scholarly journals A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

2019 ◽  
Vol 35 (5) ◽  
pp. 1415-1449
Author(s):  
Emmanuel Chasseigne ◽  
Olivier Ley ◽  
Thi Tuyen Nguyen
Keyword(s):  
A Priori ◽  
Lipschitz Estimates ◽  
Jacobi Equations

scholarly journals Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians

2019 ◽  
Vol 21 (01) ◽  
pp. 1750098 ◽  
Author(s):  
Andrea Davini
Keyword(s):  
Viscosity Solution ◽  
Viscosity Solutions ◽  
Parabolic Equations ◽  
General Class ◽  
Existence And Uniqueness ◽  
Quasilinear Equations ◽  
Uniqueness Of Solutions ◽  
Superlinear Growth ◽  
Lipschitz Estimates ◽  
Jacobi Equations

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates provided in [S. N. Armstrong and H. V. Tran, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann. 361 (2015) 647–687], as well as on viscosity solution arguments.

Aubry-Mather type at the Double Resonance

2020 ◽  
pp. 121-132
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Fixed Point ◽  
A Priori ◽  
Double Resonance ◽  
Critical Energy ◽  
Resonance Regime ◽  
First Case ◽  
Local Coordinates ◽  
Hyperbolic Fixed Point ◽  
Lipschitz Estimates ◽  
Lipschitz Estimate

This chapter proves Aubry-Mather type for the double resonance regime. It begins by considering the “non-critical energy case” and showing that the cohomologies as chosen are of Aubry-Mather type. The proof consists of two cases. In the first case, the chapter uses the almost verticality of the cylinder, and the idea is similar to the proof of Theorem 9.3. It applies the a priori Lipschitz estimates for the Aubry sets. In the second case, the chapter uses the strong Lipschitz estimate for the energy, and the idea is similar to the proof of Theorem 11.1. It then looks at the construction of the local coordinates. This is done separately near the hyperbolic fixed point (local) and away from it (global).

scholarly journals Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

2019 ◽  
Vol 21 (04) ◽  
pp. 1850018 ◽  
Author(s):  
Valentine Roos
Keyword(s):  
Jacobi Equation ◽  
A Priori ◽  
Initial Condition ◽  
First Order ◽  
Momentum Variable ◽  
Lipschitz Estimates ◽  
The Cauchy Problem ◽  
Local Lipschitz Estimates ◽  
Variational Operator ◽  
Hamilton Jacobi Equation

We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

scholarly journals Ergodic pairs for singular or degenerate fully nonlinear operators

2019 ◽  
Vol 25 ◽  
pp. 75 ◽  
Author(s):  
Isabeau Birindelli ◽  
Françoise Demengel ◽  
Fabiana Leoni
Keyword(s):  
Comparison Principle ◽  
Order Term ◽  
A Priori ◽  
Zero Order ◽  
Dirichlet Problems ◽  
Zeroth Order ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Lipschitz Estimates ◽  
Ergodic Problem

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

scholarly journals Transport equations with nonlocal diffusion and applications to Hamilton–Jacobi equations

2021 ◽  
Author(s):  
Alessandro Goffi
Keyword(s):  
Polynomial Growth ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Nonlocal Diffusion ◽  
Priori Estimates ◽  
Fokker Planck Equations ◽  
Jacobi Equations ◽  
Hölder Estimates ◽  
Fokker Planck

AbstractWe investigate regularity and a priori estimates for Fokker–Planck and Hamilton–Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $$s\in (1/2,1)$$ s ∈ ( 1 / 2 , 1 ) . As for Fokker–Planck equations, we establish integrability estimates under a fractional version of the Aronson–Serrin interpolated condition on the velocity field and Bessel regularity when the drift has low Lebesgue integrability with respect to the solution itself. Using these estimates, through the Evans’ nonlinear adjoint method we prove new integral, sup-norm and Hölder estimates for weak and strong solutions to fractional Hamilton–Jacobi equations with unbounded right-hand side and polynomial growth in the gradient. Finally, by means of these latter results, exploiting Calderón–Zygmund-type regularity for linear nonlocal PDEs and fractional Gagliardo–Nirenberg inequalities, we deduce optimal $$L^q$$ L q -regularity for fractional Hamilton–Jacobi equations.

scholarly journals Robust NonlinearH∞Control Design via Stable Manifold Method

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Yoshiki Abe ◽  
Gou Nishida ◽  
Noboru Sakamoto ◽  
Yutaka Yamamoto
Keyword(s):  
Stable Manifold ◽  
Optimal Control Problems ◽  
Design Method ◽  
A Priori ◽  
Control Design ◽  
Control Problems ◽  
Dimensional Approximation ◽  
Linearized System ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

This paper proposes a systematic numerical method for designing robust nonlinearH∞controllers without a priori lower-dimensional approximation with respect to solutions of the Hamilton-Jacobi equations. The method ensures the solutions are globally calculated with arbitrary accuracy in terms of the stable manifold method that is a solver of Hamilton-Jacobi equations in nonlinear optimal control problems. In this realization, the existence of stabilizing solutions of the Hamilton-Jacobi equations can be derived from some properties of the linearized system and the equivalent Hamiltonian system that is obtained from a transformation of the Hamilton-Jacobi equation. A numerical example is shown to validate the design method.

Application of cross correlation to HREM images of surfaces

1987 ◽  
Vol 45 ◽  
pp. 754-755
Author(s):  
D. E. Luzzi ◽  
L. D. Marks ◽  
M. I. Buckett
Keyword(s):  
Cross Correlation ◽  
A Priori ◽  
Matrix Material ◽  
Correlation Method ◽  
Accurate Knowledge ◽  
Complex Image ◽  
Fourier Filtering ◽  
The Matrix ◽  
Low Pass ◽  
Data Reduction Technique

As the HREM becomes increasingly used for the study of dynamic localized phenomena, the development of techniques to recover the desired information from a real image is important. Often, the important features are not strongly scattering in comparison to the matrix material in addition to being masked by statistical and amorphous noise. The desired information will usually involve the accurate knowledge of the position and intensity of the contrast. In order to decipher the desired information from a complex image, cross-correlation (xcf) techniques can be utilized. Unlike other image processing methods which rely on data massaging (e.g. high/low pass filtering or Fourier filtering), the cross-correlation method is a rigorous data reduction technique with no a priori assumptions.We have examined basic cross-correlation procedures using images of discrete gaussian peaks and have developed an iterative procedure to greatly enhance the capabilities of these techniques when the contrast from the peaks overlap.

Towards 1-Ångstrom-resolution STEM

1993 ◽  
Vol 51 ◽  
pp. 996-997
Author(s):  
H.S. von Harrach ◽  
D.E. Jesson ◽  
S.J. Pennycook
Keyword(s):  
High Resolution ◽  
Field Emission ◽  
Phase Contrast ◽  
Spherical Aberration ◽  
A Priori ◽  
Dark Field ◽  
Image Resolution ◽  
Objective Lens ◽  
Annular Dark Field ◽  
First Results

Phase contrast TEM has been the leading technique for high resolution imaging of materials for many years, whilst STEM has been the principal method for high-resolution microanalysis. However, it was demonstrated many years ago that low angle dark-field STEM imaging is a priori capable of almost 50% higher point resolution than coherent bright-field imaging (i.e. phase contrast TEM or STEM). This advantage was not exploited until Pennycook developed the high-angle annular dark-field (ADF) technique which can provide an incoherent image showing both high image resolution and atomic number contrast.This paper describes the design and first results of a 300kV field-emission STEM (VG Microscopes HB603U) which has improved ADF STEM image resolution towards the 1 angstrom target. The instrument uses a cold field-emission gun, generating a 300 kV beam of up to 1 μA from an 11-stage accelerator. The beam is focussed on to the specimen by two condensers and a condenser-objective lens with a spherical aberration coefficient of 1.0 mm.

Treating Velopharyngeal Inadequacy Using Bilateral Buccal Flap Revision Palatoplasty

2019 ◽  
Vol 4 (5) ◽  
pp. 878-892
Author(s):  
Joseph A. Napoli ◽  
Linda D. Vallino
Keyword(s):  
Systematic Review ◽  
Obstructive Sleep Apnea ◽  
Sleep Apnea ◽  
Cleft Lip ◽  
Cleft Lip And Palate ◽  
A Priori ◽  
Sufficient Evidence ◽  
Obstructive Sleep ◽  
Before And After ◽  
Buccal Flap

Purpose The 2 most commonly used operations to treat velopharyngeal inadequacy (VPI) are superiorly based pharyngeal flap and sphincter pharyngoplasty, both of which may result in hyponasal speech and airway obstruction. The purpose of this article is to (a) describe the bilateral buccal flap revision palatoplasty (BBFRP) as an alternative technique to manage VPI while minimizing these risks and (b) conduct a systematic review of the evidence of BBFRP on speech and other clinical outcomes. A report comparing the speech of a child with hypernasality before and after BBFRP is presented. Method A review of databases was conducted for studies of buccal flaps to treat VPI. Using the principles of a systematic review, the articles were read, and data were abstracted for study characteristics that were developed a priori. With respect to the case report, speech and instrumental data from a child with repaired cleft lip and palate and hypernasal speech were collected and analyzed before and after surgery. Results Eight articles were included in the analysis. The results were positive, and the evidence is in favor of BBFRP in improving velopharyngeal function, while minimizing the risk of hyponasal speech and obstructive sleep apnea. Before surgery, the child's speech was characterized by moderate hypernasality, and after surgery, it was judged to be within normal limits. Conclusion Based on clinical experience and results from the systematic review, there is sufficient evidence that the buccal flap is effective in improving resonance and minimizing obstructive sleep apnea. We recommend BBFRP as another approach in selected patients to manage VPI. Supplemental Material https://doi.org/10.23641/asha.9919352

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