scholarly journals MHD Equations in a Bounded Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria B. Kania
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Base Space ◽  
Dirichlet Boundary Conditions ◽  
Mhd Equations ◽  
Semigroup Approach ◽  
Mhd System

Abstract We consider the MHD system in a bounded domain Ω ⊂ ℝ N , N = 2, 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (L 2(Ω)) N × (L 2(Ω)) N . Solutions to MHD system are obtained next as a limits of that fractional approximations.

Multiple solutions for a non-homogeneous elliptic equation at the critical exponent

2004 ◽  
Vol 134 (1) ◽  
pp. 69-87 ◽  
Author(s):  
Mónica Clapp ◽  
Manuel Del Pino ◽  
Monica Musso
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Critical Exponent ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Homogeneous Elliptic Equation

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

scholarly journals A biharmonic equation with singular nonlinearity

2011 ◽  
Vol 55 (1) ◽  
pp. 155-166 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Biharmonic Equation ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Operator ◽  
The Green Function ◽  
Uniqueness Of A Solution

AbstractWe study the biharmonic equation Δ2u=u−α, 0 < α < 1, in a smooth and bounded domain Ω ⊂ ℝn,n≥ 2, subject to Dirichlet boundary conditions. Under some suitable assumptions on Ω related to the positivity of the Green function for the biharmonic operator, we prove the existence and uniqueness of a solution.

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

scholarly journals On the cubic NLS on 3D compact domains

2013 ◽  
Vol 13 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Fabrice Planchon
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Schrodinger Equation ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Dirichlet Boundary Conditions ◽  
Well Posedness ◽  
Bounded Domains ◽  
Bilinear Estimates

AbstractWe prove bilinear estimates for the Schrödinger equation on 3D domains, with Dirichlet boundary conditions. On non-trapping domains, they match the ${ \mathbb{R} }^{3} $ case, while on bounded domains they match the generic boundaryless manifold case. We obtain, as an application, global well-posedness for the defocusing cubic NLS for data in ${ H}_{0}^{s} (\Omega )$, $1\lt s\leq 3$, with $\Omega $ any bounded domain with smooth boundary.

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

Biharmonic Equations Under Dirichlet Boundary Conditions with Supercritical Growth

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Hanen Ben Omrane ◽  
Mouna Ghedamsi ◽  
Saïma Khenissy
Keyword(s):  
Boundary Conditions ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Biharmonic Equations ◽  
Smooth Bounded Domain ◽  
Uniqueness Results

AbstractWe prove nonexistence and uniqueness results of solutions for biharmonic equations under the Dirichlet boundary conditions on a smooth bounded domain. We carry on the work in [

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

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

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