scholarly journals The Poisson equation in homogeneous Sobolev spaces

2004 ◽  
Vol 2004 (36) ◽  
pp. 1909-1921
Author(s):  
Tatiana Samrowski ◽  
Werner Varnhorn
Keyword(s):  
Sobolev Space ◽  
Poisson Equation ◽  
Sobolev Spaces ◽  
Exterior Domain ◽  
Smooth Boundary ◽  
Boundary Values ◽  
Homogeneous Sobolev Space ◽  
The Poisson Equation ◽  
Homogeneous Sobolev Spaces ◽  
External Forces

We consider Poisson's equation in ann-dimensional exterior domainG(n≥2)with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certainLq(G)-spaces there exists a solution in the homogeneous Sobolev spaceS2,q(G), containing functions being local inLq(G)and having second-order derivatives inLq(G)Concerning the uniqueness of this solution we prove that the corresponding nullspace has the dimensionn+1, independent ofq.

scholarly journals Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

2021 ◽  
Vol 47 (1) ◽  
pp. 203-235
Author(s):  
Feng Liu ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Maximal Function ◽  
Pointwise Estimates ◽  
Boundary Values ◽  
First Order ◽  
Littlewood Maximal Function ◽  
Weak Derivatives ◽  
Derivatives Of

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

Bilinear forms on non-homogeneous Sobolev spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 995-1026
Author(s):  
Carme Cascante ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Form ◽  
Positive Measure ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

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