scholarly journals BIFURCATION, A-PRIORI BOUND AND NEGATIVE SOLUTIONS FOR THE COMPLEX HESSIAN EQUATION

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Hua Luo ◽  
◽  
Guowei Dai ◽  
Keyword(s):  
A Priori ◽  
Hessian Equation ◽  
A Priori Bound

scholarly journals On Jensen-type Inequalities for Nonsmooth Radial Scattering Solutions of a Loglog Energy-Supercritical Schrödinger Equation

2018 ◽  
Vol 2020 (8) ◽  
pp. 2501-2541
Author(s):  
Tristan Roy
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
A Priori ◽  
Time Interval ◽  
Type Estimate ◽  
A Priori Bound ◽  
Long Time ◽  
Supercritical Nonlinearity ◽  
Gamma S ◽  
Radial Scattering

Abstract We prove scattering of solutions of the loglog energy-supercritical Schrödinger equation $i \partial _{t} u + \triangle u = |u|^{\frac{4}{n-2}} u g(|u|)$ with $g(|u|) := \log ^{\gamma } {( \log{(10+|u|^{2})} )}$, $0 < \gamma < \gamma _{n}$, n ∈ {3, 4, 5}, and with radial data $u(0) := u_{0} \in \tilde{H}^{k}:= \dot{H}^{k} (\mathbb{R}^{n})\,\cap\,\dot{H}^{1} (\mathbb{R}^{n})$, where $\frac{n}{2} \geq k> 1 \left(\text{resp.}\,\frac{4}{3}> k > 1\right)$ if n ∈ {3, 4} (resp. n = 5). The proof uses concentration techniques (see e.g., [ 2, 12]) to prove a long-time Strichartz-type estimate on an arbitrarily long time interval J depending on an a priori bound of some norms of the solution, combined with an induction on time of the Strichartz estimates in order to bound these norms a posteriori (see e.g., [ 8, 10]). We also revisit the scattering theory of solutions with radial data in $\tilde{H}^{k}$, $k> \frac{n}{2}$, and n ∈ {3, 4}; more precisely, we prove scattering for a larger range of $\gamma$ s than in [ 10]. In order to control the barely supercritical nonlinearity for nonsmooth solutions, that is, solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

scholarly journals Oscillation of Half-Linear Differential Equations with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Simona Fišnarová ◽  
Robert Mařík
Keyword(s):  
Differential Equation ◽  
A Priori ◽  
Nonoscillatory Solution ◽  
Neutral Equations ◽  
Oscillation Criteria ◽  
A Priori Bound ◽  
Linear Delay ◽  
Linear Differential ◽  
Delay Differential ◽  
Equations With Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

scholarly journals Optimal Algorithms for Computing Average Temperatures

2019 ◽  
Vol 5 (1) ◽  
pp. 34-44
Author(s):  
S. Foucart ◽  
M. Hielsberg ◽  
G. L. Mullendore ◽  
G. Petrova ◽  
P. Wojtaszczyk
Keyword(s):  
Optimal Algorithm ◽  
A Priori ◽  
Global Temperature ◽  
Data Sets ◽  
Average Temperature ◽  
Generic Models ◽  
Average Global Temperature ◽  
A Priori Bound ◽  
Average Precipitation

AbstractA numerical algorithm is presented for computing average global temperature (or other quantities of interest such as average precipitation) from measurements taken at speci_ed locations and times. The algorithm is proven to be in a certain sense optimal. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature. This a priori bound involves a computable compatibility constant which assesses the quality of the measurements for the chosen model. The optimal algorithm is constructed by solving a convex minimization problem and involves a set of functions selected a priori in relation to the model. It is shown that the solution promotes sparsity and hence utilizes a smaller number of well-chosen data sites than those provided. The algorithm is then applied to canonical data sets and mathematically generic models for the computation of average temperature and average precipitation over given regions and given time intervals. A comparison is provided between the proposed algorithms and existing methods.

AN UPPER BOUND FOR THE NUMBER OF VASSILIEV KNOT INVARIANTS

1994 ◽  
Vol 03 (02) ◽  
pp. 141-151 ◽  
Author(s):  
S. V. CHMUTOV ◽  
S. V. DUZHIN
Keyword(s):  
Upper Bound ◽  
A Priori ◽  
Knot Invariants ◽  
A Priori Bound

We prove that the number of independent Vassiliev knot invariants of order n is less than (n − 1)! — thus strengthening the a priori bound (2n − 1)!!

scholarly journals A note on strongly nonlinear parabolic variational inequalities

2017 ◽  
Vol 2 (2) ◽  
pp. 443-448 ◽  
Author(s):  
A. T. El-Dessouky
Keyword(s):  
Variational Inequalities ◽  
Weak Solutions ◽  
A Priori ◽  
Nonlinear Perturbation ◽  
Existence Of Weak Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
A Priori Bound ◽  
Derivatives Of ◽  
Quasilinear Parabolic

AbstractWe prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions.

Liénard systems with several limit cycles

1987 ◽  
Vol 102 (3) ◽  
pp. 565-572 ◽  
Author(s):  
N. G. Lloyd
Keyword(s):  
Periodic Solutions ◽  
Limit Cycles ◽  
Bounded Solution ◽  
A Priori ◽  
Degree Theory ◽  
Extensive Literature ◽  
Liénard Systems ◽  
A Priori Bound ◽  
Image Position

There is an extensive literature on Liénard's equationand numerous criteria for the existence of limit cycles have been developed: see the survey of Staude[7], for example. Broadly speaking, such results are proved in one of two ways: a bounded solution is shown to exist and the Poincaré–Bendixson theorem used, or an ‘a priori’ bound for periodic solutions is obtained and the methods of degree theory utilized.

scholarly journals On estimates for the Fu–Yau generalization of a Strominger system

2019 ◽  
Vol 2019 (751) ◽  
pp. 243-274 ◽  
Author(s):  
Duong H. Phong ◽  
Sebastien Picard ◽  
Xiangwen Zhang
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Higher Dimensions ◽  
Hessian Equation ◽  
Right Hand ◽  
Priori Estimates ◽  
Dimension 2

AbstractWe study an equation proposed by Fu and Yau as a natural n-dimensional generalization of a Strominger system that they solved in dimension 2. It is a complex Hessian equation with right-hand side depending on gradients. Building on the methods of Fu and Yau, we obtain {C^{0}}, {C^{2}}, and {C^{2,\alpha}} a priori estimates. We also identify difficulties in extending the Fu–Yau arguments for non-degeneracy from dimension 2 to higher dimensions.

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