scholarly journals Fractional Laplacian, homogeneous Sobolev spaces and their realizations

2020 ◽  
Vol 199 (6) ◽  
pp. 2243-2261 ◽  
Author(s):  
Alessandro Monguzzi ◽  
Marco M. Peloso ◽  
Maura Salvatori
Keyword(s):  
Sobolev Spaces ◽  
Fractional Laplacian ◽  
Homogeneous Sobolev Spaces

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})}.

scholarly journals Bilinear forms on homogeneous Sobolev spaces

2018 ◽  
Vol 457 (1) ◽  
pp. 722-750
Author(s):  
Carme Cascante ◽  
Joan Fàbrega ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

scholarly journals Traces for homogeneous Sobolev spaces in infinite strip-like domains

2019 ◽  
Vol 277 (7) ◽  
pp. 2288-2380 ◽  
Author(s):  
Giovanni Leoni ◽  
Ian Tice
Keyword(s):  
Sobolev Spaces ◽  
Infinite Strip ◽  
Homogeneous Sobolev Spaces

scholarly journals A density result for homogeneous Sobolev spaces

2021 ◽  
Vol 213 ◽  
pp. 112501
Author(s):  
Debanjan Nandi
Keyword(s):  
Sobolev Spaces ◽  
Density Result ◽  
Homogeneous Sobolev Spaces

scholarly journals The principal transmission condition

2021 ◽  
Vol 4 (4) ◽  
pp. 1-33
Author(s):  
Gerd Grubb ◽  
Keyword(s):  
Sobolev Spaces ◽  
Fractional Laplacian ◽  
Pseudodifferential Operators ◽  
Preceding Paper ◽  
Limited Range ◽  
Transmission Condition ◽  
Large Solutions ◽  
Homogeneous Dirichlet Problem ◽  
Integration By Parts Formula ◽  
Regularity Results

<abstract><p>The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a &gt; 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.</p></abstract>

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