An explicit bound for the Poincaré constant on a Lipschitz domain

2017 ◽  
Vol 112 ◽  
pp. 87-97
Author(s):  
A. F. M. ter Elst ◽  
Keith Ruddell
Keyword(s):  
Lipschitz Domain ◽  
Explicit Bound ◽  
Poincaré Constant

scholarly journals On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

2021 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Luigi Provenzano
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Lipschitz Domain ◽  
Spectral Properties ◽  
Explicit Representation ◽  
Lipschitz Domains ◽  
Dirichlet Problems ◽  
Trace Space ◽  
In Series ◽  
Definition Of

AbstractWe consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

scholarly journals The Differentiability of the Drag with Respect to the Variations of a Lipschitz Domain in a Navier--Stokes Flow

1997 ◽  
Vol 35 (2) ◽  
pp. 626-640 ◽  
Author(s):  
Juan Antonio Bello ◽  
Enrique Fernández-Cara ◽  
Jérôme Lemoine ◽  
Jacques Simon
Keyword(s):  
Stokes Flow ◽  
Lipschitz Domain ◽  
Navier Stokes

scholarly journals An Elementary Proof of Jin's Theorem with a Bound

10.37236/3846 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Mauro Di Nasso
Keyword(s):  
Ergodic Theory ◽  
Elementary Proof ◽  
Nonstandard Analysis ◽  
Measure Theory ◽  
Difference Sets ◽  
Explicit Bound ◽  
Finite Set

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of difference sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultrafilters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If $A$ and $B$ are sets of integers having positive upper Banach densities $a$ and $b$ respectively, then there exists a finite set $F$ of cardinality at most $1/ab$ such that $(A-B)+F$ covers arbitrarily long intervals.

ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 701-716 ◽  
Author(s):  
REMIGIO RUSSO ◽  
ALFONSINA TARTAGLIONE
Keyword(s):  
Boundary Layer ◽  
Weak Solution ◽  
Lipschitz Domain ◽  
Stokes System ◽  
Navier Stokes ◽  
Robin Problem ◽  
Layer Potentials ◽  
Very Weak Solution ◽  
Stokes Systems ◽  
Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

scholarly journals Kelvin transform for α-harmonic functions in regular domains

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Krzysztof Michalik ◽  
Michał Ryznar
Keyword(s):  
Lipschitz Domain ◽  
Harmonic Functions ◽  
Stable Processes ◽  
Kelvin Transform

AbstractWe investigate conditional stable processes in a Lipschitz domain

scholarly journals On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 77
Author(s):  
Vincenzo Coscia
Keyword(s):  
Asymptotic Behavior ◽  
Lipschitz Domain ◽  
Three Dimensional ◽  
Finite Energy ◽  
Asymptotic Behavior Of Solutions ◽  
Exterior Domains ◽  
Displacement Problem ◽  
Linear Elastostatics

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain.

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