Estimate of the deformation of a strictly convex domain as a function of the change in the relative metric of its boundary

1991 ◽  
Vol 31 (5) ◽  
pp. 711-716
Author(s):  
V. A. Aleksandrov
Keyword(s):  
Convex Domain ◽  
Strictly Convex ◽  
Relative Metric

scholarly journals The Heisenberg group acts on a strictly convex domain

2017 ◽  
Vol 21 (4) ◽  
pp. 101-104 ◽  
Author(s):  
Daryl Cooper
Keyword(s):  
Heisenberg Group ◽  
Convex Domain ◽  
Strictly Convex

On the Characteristic function of a Strictly Convex Domain and the Fubini-Pick Invariant

1988 ◽  
Vol 13 (3-4) ◽  
pp. 367-378 ◽  
Author(s):  
Takeshi Sasaki
Keyword(s):  
Characteristic Function ◽  
Convex Domain ◽  
Strictly Convex ◽  
Pick Invariant

Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

1973 ◽  
Vol 74 (1) ◽  
pp. 107-116 ◽  
Author(s):  
Vishwa Chander Dumir ◽  
Dharam Singh Khassa
Keyword(s):  
Lower Bound ◽  
Lebesgue Measure ◽  
Convex Domain ◽  
Gauge Function ◽  
Symmetric Convex ◽  
Convex Domains ◽  
Strictly Convex ◽  
Point Set ◽  
Image Position

Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined bywhere V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured thatwhere ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

scholarly journals SOLUTIONS WITH MULTIPLE ALTERNATE SIGN PEAKS ALONG A BOUNDARY GEODESIC TO A SEMILINEAR DIRICHLET PROBLEM

2014 ◽  
Vol 16 (01) ◽  
pp. 1350020 ◽  
Author(s):  
TERESA D'APRILE ◽  
ANGELA PISTOIA
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Convex Domain ◽  
Closed Curve ◽  
Positive Parameter ◽  
Two Dimensional ◽  
Strictly Convex ◽  
Small Positive Parameter ◽  
Boundary Geodesic ◽  
Odd Nonlinearity

We study the existence of sign-changing multiple interior spike solutions for the following Dirichlet problem [Formula: see text] where Ω is a smooth and bounded domain of ℝN, ε is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. In particular we prove that if Ω has a plane of symmetry and its intersection with the plane is a two-dimensional strictly convex domain, then, provided that k is even and sufficiently large, a k-peak solution exists with alternate sign peaks aligned along a closed curve near a geodesic of ∂Ω.

scholarly journals Sectional Curvature of Projective Invariant Metrics on a Strictly Convex Domain

1996 ◽  
Vol 19 (2) ◽  
pp. 419-433 ◽  
Author(s):  
Takeshi SASAKI ◽  
Takeshi YAGI
Keyword(s):  
Sectional Curvature ◽  
Convex Domain ◽  
Invariant Metrics ◽  
Strictly Convex ◽  
Projective Invariant

scholarly journals Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

2021 ◽  
Vol 11 (1) ◽  
pp. 321-356
Author(s):  
Haitao Wan ◽  
Yongxiu Shi ◽  
Wei Liu
Keyword(s):  
Dirichlet Problem ◽  
Convex Domain ◽  
Boundary Behavior ◽  
Strictly Convex ◽  
Convex Solution ◽  
Regularly Varying ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

Abstract In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation  det ( D 2 u ) = b ( x ) g ( − u ) , u < 0  in  Ω  and  u = 0  on  ∂ Ω , $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N (N ≥ 2), b ∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is normalized regularly varying at zero with index −γ(γ>1). Our results reveal the refined influence of the highest and the lowest values of the (N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.

On stable CMC free-boundary surfaces in a strictly convex domain of a bi-invariant Lie group

2020 ◽  
Vol 31 (11) ◽  
pp. 2050086
Author(s):  
Ezequiel Barbosa ◽  
Farley Santana ◽  
Abhitosh Upadhyay
Keyword(s):  
Free Boundary ◽  
Convex Domain ◽  
Lie Group ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Boundary Surface ◽  
Connected Components ◽  
Geodesic Ball ◽  
Invariant Metric ◽  
Strictly Convex

Let [Formula: see text] be a three-dimensional Lie group with a bi-invariant metric. Consider [Formula: see text] a strictly convex domain in [Formula: see text]. We prove that if [Formula: see text] is a stable CMC free-boundary surface in [Formula: see text] then [Formula: see text] has genus either 0 or 1, and at most three boundary components. This result was proved by Nunes [I. Nunes, On stable constant mean curvature surfaces with free-boundary, Math. Z. 287(1–2) (2017) 73–479] for the case where [Formula: see text] and by R. Souam for the case where [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], excluding the possibility of [Formula: see text] having three boundary components. Besides [Formula: see text] and [Formula: see text], our result also apply to the spaces [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. When [Formula: see text] and [Formula: see text] is a geodesic ball with radius [Formula: see text], we obtain that if [Formula: see text] is stable then [Formula: see text] is a totally umbilical disc. In order to prove those results, we use an extended stability inequality and a modified Hersch type balancing argument to get a better control on the genus and on the number of connected components of the boundary of the surfaces.

scholarly journals Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior

2019 ◽  
Vol 9 (1) ◽  
pp. 729-744 ◽  
Author(s):  
Xuemei Zhang ◽  
Meiqiang Feng
Keyword(s):  
Asymptotic Behavior ◽  
Convex Domain ◽  
Regular Variation ◽  
Blow Up ◽  
Variation Theory ◽  
Type Condition ◽  
Strictly Convex ◽  
Convex Solution ◽  
Monge Ampere ◽  
Standard Techniques

Abstract Consider the boundary blow-up Monge-Ampère problem $$\begin{array}{} \displaystyle M[u]=K(x)f(u) \mbox{ for } x \in {\it\Omega},\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial {\it\Omega})\rightarrow 0. \end{array}$$ Here M[u] = det (uxixj) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in ℝN (N ≥ 2). Under K(x) satisfying appropriate conditions, we first prove that the boundary blow-up Monge-Ampère problem has a strictly convex solution if and only if f satisfies Keller-Osserman type condition. Then the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Finally, if f does not satisfy Keller-Osserman type condition, we show the existence of strictly convex solutions under different conditions on K(x). The proof combines standard techniques based upon the sub-supersolution method with non-standard arguments, such as the Karamata regular variation theory.

Estimating Large-Scale Tree Logit Models via a Difference of Strictly Convex Functions

2018 ◽  
Author(s):  
Srikanth Jagabathula ◽  
Paat Rusmevichientong
Keyword(s):  
Convex Functions ◽  
Large Scale ◽  
Logit Models ◽  
Strictly Convex
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