AN OPTIMIZATION PROBLEM RELATED TO THE BEST SOBOLEV TRACE CONSTANT IN THIN DOMAINS

2008 ◽  
Vol 10 (05) ◽  
pp. 633-650 ◽  
Author(s):  
JULIÁN FERNÁNDEZ BONDER ◽  
JULIO D. ROSSI ◽  
CAROLA-BIBIANE SCHÖNLIEB
Keyword(s):  
Eigenvalue Problem ◽  
Minimization Problem ◽  
Optimization Problem ◽  
Limit Problem ◽  
Thin Domains ◽  
Best Constant ◽  
Bounded Smooth Domain ◽  
Neumann Eigenvalue ◽  
Bounded Smooth ◽  
Sobolev Trace Embedding

Let Ω ⊂ ℝNbe a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) ↪ Lq(∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem [Formula: see text] for functions that verify u|A= 0. It is known that there exists an optimal hole that minimizes the best constant SAamong subsets of Ω of the prescribed volume.In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain.

scholarly journals THE BEST SOBOLEV TRACE CONSTANT IN PERIODIC MEDIA FOR CRITICAL AND SUBCRITICAL EXPONENTS

2009 ◽  
Vol 51 (3) ◽  
pp. 619-630
Author(s):  
JULIÁN FERNÁNDEZ BONDER ◽  
RAFAEL ORIVE ◽  
JULIO D. ROSSI
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Limit Problem ◽  
Neumann Boundary Conditions ◽  
Periodic Media ◽  
Best Constant ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Trace Embedding

AbstractIn this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

scholarly journals The best Sobolev trace constant in domains with holes for critical or subcritical exponents

2007 ◽  
Vol 49 (2) ◽  
pp. 213-230 ◽  
Author(s):  
J. Fernandezbonder ◽  
R. Orive ◽  
J. D. Rossi
Keyword(s):  
Critical Size ◽  
Limit Problem ◽  
Smooth Domain ◽  
Best Constant ◽  
Bounded Smooth Domain ◽  
Extra Term ◽  
Bounded Smooth ◽  
Sobolev Trace Embedding

AbstractIn this paper we study the best constant in the Sobolev trace embedding H1 (Ω) →Lq(∂Ω) in a bounded smooth domain for 1 < q < 2+ = 2(N - 1)/(N - 2), that is, critical or subcritical q. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary.

First Variations of the Best Sobolev Trace Constant with Respect to the Domain

2008 ◽  
Vol 51 (1) ◽  
pp. 140-145 ◽  
Author(s):  
Julio D. Rossi
Keyword(s):  
Smooth Domain ◽  
Best Constant ◽  
First Variation ◽  
Critical Domain ◽  
Bounded Smooth Domain ◽  
The First Variation ◽  
Bounded Smooth ◽  
Sobolev Trace Embedding

AbstractIn this paper we study the best constant of the Sobolev trace embedding H1(Ω) → L2(∂Ω), where Ω is a bounded smooth domain in ℝN. We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.

scholarly journals The eigenvalue problem for thep-Laplacian-like equations

2003 ◽  
Vol 2003 (9) ◽  
pp. 575-586 ◽  
Author(s):  
Zu-Chi Chen ◽  
Tao Luo
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Domain ◽  
Multiplicity Result ◽  
Bounded Smooth Domain ◽  
Capillary Phenomena ◽  
Bounded Smooth

We consider the eigenvalue problem for the followingp-Laplacian-like equation:−div(a(|Du|p)|Du|p−2Du)=λf(x,u)inΩ,u=0on∂Ω, whereΩ⊂ℝnis a bounded smooth domain. Whenλis small enough, a multiplicity result for eigenfunctions are obtained. Two examples from nonlinear quantized mechanics and capillary phenomena, respectively, are given for applications of the theorems.

MULTIPLE SIGN CHANGING SOLUTIONS OF A QUASILINEAR ELLIPTIC EIGENVALUE PROBLEM INVOLVING THE p-LAPLACIAN

2004 ◽  
Vol 06 (02) ◽  
pp. 245-258 ◽  
Author(s):  
THOMAS BARTSCH ◽  
ZHAOLI LIU
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Domain ◽  
Oscillatory Behaviour ◽  
Bounded Smooth Domain ◽  
Elliptic Eigenvalue Problem ◽  
Positive Multiple ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic

We consider the eigenvalue problem [Formula: see text] where Ω⊂ℝN is a bounded smooth domain and Δpu denotes the p-Laplacian, 1<p<+∞; λ>0 is a parameter. The nonlinearity f is required to have an oscillatory behaviour. We prove the existence of multiple positive, multiple negative, and in particular, of multiple sign changing solutions depending on λ.

scholarly journals Towers of Nodal Bubbles for the Bahri–Coron Problem in Punctured Domains

2018 ◽  
Vol 2019 (19) ◽  
pp. 5953-5974
Author(s):  
Mónica Clapp ◽  
Jorge Faya ◽  
Filomena Pacella
Keyword(s):  
Blow Up ◽  
Critical Sobolev Exponent ◽  
Limit Problem ◽  
Smooth Domain ◽  
Critical Problem ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Sign Changing Solutions

Abstract Let Ω be a bounded smooth domain in $\mathbb {R}^{N}$ which contains a ball of radius R centered at the origin, N ≥ 3. Under suitable symmetry assumptions, for each δ ∈ (0, R), we establish the existence of a sequence (um, δ) of nodal solutions to the critical problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u\text{ in }\Omega_{\delta}:=\{x\in\Omega :\left\vert x\right\vert>\delta\},\quad u=0\text{ on }\partial \Omega_{\delta},\nonumber\end{align*}$$ where $2^{\ast }:=\frac {2N}{N-2}$ is the critical Sobolev exponent. We show that, if Ω is strictly star-shaped then, for each $m\in \mathbb {N},$ the solutions um, δ concentrate and blow up at 0, as $\delta \rightarrow 0,$ and their limit profile is a tower of nodal bubbles, that is, it is a sum of rescaled nonradial sign-changing solutions to the limit problem $$\begin{align*}-\Delta u=|u|^{2^{\ast}-2}u, \quad u\in D^{1,2}(\mathbb{R}^{N}),\nonumber\end{align*}$$ centered at the origin.

Long-time dynamics of a class of nonlocal extensible beams with time delay

2021 ◽  
pp. 108128652110194
Author(s):  
Fengjuan Meng ◽  
Cuncai Liu ◽  
Chang Zhang
Keyword(s):  
Time Delay ◽  
Stability Property ◽  
Beam Equation ◽  
Exponential Attractors ◽  
Time Dynamics ◽  
Bounded Smooth Domain ◽  
Finite Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
Long Time Dynamics

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.

Further study of a fourth-order elliptic equation with negative exponent

2011 ◽  
Vol 141 (3) ◽  
pp. 537-549 ◽  
Author(s):  
Zongming Guo ◽  
Zhongyuan Liu
Keyword(s):  
Fourth Order ◽  
Blow Up ◽  
Smooth Domain ◽  
Regular Solutions ◽  
Order Problem ◽  
Bounded Smooth Domain ◽  
Main Technique ◽  
Fourth Order Elliptic Equation ◽  
Fourth Order Problem ◽  
Bounded Smooth

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

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