Local minimizers of the Willmore functional

Analysis ◽  
2015 ◽  
Vol 35 (2) ◽  
Author(s):  
Florian Skorzinski
Keyword(s):  
Critical Point ◽  
Local Minimizer ◽  
Positive Definite ◽  
Second Derivative ◽  
Sufficient Condition ◽  
Conformal Transformations ◽  
Willmore Functional ◽  
Local Minimizers

AbstractSince the Willmore functional is invariant with respect to conformal transformations and reparametrizations, the kernel of the second derivative of the functional at a critical point will always contain a subspace generated by these transformations. We prove that the second derivative being positive definite outside this space is a sufficient condition for a critical point to be a local minimizer.

The Application of Fuzzification for Solving Quadratic Functions

2021 ◽  
Author(s):  
Lunshan Gao
Keyword(s):  
Approximate Solution ◽  
Approximation Algorithm ◽  
Critical Point ◽  
Numerical Experiments ◽  
Optimization Problems ◽  
Local Minimizer ◽  
Quadratic Functions ◽  
Computational Time ◽  
Computational Results ◽  
Average Computational Time

Abstract This paper describes an approximation algorithm for solving standard quadratic optimization problems(StQPs) over the standard simplex by using fuzzification technique. We show that the approximate solution of the algorithm is an epsilon -critical point and satisfies epsilon-delta condition. The algorithm is compared with IBM ILOG CPLEX (short for CPLEX). The computational results indicate that the new algorithm is faster than CPLEX. Especially for infeasible problems. Furthermore, we calculate 100 instances for different size StQP problems. The numerical experiments show that the average computational time of the new algorithm for calculating the first local minimizer is in BigO(n) when the size of the problems is less or equal to 450.

Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth

2003 ◽  
Vol 133 (6) ◽  
pp. 1249-1262 ◽  
Author(s):  
Menita Carozza ◽  
Antonia Passarelli di Napoli
Keyword(s):  
Partial Regularity ◽  
Local Minimizer ◽  
Quadratic Growth ◽  
Local Minimizers

We prove partial regularity for local minimizers of quasiconvex integrals of the form I(v) = ∫ΩF(Dv(x))dx, where the integrand f(ξ) has sub-quadratic growth, i.e. |F(ξ)| ≤ L(1 + |ξ|p), with 1 < p < 2. A function u ∈ W1,p(Ω;RN) is a W1,q(Ω;RN) local minimizer of I(v) if there exists a δ > 0 such that I(u) ≤ I(v) whenever v and ‖Dv − Du‖q ≤ δ.

A more conclusive and more inclusive second derivative test

2020 ◽  
Vol 104 (560) ◽  
pp. 247-254
Author(s):  
Ronald Skurnick ◽  
Christopher Roethel
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Local Maximum ◽  
Number Line ◽  
Second Derivative ◽  
First Derivative ◽  
Line Test ◽  
First Derivative Test ◽  
Derivative Test

Given a differentiable function f with argument x, its critical points are those values of x, if any, in its domain for which either f′ (x) = 0 or f′ (x) is undefined. The first derivative test is a number line test that tells us, definitively, whether a given critical point, x = c, of f(x) is a local maximum, a local minimum, or neither. The second derivative test is not a number line test, but can also be applied to classify the critical points of f(x). Unfortunately, the second derivative test is, under certain conditions, inconclusive.

scholarly journals Gap Phenomenon of an Abstract Willmore Type Functional of Hypersurface in Unit Sphere

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanqi Zhu ◽  
Jin Liu ◽  
Guohua Wu
Keyword(s):  
Critical Point ◽  
Unit Sphere ◽  
Variational Equation ◽  
Integral Formula ◽  
Willmore Functional ◽  
Type Integral

For ann-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type calledWn,F-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called theWn,F-Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples ofWn,F-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.

Local minimality of the ball for the Gaussian perimeter

2019 ◽  
Vol 12 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Domenico Angelo La Manna
Keyword(s):  
Local Minimizer ◽  
Small Radius ◽  
Second Variation ◽  
Local Minimizers ◽  
Local Minimality ◽  
The Second Variation

AbstractWe prove that balls centered at the origin and with small radius are stable local minimizers of the Gaussian perimeter among all symmetric sets. Precisely, using the second variation of the Gaussian perimeter, we show that if the radius is smaller than{\sqrt{n+1}}, then the ball is a local minimizer, while if it is larger, the ball is not a local minimizer.

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