Rearrangements of functions on unbounded domains

1994 ◽  
Vol 124 (4) ◽  
pp. 621-644 ◽  
Author(s):  
R. J. Douglas
Keyword(s):  
Variational Problem ◽  
Unbounded Domain ◽  
Extreme Points ◽  
Unbounded Domains ◽  
Compact Set ◽  
Weakly Compact ◽  
Weak Closure ◽  
Weakly Compact Set ◽  
Steady Vortices

A characterisation is provided for the weak closure of the set of rearrangements of a function on an unbounded domain. The extreme points of this convex, weakly compact set are classified. This result is used to study the maximising sequences of a variational problem for steady vortices.

scholarly journals Some fixed point theorems on non-convex sets

2017 ◽  
Vol 18 (2) ◽  
pp. 377
Author(s):  
Mohanasundaram Radhakrishnan ◽  
S. Rajesh ◽  
Sushama Agrawal
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Convex Sets ◽  
Compact Set ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Weakly Compact Set

<span style="color: #000000;">In this paper, we prove that if </span><span style="color: #008000;">$K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonempty</span><span style="color: #000000;"> weakly compact set in a </span><span style="text-decoration: underline; color: #000000;">Banach</span><span style="color: #000000;"> space </span><span style="color: #008000;">$X$</span><span style="color: #000000;">, </span><span style="color: #008000;">$T:K\to K$</span><span style="color: #000000;"> is a </span><span style="text-decoration: underline; color: #000000;">nonexpansive</span><span style="color: #000000;"> map satisfying </span><span style="color: #008000;">$\frac{x+Tx}{2}\in K$</span><span style="color: #000000;"> for all </span><span style="color: #008000;">$x\in K$</span><span style="color: #000000;"> and if </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> is </span><span style="color: #008000;">$3-$</span><span style="color: #000000;">uniformly convex or </span><span style="color: #008000;">$X$</span><span style="color: #000000;"> has the </span><span style="text-decoration: underline; color: #000000;">Opial</span><span style="color: #000000;"> property, then </span><span style="color: #008000;">$T$</span><span style="color: #000000;"> has a fixed point in </span><span style="color: #008000;">$K.$ <br /></span>

Every weakly compact set can be uniformly embedded into a reflexive Banach space

2009 ◽  
Vol 25 (7) ◽  
pp. 1109-1112 ◽  
Author(s):  
Li Xin Cheng ◽  
Qing Jin Cheng ◽  
Zheng Hua Luo ◽  
Wen Zhang
Keyword(s):  
Banach Space ◽  
Reflexive Banach Space ◽  
Compact Set ◽  
Weakly Compact ◽  
Weakly Compact Set

Monotonicity and symmetry of positive solutions to fractional p-Laplacian equation

2021 ◽  
pp. 2150005
Author(s):  
Wei Dai ◽  
Zhao Liu ◽  
Pengyan Wang
Keyword(s):  
Dirichlet Problem ◽  
Nonlinear Equations ◽  
Positive Solutions ◽  
Unbounded Domain ◽  
Fractional Laplacian ◽  
Unbounded Domains ◽  
Laplacian Operator ◽  
Maximum Principles ◽  
Laplacian Equation ◽  
Laplacian Equations

In this paper, we are concerned with the following Dirichlet problem for nonlinear equations involving the fractional [Formula: see text]-Laplacian: [Formula: see text] where [Formula: see text] is a bounded or an unbounded domain which is convex in [Formula: see text]-direction, and [Formula: see text] is the fractional [Formula: see text]-Laplacian operator defined by [Formula: see text] Under some mild assumptions on the nonlinearity [Formula: see text], we establish the monotonicity and symmetry of positive solutions to the nonlinear equations involving the fractional [Formula: see text]-Laplacian in both bounded and unbounded domains. Our results are extensions of Chen and Li [Maximum principles for the fractional p-Laplacian and symmetry of solutions, Adv. Math. 335 (2018) 735–758] and Cheng et al. [The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017) 1750018].

Least energy solutions for elliptic equations in unbounded domains

1996 ◽  
Vol 126 (1) ◽  
pp. 195-208 ◽  
Author(s):  
Manuel A. del Pino ◽  
Patricio L. Felmer
Keyword(s):  
Critical Point ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Asymptotic Results ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Image Position ◽  
Least Energy Solutions ◽  
Least Energy

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the formwhere Ω is an unbounded domain in RN and f is a C1 function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

scholarly journals Exponential decay for the damped wave equation in unbounded domains

2016 ◽  
Vol 18 (06) ◽  
pp. 1650012 ◽  
Author(s):  
Nicolas Burq ◽  
Romain Joly
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Exponential Decay ◽  
Unbounded Domain ◽  
Unbounded Domains ◽  
Geometric Control ◽  
Damped Wave Equation ◽  
Logarithmic Decay ◽  
And Control

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

scholarly journals On the Mixed Dirichlet--Farwig biharmonic problem in exterior domains

2019 ◽  
Vol 16 ◽  
pp. 8322-8329
Author(s):  
Hovik A. Matevossian
Keyword(s):  
Elliptic Boundary ◽  
Unbounded Domains ◽  
Generalized Solutions ◽  
Asymptotic Behavior Of Solutions ◽  
Variation Principle ◽  
Compact Set ◽  
Exterior Domains ◽  
Dirichlet Integral ◽  
Elliptic Boundary Value ◽  
Biharmonic Problem

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Dirichlet--Farwig biharmonic problem in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight $|x|^a$. Admitting different boundary conditions, we used the variation principle and depending on the value of the parameter $a$, we obtained uniqueness (non-uniqueness) theorems of the problem or present exact formulas for the dimension of the space of solutions.

scholarly journals Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Salah Missaoui
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Unbounded Domain ◽  
Three Dimensional ◽  
Compact Set ◽  
Klein Gordon ◽  
Regularity Of The Attractor ◽  
Schrödinger System

<p style='text-indent:20px;'>The main goal of this paper is to study the asymptotic behavior of a coupled Klein-Gordon-Schrödinger system in three dimensional unbounded domain. We prove the existence of a global attractor of the systems of the nonlinear Klein-Gordon-Schrödinger (KGS) equations in <inline-formula><tex-math id="M2">\begin{document}$ H^1({\mathbb R}^3)\times H^1({\mathbb R}^3)\times L^2({\mathbb R}^3) $\end{document}</tex-math></inline-formula> and more particularly that this attractor is in fact a compact set of <inline-formula><tex-math id="M3">\begin{document}$ H^2({\mathbb R}^3)\times H^2({\mathbb R}^3)\times H^1({\mathbb R}^3) $\end{document}</tex-math></inline-formula>.</p>

(D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets

1980 ◽  
Vol 32 (3) ◽  
pp. 703-713 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Extreme Point ◽  
Type Theorem ◽  
Extreme Points ◽  
Compact Set ◽  
Dimensional Simplex ◽  
Starshaped Set ◽  
Starshaped Sets

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.

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