Dirichlet Boundary Research Articles

Control problems in the coefficients and the domain for linear elliptic equations

In the present work we are interested in an optimal design problem for a linear elliptic state equation with a homogeneous boundary Dirichlet condition. The control variables correspond to the coefficients of the diffusion term and the open set where the equation is posed. From the application point of view these variables represent the layout of the materials composing the corresponding domain and its shape. We obtain a relaxed formulation of the problem, the optimality conditions, and we provide a numerical algorithm to solve it. Some numerical simulations are also carried out.

On the solution of evolution p-Laplace equation with memory term and unknown boundary Dirichlet condition

On the solution of evolution p-Laplace equation with memory term and unknown boundary Dirichlet condition

Robust convergence result of discontinuous Galerkin stabilization method for two-dimensional reaction–diffusion equation with discontinuous source term

A reaction–diffusion problem with discontinuous source term and Dirichlet's boundary conditions on the unit square is considered in this paper. The proposed problem has been discretized using a combination of standard Galerkin finite element method (FEM) and non-symmetric discontinuous Galerkin finite element method with an interior penalty (NIPG) with bilinear elements. Layer adapted mesh of Shishkin type has been utilized to discretize the domain. Standard Galerkin FEM is applied on the layer part of the domain where the domain is dense enough and NIPG is applied to the outside layer part. By means of special choice of discontinuity-penalization parameters, the scheme is proved to be uniformly convergent of order O ( ε 1 / 4 N − 1 + N − 1 ln ⁡ N ) . Numerical tests are carried out in support of theoretical findings.

Truncation Error Analysis for an Accurate Nonlocal Approximation to Manifold Poisson Models with Dirichlet Boundary

Truncation Error Analysis for an Accurate Nonlocal Approximation to Manifold Poisson Models with Dirichlet Boundary

Singular solutions of a fractional Dirichlet problem in a punctured domain

Let D be a bounded regular domain in Rn (n ? 3) containing 0, 0 < ? < 2, and ? < 1. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation (??|D)?2 u = a(x)u?(x) in D\{0}, with the boundary Dirichlet conditions lim |x|?0 |x|n??u(x) = 0 and lim x??D ?(x)2??u(x) = 0, where (??|D)?/2 is the fractional Laplace associated to the subordinate killed Brownian motion process in D and ?(x) = dist ( x, ?D) denotes the Euclidean distance between x and ?D. The function a is a positive continuous function in D\{0}, which may be singular at x = 0 and/or at the boundary ?D satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove the existence and global asymptotic behavior of a positive continuous solution on ?D\{0}. We will use some potential theory arguments and Karamata regular variation theory tools.

Sign-changing solutions for a weighted Schrödinger–Kirchhoff equation with double exponential nonlinearities growth

This work is devoted to the existence of least energy nodal solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball [Formula: see text] of [Formula: see text]. The nonlinearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. By using the constrained minimization in Nehari set, the quantitative deformation Lemma and degree theory results, we prove our existence result.

A second-order nonlocal approximation for Poisson model with Dirichlet boundary

A second-order nonlocal approximation for Poisson model with Dirichlet boundary

Generation of Electric Current by Magnetic Field at the Boundary: Quantum Scale Anomaly Versus Semiclassical Meissner Current Outside of the Conformal Limit

AbstractThe scale (conformal) anomaly can generate an electric current near the boundary of a system in the presence of a static magnetic field. The magnitude of this magnetization current, produced at zero temperature and in the absence of matter, is proportional to a beta function associated with the renormalization of the electric charge. Using first‐principle lattice simulations, this paper investigates how the breaking of the scale symmetry affects this “scale magnetic effect” near a Dirichlet boundary in scalar quantum electrodynamics (Abelian Higgs model). This study demonstrates the interplay of the generated current with vortex excitations both in symmetric (normal) and broken (superconducting) phases and compares the results with the anomalous current produced in the conformal, scale‐invariant regime. Possible experimental signatures of the effect in Dirac semimetals are discussed.

Weighted fourth order equation of Kirchhoff type in dimension 4 with non-linear exponential growth

In this work, we are concerned with the existence of a ground state solution for a Kirchhoff weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb{R}^{4}$. The nonlinearities have critical growth in view of Adams' inequalities. To prove the existence result, we use Pass Mountain Theorem. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term $f$. The associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

In this paper, we consider the linearized compressible barotropic Navier‐Stokes system in a bounded interval with a time‐varying delay term acting in the Dirichlet boundary or internal feedback of the hyperbolic component. Assuming some suitable conditions on the time‐dependent delay term and the coefficients of feedback (delayed or not), we study the exponential stability of the concerned hyperbolic‐parabolic system. Due to the presence of the time‐varying delay term, the corresponding spatial operator is also time dependent. Using classical semigroup theory with Kato's variable norm approach, we first show the existence and uniqueness of the Navier‐Stokes system with time delay, acting in the boundary or interior. Next, we prove the two stabilization results by means of interior delay feedback and boundary delay. In both cases, we establish the exponential stability results by introducing some suitable functional energy and using the Lyapunov function approach.

Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals

Abstract Let I I be a bounded interval of R {\mathbb{R}} and λ 1 ( I ) {\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 {(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) 0\leqslant \alpha \lt {\lambda }_{1}(I) , there holds sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ , \mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.

Distributed mathematical model for simulating temperature profile in landfill

Elevated Landfill temperatures have an undesirable effect on landfill cover, stability, slope and leachate migration pattern. Thus, to predict the temperature profile in the landfill a distributed numerical model using MacCormack finite difference method is developed. The developed model considers stratification of the upper and lower layers of the waste as new and old waste by assigning different values of heat generation for aerobic and anaerobic processes. Further, as the new layers of the waste get accumulated over the older layers, the density, moisture content and hydraulic conductivity of the underlying waste layers get modified. The mathematical model utilizes a predictor–corrector approach with a Dirichlet boundary at the surface and no flow condition at the bottom. The developed model is applied to the Gazipur site located in Delhi in India. A correlation coefficient of 0.8 and 0.73 is obtained between the simulated and observed temperatures in calibration and validation respectively. The result shows that the temperature at all the depths and in all the seasons was found to be higher than the atmospheric temperature. The maximum difference of 333 °C was observed in December, and the minimum difference of 22 °Cs was observed in June. The temperature rise is higher in the upper waste layers as it undergoes aerobic degradation. The locus of the maximum temperature gets modified with moisture movement. Since the developed model shows a good agreement with the field observation, it can be used to predict the temperature variation within the landfill under different climatic conditions.

Fully discrete heterogeneous multiscale method for parabolic problems with multiple spatial and temporal scales

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates.

Blow-up phenomena for a p(x)-biharmonic heat equation with variable exponent

In this paper, we deal with a p(x)-biharmonic heat equation with variable exponent under Dirichlet boundary and initial condition. We prove the blow up of solutions under suitable conditions.

On the parabolic Cauchy problem for quantum graphs with vertex noise

We investigate the parabolic Cauchy problem associated with quantum graphs including Lipschitz or polynomial type nonlinearities and additive Gaussian noise perturbed vertex conditions. The vertex conditions are the standard continuity and Kirchhoff assumptions in each vertex. In the case when only Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a mild solution with continuous paths in the standard state space $\mathcal{H}$ of square integrable functions on the edges. We also show that the solution is Markov and Feller. Furthermore, assuming that the vertex values of the normalized eigenfunctions of the self-adjoint operator governing the problem are uniformly bounded, we show that the mild solution has continuous paths in the fractional domain space associated with the Hamiltonian operator, $\mathcal{H}_{\alpha}$ for $\alpha<\frac{1}{4}$. This is the case when the Hamiltonian operator is the standard Laplacian perturbed by a potential. We also show that if noise is present in both type of vertex conditions, then the problem admits a mild solution with continuous paths in the fractional domain space $\mathcal{H}_{\alpha}$ with $\alpha<-\frac{1}{4}$ only. These regularity results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in case of a single interval and classical boundary Dirichlet or Neumann noise.

Limits of solutions to the semilinear plate equation with small parameter

We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.

Solution to Fractional Integro-differential Equation with Unknown Flux on the Dirichlet Boundary

Solution to Fractional Integro-differential Equation with Unknown Flux on the Dirichlet Boundary

Ground state solutions for weighted $N$-Laplacian problem with exponential non linear growth

In this paper, we establish the existence of a positive ground state solution for a weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb R^N,$ $N > 2$. The nonlinearity of the equation is critical or subcritical growth in view of Trudinger-Moser inequalities. In order to obtain our existence result we used minimax techniques combined with Trudinger-Moser inequalities. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check compactness levels.

Output feedback stabilization of an ODE-Reaction–Diffusion PDE cascade with a long interconnection delay

This paper addresses the output feedback stabilization of an ODE–PDE cascade. The PDE takes the form of a 1-D reaction–diffusion equation. The output of the ODE enters into the PDE via a Dirichlet/Neumann/Robin boundary condition. It is assumed that both the output of the ODE and the output of the PDE, selected as a boundary Dirichlet trace, are available for feedback control. The proposed control strategy takes the form of a finite-dimensional observer-based controller. Under a suitable structural controllability property, we show that the reported control strategy achieves the exponential stabilization of the plant when the order of the observer is selected large enough. We then demonstrate how such a control strategy can be adapted and augmented with a predictor component in order to achieve the stabilization of the above mentioned PDE–ODE cascade when the output of the ODE enters into the PDE with an arbitrarily long delay.

Optimizing pump-and-treat method by considering important remediation objectives

The efficiency of groundwater remediation by pump-and-treat (PAT) method is affected by several components. The most important of these components include the pumping wells location, pumping rate, and remediation period. In this research, hybrid optimization-simulation models were developed to find the appropriate groundwater remediation strategy by PAT method. The GA-FEM and NSGA-II-FEM models were used to solve four optimization problems for a hypothetical and real aquifer. These optimization problems were investigated from one objective problem to a four-objectives problem. In the multi-objective problems, in each step, one objective function is added to the previous set of objective functions. In the one-objective case, the objective function was defined as minimizing the contaminant concentration by pumping at a constant rate, while in the two-objectives problem, minimizing the drawdown of groundwater head by pumping at a constant rate was added. In the three-objectives problem, the pumping rate was variable and the mean pumping rate from all the wells is minimized. Finally, minimizing the remediation period is added in the four-objective case. The results indicated that locating the pumping wells in the path of the contaminant flow and close to the source improves the efficiency of the PAT system. The wells with higher pumping rates would be in the path of contamination flow and the wells with lower pumping rates should be located in nodes near the Dirichlet boundary. It is concluded that the remediation period in the hypothetical and real aquifer cannot be less than almost 3000 and 760 days, respectively. Finally, it can be said, the most important component in choosing the proper PAT strategy is the proper location of pumping wells.