Localization of the LQR Feedback Kernel in Spatially-Invariant Problems over Sobolev Spaces

2020 ◽  
Author(s):  
Emily Jensen ◽  
Jonathan P. Epperlein ◽  
Bassam Bamieh
Keyword(s):  
Sobolev Spaces

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

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.

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

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 The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

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