scholarly journals Optimization Related to Some Nonlocal Problems of Kirchhoff Type

2016 ◽  
Vol 68 (3) ◽  
pp. 521-540 ◽  
Author(s):  
Behrouz Emamizadeh ◽  
Amin Farjudian ◽  
Mohsen Zivari-Rezapour
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Minimization Problem ◽  
Optimization Problems ◽  
Nonlocal Boundary ◽  
Energy Functional ◽  
The Other ◽  
Maximization Problem ◽  
Kirchhoff Type

AbstractIn this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton, we are able to show that both problems are solvable and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions.The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type that is stable. Some numerical results are included to conûrm the theory.

scholarly journals On the supremal version of the Alt–Caffarelli minimization problem

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Graziano Crasta ◽  
Ilaria Fragalà
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Recent Work ◽  
Free Boundary Problem ◽  
Minimization Problem ◽  
Companion Paper ◽  
Infinity Laplacian ◽  
Link Type ◽  
Bernoulli Free Boundary

Abstract This is a companion paper to our recent work [Bernoulli free boundary problem for the infinity Laplacian, preprint 2018, https://arxiv.org/abs/1804.08573]. Here we consider its variational side, which corresponds to the supremal version of the Alt–Caffarelli minimization problem.

Uniqueness of solution to a free boundary problem from combustion with transport

MAT Serie A ◽  
2001 ◽  
Vol 5 ◽  
pp. 37-41
Author(s):  
Claudia Lederman ◽  
Juan Luis Vázquez ◽  
Noemí Wolanski
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Uniqueness Of Solution ◽  
A Free Boundary Problem

EXISTENCE OF PERIODIC SOLUTIONS FOR A FREE BOUNDARY PROBLEM OF HYPERBOLIC TYPE

2008 ◽  
Vol 05 (04) ◽  
pp. 785-806
Author(s):  
KAZUAKI NAKANE ◽  
TOMOKO SHINOHARA
Keyword(s):  
Mathematical Model ◽  
Free Boundary ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Physical Phenomenon ◽  
Hyperbolic Type ◽  
Existence Of Periodic Solutions ◽  
A Free Boundary Problem ◽  
Thin Tape

A free boundary problem that arises from the physical phenomenon of "peeling a thin tape from a domain" is treated. In this phenomenon, the movement of the tape is governed by a hyperbolic equation and is affected by the peeling front. We are interested in the behavior of the peeling front, especially, the phenomenon of self-excitation vibration. In the present paper, a mathematical model of this phenomenon is proposed. The cause of this vibration is discussed in terms of adhesion.

An evolutional free-boundary problem of a reaction–diffusion–advection system

2017 ◽  
Vol 147 (3) ◽  
pp. 615-648 ◽  
Author(s):  
Ling Zhou ◽  
Shan Zhang ◽  
Zuhan Liu
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Reaction Diffusion ◽  
Spreading Speed ◽  
Heterogeneous Environment ◽  
Long Run ◽  
Strong Competition ◽  
Asymptotic Spreading

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

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