-de Sitter spacetime considering the presence of two flat boundaries perpendicular to it. In this setup, the Robin boundary conditions are imposed on the scalar charged quantum field on the boundaries. The particular cases of Dirichlet and Neumann boundary conditions are studied separately. Due to the coupling of the quantum scalar field with the classical gauge field, corresponding to a magnetic flux running along the string’s core, a nonzero vacuum expectation value for the current density operator along the azimuthal direction is induced. The two boundaries divide the space in three regions with different properties of the vacuum states. In this way, our main objective is to calculate the induced currents in these three regions. In order to develop this analysis we calculate, for both regions, the positive frequency Wightman functions. Because the vacuum bosonic current in dS space has been investigated before, in this paper we consider only the contributions induced by the boundaries. We show that for each region the azimuthal current densities are odd functions of the magnetic flux along the string. To probe the correctness of our results, we take the particular cases and analyze some asymptotic limits of the parameters of the model. Also some graphs are presented exhibiting the behavior of the current with relevant physical parameter of the system.

Water wave interactions with surface-piercing vertical barriers in a rectangular tank: Connections with Bloch waves and quasimodes

Eigenmodes are studied for a fluid-filled rectangular tank containing one or more vertical barriers, and on which either Dirichlet or Neumann boundary conditions are prescribed on the lateral walls. In the case where the tank contains a single barrier, the geometry of the tank is equivalent to the unit cell of the cognate periodic array, and its eigenmodes are equivalent to standing Bloch waves. As the submergence depth of the barrier increases, it is shown that the passbands (i.e. frequency intervals in which the periodic array supports Bloch waves) become thinner, and that this effect becomes stronger at higher frequencies. The eigenmodes of a uniform array of vertical barriers in a rectangular tank are also considered. They are found to be a superposition of left- and right-propagating Bloch waves, which couple together at the lateral walls of the tank. A homotopy procedure is used to relate the eigenmodes to the quasimodes of the same uniform array in a fluid of infinite horizontal extent, and the quasimodes are shown to govern the response of the array to incident waves. Qualitative features of the mode shapes are typically preserved by the homotopy, which suggests that the resonant responses of the array in an infinite fluid can be understood in terms of modes of the array in a finite tank.

Coexistence states for a prey-predator model with self-diffusion and predator-taxis

This paper is concerned with the stationary problem of a prey-predator model with self-diffusion and predator-taxis. Under homogeneous Dirichlet boundary conditions, the bifurcation approach is used to derive the sufficient conditions for the existence of coexistence states. Our analysis shows that the coupling of self-diffusion and predator-taxis cause some difficulties in establishing a priori estimates for the solutions.