Characterizing Existence of Minimizers and Optimality to Nonconvex Quadratic Integrals

2021 ◽  
Author(s):  
Fabián Flores-Bazán ◽  
Luis González-Valencia
Keyword(s):  
Existence Of Minimizers

Existence of Minimizers on Drops

2013 ◽  
Vol 23 (2) ◽  
pp. 1154-1166 ◽  
Author(s):  
R. Correa ◽  
P. Gajardo ◽  
L. Thibault ◽  
D. Zagrodny
Keyword(s):  
Existence Of Minimizers

scholarly journals Existence and nonexistence in the liquid drop model

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Rupert L. Frank ◽  
Phan Thành Nam
Keyword(s):  
Liquid Drop ◽  
Riesz Potential ◽  
Heavy Nuclei ◽  
Liquid Drop Model ◽  
Optimal Range ◽  
Nonexistence Result ◽  
Existence Of Minimizers ◽  
Existence And Nonexistence

AbstractWe revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

scholarly journals Symmetric double bubbles in the Grushin plane

2019 ◽  
Vol 25 ◽  
pp. 77
Author(s):  
Valentina Franceschi ◽  
Giorgio Stefani
Keyword(s):  
Direct Method ◽  
Regularity Theory ◽  
Contact Interface ◽  
Double Bubble ◽  
Grushin Plane ◽  
Existence Of Minimizers ◽  
Double Bubbles ◽  
The Given ◽  
Suitable Change ◽  
Horizontal Interface

We address the double bubble problem for the anisotropic Grushin perimeter Pα, α ≥ 0, and the Lebesgue measure in ℝ2, in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in Foisy et al. [Pacific J. Math. 159 (1993) 47–59], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in Monti and Morbidelli [J. Geom. Anal. 14 (2004) 355–368], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

Minimizing the Lp norm of the curl of vector fields in a multi-connected domain

2017 ◽  
Vol 28 (01) ◽  
pp. 1750004
Author(s):  
Junichi Aramaki
Keyword(s):  
Connected Domain ◽  
Vector Fields ◽  
Three Dimensional ◽  
Tangential Component ◽  
Lp Norm ◽  
Existence Of Minimizers

We study the problem of minimizing the [Formula: see text] norm of the curl of vector fields in a three-dimensional, bounded multi-connected domain with a prescribed tangential component on the boundary. We then prove the existence of minimizers and estimate them.

scholarly journals The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers

2015 ◽  
Vol 66 (4) ◽  
pp. 1671-1693 ◽  
Author(s):  
Patrizio Neff ◽  
Johannes Lankeit ◽  
Ionel-Dumitrel Ghiba ◽  
Robert Martin ◽  
David Steigmann
Keyword(s):  
Strain Energy ◽  
Logarithmic Strain ◽  
Energy Part ◽  
Existence Of Minimizers

scholarly journals DIRICHLET PROBLEMS FOR THE 1-LAPLACE OPERATOR, INCLUDING THE EIGENVALUE PROBLEM

2007 ◽  
Vol 09 (04) ◽  
pp. 515-543 ◽  
Author(s):  
BERND KAWOHL ◽  
FRIEDEMANN SCHURICHT
Keyword(s):  
Eigenvalue Problem ◽  
Laplace Operator ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Dirichlet Boundary Conditions ◽  
Necessary Condition ◽  
Dirichlet Problems ◽  
Lagrange Equations ◽  
Existence Of Minimizers ◽  
Formal Limit

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to [Formula: see text], we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed [Formula: see text]-norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler–Lagrange equation as necessary condition for minimizers. Here the degenerate expression Du/|Du| has to be replaced by a suitable vector field [Formula: see text] to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler–Lagrange equations have to be satisfied.

scholarly journals Existence of Minimizers for NonLevel Convex Supremal Functionals

2014 ◽  
Vol 52 (5) ◽  
pp. 3341-3370 ◽  
Author(s):  
Ana Margarida Ribeiro ◽  
Elvira Zappale
Keyword(s):  
Supremal Functionals ◽  
Existence Of Minimizers
Sign in / Sign up

Export Citation Format

Share Document