scholarly journals The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces

2021 ◽  
Author(s):  
Li Chen ◽  
José María Martell ◽  
Cruz Prisuelos-Arribas
Keyword(s):  
Elliptic Operators ◽  
Square Function ◽  
The Novel ◽  
Space Theory ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
First Order ◽  
Measurable Coefficients ◽  
First Order Method ◽  
Unweighted Case

AbstractThis paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.

scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

scholarly journals ON SUPERLUMINAL FERMIONS WITHIN THE SECOND DERIVATIVE EQUATION

2012 ◽  
Vol 27 (14) ◽  
pp. 1250081 ◽  
Author(s):  
S. I. KRUGLOV
Keyword(s):  
Energy Momentum Tensor ◽  
Canonical Quantization ◽  
Vacuum Expectation ◽  
Momentum Tensor ◽  
Order Equation ◽  
Quantum Mechanical ◽  
Projection Operators ◽  
The Novel ◽  
First Order ◽  
Electromagnetic Interactions

We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy–momentum tensor are found. The minimal and nonminimal electromagnetic interactions of fermions are considered and Schrödinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.

scholarly journals Cosmology from a new nonconservative gravity

2018 ◽  
Vol 27 (06) ◽  
pp. 1841006 ◽  
Author(s):  
Júlio C. Fabris ◽  
Hermano Velten ◽  
Thiago R. P. Caramês ◽  
Matheus J. Lazo ◽  
Gastão S. F. Frederico
Keyword(s):  
Fluid Model ◽  
Gravitational Theory ◽  
Dissipative Process ◽  
The Novel ◽  
Least Action Principle ◽  
Dynamical Equations ◽  
First Order ◽  
Least Action ◽  
Dissipative Effects ◽  
Cosmic Background

In this paper, we present a cosmological model arising from a nonconservative gravitational theory proposed in [M. J. Lazo, J. Paiva, J. T. S. Amaral and G. S. F. Frederico, Phys. Rev. D 95 (2017) 101501.] The novel feature where comparing with previous implementations of dissipative effects in gravity is the possible arising of such phenomena from a least action principle, so they are of a purely geometric nature. We derive the dynamical equations describing the behavior of the cosmic background, considering a single fluid model composed by pressureles matter, whereas the dark energy is conceived as an outcome of the “geometric” dissipative process emerging in the model. Besides, adopting the synchronous gauge, we obtain the first-order perturbative equations which shall describe the evolution of the matter perturbations within the linear regime.

Transient Response of Laminated Plates Subject to Close Proximity Explosions

1999 ◽  
Author(s):  
John M. Coggin ◽  
Jeffrey M. K. Chock ◽  
Rakesh K. Kapania ◽  
Eric R. Johnson
Keyword(s):  
Transient Response ◽  
Composite Plates ◽  
Laminated Plates ◽  
Blast Load ◽  
The Novel ◽  
Element Mesh ◽  
Simply Supported ◽  
First Order ◽  
Close Proximity ◽  
Load Types

Abstract We study the transient response of simply supported composite plates subject to close proximity explosions. Many studies are currently availiable in which the blast load is applied uniformly across the plate; and is described by step, N-pulse, or Friedlander equations. The novel aspect considered here is the case for which the blast pressure is due to a close proximity explosion, and is therefore taken to be both spatially and temporally varying. Two methods for calculating blast pressures are developed for arbitrary blast size and distance. A FORTAN program is described that automates the application of an arbitrary blast load to a generic finite element mesh. Modal superposition and NASTRAN solution procedures are verified for several load types and stacking sequences. Results are obtained within the framework of classical and first order plate theories for a variety of parameters including; stacking sequence, blast size, blast distance, and blast calculation method.

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