The resolvent problem for the Stokes system in exterior domains: uniqueness and non-regularity in Hölder spaces

1992 ◽  
Vol 122 (1-2) ◽  
pp. 1-10 ◽  
Author(s):  
Paul Deuring
Keyword(s):  
Boundary Conditions ◽  
Existence Result ◽  
Stokes System ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Exterior Domains ◽  
Hölder Spaces ◽  
Image Position ◽  
Holder Spaces ◽  
Resolvent Problem

SynopsisWe consider the resolvent problem for the Stokes system in exterior domains, under Dirichlet boundary conditions:where Ω is a bounded domain in ℝ3. It will be shown that in general there is no constant C > 0 such that for with , div u = 0, and for with . However, if a solution (u, π) of problem (*) exists, it is uniquely determined, provided that u(x) and ∇π(x) decay for large values of |x|. These assertions imply a non-existence result in Hölder spaces.

scholarly journals Well Posedness for a Class of Flexible Structure in Hölder Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Claudio Cuevas ◽  
Carlos Lizama
Keyword(s):  
Boundary Conditions ◽  
External Force ◽  
Dirichlet Boundary ◽  
Flexible Structures ◽  
Dirichlet Boundary Conditions ◽  
Flexible Structure ◽  
Well Posedness ◽  
Hölder Spaces ◽  
Holder Spaces ◽  
Well Posed

We characterize well-posedness in Hölder spaces for an abstract version of the equation(∗) u′′+λu′′′=c2(Δu+μΔu′)+fwhich model thevibrationsof flexible structures possessing internal material damping and external forcef. As a consequence, we show that in case of the Laplacian with Dirichlet boundary conditions, equation(∗)is always well-posed provided0<λ<μ.

scholarly journals A note on the characterisation of eigencurves for certain two parameter eigenvalue problems in ordinary differential equations

1993 ◽  
Vol 35 (1) ◽  
pp. 63-67 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Weight Function ◽  
Analytic Functions ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
Survey Paper ◽  
Image Position ◽  
Two Parameter

We are interested in two parameter eigenvalue problems of the formsubject to Dirichlet boundary conditionsThe weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurvesin the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

scholarly journals On a singular sublinear polyharmonic problem

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Sonia Ben Othman
Keyword(s):  
Fixed Point ◽  
Green Function ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Unit Ball ◽  
Existence Result ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Combined Effects ◽  
Polyharmonic Equations

This paper deals with a class of singular nonlinear polyharmonic equations on the unit ballBinℝn (n≥2)where the combined effects of a singular and a sublinear term allow us by using the Schauder fixed point theorem to establish an existence result for the following problem:(−Δ)mu=φ(⋅,u)+ψ(⋅,u)inB(in the sense of distributions),u>0,lim⁡|x|→1u(x)/(1−|x|)m−1=0. Our approach is based on estimates for the polyharmonic Green function onBwith zero Dirichlet boundary conditions.

scholarly journals The Stokes resolvent problem: optimal pressure estimates and remarks on resolvent estimates in convex domains

2020 ◽  
Vol 59 (5) ◽  
Author(s):  
Patrick Tolksdorf
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Operator Norm ◽  
Dirichlet Boundary Conditions ◽  
Gradient Estimates ◽  
Resolvent Estimates ◽  
Famous Result ◽  
Neumann Type ◽  
Type Boundary ◽  
Resolvent Problem

Abstract The Stokes resolvent problem $$\lambda u - \Delta u + \nabla \phi = f$$ λ u - Δ u + ∇ ϕ = f with $${\text {div}}(u) = 0$$ div ( u ) = 0 subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of $$\mathrm {L}^2_{\sigma } (\Omega ) \ni f \mapsto \phi \in \mathrm {L}^2 (\Omega )$$ L σ 2 ( Ω ) ∋ f ↦ ϕ ∈ L 2 ( Ω ) decays like $$|\lambda |^{- 1 / 2}$$ | λ | - 1 / 2 which agrees exactly with the scaling of the equation. In comparison to that, the operator norm of this mapping under Dirichlet boundary conditions decays like $$|\lambda |^{- \alpha }$$ | λ | - α for $$0 \le \alpha \le 1 / 4$$ 0 ≤ α ≤ 1 / 4 and we show optimality of this rate, thereby, violating the natural scaling of the equation. In the second part of this article, we investigate the Stokes resolvent problem subject to homogeneous Neumann-type boundary conditions if the underlying domain $$\Omega $$ Ω is convex. Invoking a famous result of Grisvard (Elliptic problems in nonsmooth domains. Monographs and studies in mathematics, Pitman, 1985), we show that weak solutions u with right-hand side $$f \in \mathrm {L}^2 (\Omega ; {\mathbb {C}}^d)$$ f ∈ L 2 ( Ω ; C d ) admit $$\mathrm {H}^2$$ H 2 -regularity and further prove localized $$\mathrm {H}^2$$ H 2 -estimates for the Stokes resolvent problem. By a generalized version of Shen’s $$\mathrm {L}^p$$ L p -extrapolation theorem (Shen in Ann Inst Fourier (Grenoble) 55(1):173–197, 2005) we establish optimal resolvent estimates and gradient estimates in $$\mathrm {L}^p (\Omega ; {\mathbb {C}}^d)$$ L p ( Ω ; C d ) for $$2d / (d + 2)< p < 2d / (d - 2)$$ 2 d / ( d + 2 ) < p < 2 d / ( d - 2 ) (with $$1< p < \infty $$ 1 < p < ∞ if $$d = 2$$ d = 2 ). This interval is larger than the known interval for resolvent estimates subject to Dirichlet boundary conditions (Shen in Arch Ration Mech Anal 205(2):395–424, 2012) on general Lipschitz domains.

On the existence of multiple positive solutions to some superlinear systems

2012 ◽  
Vol 142 (1) ◽  
pp. 39-59 ◽  
Author(s):  
M. Chhetri ◽  
S. Raynor ◽  
S. Robinson
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Convex Shape ◽  
Smooth Bounded Domain ◽  
Image Position

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the formon a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

scholarly journals On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

2019 ◽  
Vol 150 (4) ◽  
pp. 2025-2054
Author(s):  
Piotr Kalita ◽  
Piotr Zgliczyński
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Dirichlet Boundary Conditions ◽  
Image Position ◽  
Space Interval ◽  
Bounded Trajectory

AbstractWe study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

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.

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