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

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.

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-2008-016-5 ◽

2008 ◽

Vol 51 (1) ◽

pp. 140-145 ◽

Cited By ~ 7

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.

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199708002922 ◽

2008 ◽

Vol 10 (05) ◽

pp. 633-650 ◽

Cited By ~ 1

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.

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

Glasgow Mathematical Journal ◽

10.1017/s0017089509990048 ◽

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).

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

International Mathematics Research Notices ◽

10.1093/imrn/rnx289 ◽

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.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210509001061 ◽

2011 ◽

Vol 141 (3) ◽

pp. 537-549 ◽

Cited By ~ 2

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 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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A semilinear problem with a gradient term in the nonlinearity

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2021110 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Ignacio Guerra

Keyword(s):

Unit Ball ◽

Radial Solution ◽

The Other ◽

Smooth Domain ◽

Gradient Term ◽

Radial Solutions ◽

Bounded Smooth Domain ◽

Other Hand ◽

Bounded Smooth ◽

Semilinear Problem

We consider the following semilinear problem with a gradient term in the nonlinearity<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u>0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0<\lambda<\tilde \lambda $\end{document}</tex-math></inline-formula>.

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Optimal global asymptotic behaviour of the solution to a class of singular Dirichlet problems

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.52 ◽

2020 ◽

pp. 1-19

Author(s):

Zhijun Zhang

Keyword(s):

Asymptotic Behaviour ◽

Unique Solution ◽

Blow Up ◽

Critical Values ◽

Smooth Domain ◽

Dirichlet Problems ◽

Bounded Smooth Domain ◽

Bounded Smooth

This paper is mainly concerned with the global asymptotic behaviour of the unique solution to a class of singular Dirichlet problems − Δu = b(x)g(u), u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded smooth domain in ℝ n , g ∈ C1(0, ∞) is positive and decreasing in (0, ∞) with $\lim _{s\rightarrow 0^+}g(s)=\infty$ , b ∈ Cα(Ω) for some α ∈ (0, 1), which is positive in Ω, but may vanish or blow up on the boundary properly. Moreover, we reveal the asymptotic behaviour of such a solution when the parameters on b tend to the corresponding critical values.

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A strange bounded smooth domain of holomorphy

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1976-13964-x ◽

1976 ◽

Vol 82 (1) ◽

pp. 74-77 ◽

Cited By ~ 9

Author(s):

Klas Diederich ◽

John Erik Fornaess

Keyword(s):

Smooth Domain ◽

Bounded Smooth Domain ◽

Domain Of Holomorphy ◽

Bounded Smooth

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HARDY-SOBOLEV INEQUALITY IN H1(Ω) AND ITS APPLICATIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000713 ◽

2002 ◽

Vol 04 (03) ◽

pp. 409-434 ◽

Cited By ~ 19

Author(s):

ADIMURTHI

Keyword(s):

Boundary Condition ◽

Sobolev Inequality ◽

Neumann Boundary Condition ◽

Remainder Term ◽

Blow Up ◽

Neumann Boundary ◽

Smooth Domain ◽

Bounded Smooth Domain ◽

Eigen Value ◽

Bounded Smooth

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H1(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

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