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.

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.

A C1,α$C^{1,\alpha}$ partial regularity result for integral functionals with p⁢(x)$p(x)$-growth condition

2016 ◽  
Vol 9 (4) ◽  
pp. 395-407 ◽  
Author(s):  
Flavia Giannetti
Keyword(s):  
Growth Condition ◽  
Partial Regularity ◽  
Regularity Result ◽  
Integral Functionals ◽  
Local Minimizers ◽  
Zygmund Class ◽  
Partial Regularity Result

AbstractWe establish${C^{1,\alpha}}$partial regularity for the local minimizers of integral functionals of the type$\mathcal{F}(u;\Omega):=\int_{\Omega}(1+|Du|^{2})^{\frac{p(x)}{2}}\,dx,$where the gradient of the exponent function${p(\,\cdot\,)\geq 2}$belongs to a suitable Orlicz–Zygmund class.

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.

Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth

2009 ◽  
Vol 139 (3) ◽  
pp. 595-621 ◽  
Author(s):  
Sabine Schemm ◽  
Thomas Schmidt
Keyword(s):  
Calculus Of Variations ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Local Minimizers ◽  
Image Position

We consider strictly quasiconvex integralsin the multi-dimensional calculus of variations. For the C2-integrand f : ℝNn → ℝ we impose (p, q)-growth conditionswith γ, Γ > 0 and 1 < p ≤ q < min {p + 1/n, p(2n − 1)/(2n − 2)}. Under these assumptions we prove partial C1, αloc-regularity for strong local minimizers of F and the associated relaxed functional F.

Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations

2001 ◽  
Vol 131 (1) ◽  
pp. 155-184 ◽  
Author(s):  
Ali Taheri
Keyword(s):  
Field Theory ◽  
Calculus Of Variations ◽  
Sufficient Conditions ◽  
Local Minimizer ◽  
Local Minimizers ◽  
Indirect Approach ◽  
The Subject ◽  
Multiple Integrals ◽  
Image Position ◽  
The Right

Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functional over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.

scholarly journals Global Search Strategies for Solving Multilinear Least-Squares Problems

2012 ◽  
Vol 16 ◽  
pp. 12 ◽  
Author(s):  
Mats Andersson ◽  
Oleg Burdakov ◽  
Hans Knutsson ◽  
Spartak Zikrin
Keyword(s):  
Least Squares ◽  
Large Scale ◽  
Search Strategy ◽  
Local Minimizer ◽  
Global Search ◽  
Local Minimizers ◽  
Least Squares Problems ◽  
Linear Least Squares Problem ◽  
The Difference ◽  
Matrix Vector

The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by the results of numerical experiments performed for some problems related to the design of filter networks.  

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