Estimates for the energy of the solutions to elliptic Dirichlet problems on convex domains

2004 ◽  
Vol 134 (1) ◽  
pp. 89-107 ◽  
Author(s):  
Graziano Crasta
Keyword(s):  
Boundary Conditions ◽  
Isoperimetric Inequality ◽  
Variational Formulation ◽  
Dirichlet Boundary ◽  
Main Tool ◽  
Dirichlet Boundary Conditions ◽  
Convex Domains ◽  
Dirichlet Problems ◽  
The Poisson Equation ◽  
Minkowski Theorem

We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain Ω ⊂ Rn. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of Ω. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.

scholarly journals Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 647
Author(s):  
Michał Bełdziński ◽  
Marek Galewski ◽  
Igor Kossowski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Non Linear ◽  
The Stability

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.

Multiplicity Results for a Class of Asymptotically Linear Elliptic Problems With Resonance and Applications to Problems With Measure Data

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Alberto Ferrero ◽  
Claudio Saccon
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Dirichlet Boundary Conditions ◽  
Measure Data ◽  
Dirichlet Problems ◽  
Asymptotically Linear ◽  
Multiplicity Results ◽  
Existence And Multiplicity

AbstractWe study existence and multiplicity results for solutions of elliptic problems of the type -Δu = g(x; u) in a bounded domain Ω with Dirichlet boundary conditions. The function g(x; s) is asymptotically linear as |s| → +∞. Also resonant situations are allowed. We also prove some perturbation results for Dirichlet problems of the type -Δu = g

Finite Difference Method with Dirichlet Problems of 2D Laplace’s Equation in Elliptic Domain

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Ubaid Ullah ◽  
Muhammad Saleem Chandio
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Boundary Conditions ◽  
Difference Method ◽  
Dirichlet Boundary ◽  
Numerical Solutions ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Dirichlet Problems ◽  
Study Objective

<p>In this study finite difference method (FDM) is used with Dirichlet boundary conditions on rectangular domain to solve the 2D Laplace equation. The chosen body is elliptical, which is discretized into square grids. The finite difference method is applied for numerical differentiation of the observed example of rectangular domain with Dirichlet boundary conditions. The obtained numerical results are<br />compared with analytical solution. The obtained results show the efficiency of the FDM and settled with the obtained exact solution. The study objective is to check the accuracy of FDM for the numerical solutions of elliptical bodies of 2D Laplace equations. The study contributes to find the heat (temperature) distribution inside a regular rectangular elliptical discretized body.</p>

Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra

2020 ◽  
Author(s):  
Vitoriano Ruas
Keyword(s):  
Boundary Conditions ◽  
Error Estimates ◽  
Degrees Of Freedom ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Dirichlet Problems ◽  
Simple Alternative ◽  
Curved Domains

Abstract In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

scholarly journals Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions

2021 ◽  
pp. 104123
Author(s):  
Firdous A. Shah ◽  
Mohd Irfan ◽  
Kottakkaran S. Nisar ◽  
R.T. Matoog ◽  
Emad E. Mahmoud
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Wavelet Method ◽  
Telegraph Equations
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