scholarly journals Shape Perturbation of Grushin Eigenvalues

2021 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Paolo Luzzini ◽  
Paolo Musolino
Keyword(s):  
Symmetric Functions ◽  
Type Theorem ◽  
Dirichlet Boundary Conditions ◽  
Bifurcation Phenomenon ◽  
Multiple Eigenvalues ◽  
Shape Perturbation ◽  
Overdetermined Problems ◽  
Bounded Open Subset ◽  
Domain Perturbations ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractWe consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Zhao Liu ◽  
Wei Dai
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Type Theorem ◽  
Dirichlet Boundary Conditions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harmonic System

AbstractIn this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝwherewhereis the Green’s function in ℝ

scholarly journals Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions

2020 ◽  
Author(s):  
Denis Bonheure ◽  
Ederson Moreira dos Santos ◽  
Enea Parini ◽  
Hugo Tavares ◽  
Tobias Weth
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Symmetric Functions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Nodal Solution ◽  
Axially Symmetric ◽  
Nodal Solutions ◽  
Dirichlet Eigenvalue ◽  
Least Energy

Abstract We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen–Cahn type $f(s)=\lambda (s-|s|^{p-1}s)$ with $p>1$ and $\lambda>\lambda _2(\Omega )$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e., sign changing) solution and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega $ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen–Cahn type nonlinearities in case $\Omega $ is either a ball or a square. In particular, we give a complete description of the solution set for $\lambda \sim \lambda _2(\Omega )$, computing the Morse index of the solutions.

scholarly journals The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Peter Somora
Keyword(s):  
Lower Bound ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Root Functions ◽  
Number Of Zeros ◽  
Variational Solutions ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractWe consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.

scholarly journals Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Anyin Xia ◽  
Xianxiang Pu ◽  
Shan Li
Keyword(s):  
Porous Medium ◽  
Global Regularity ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Singularity Analysis ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Medium Equation ◽  
Regularity Results ◽  
Homogeneous Dirichlet Boundary Conditions

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5

2009 ◽  
Vol 11 (01) ◽  
pp. 59-69 ◽  
Author(s):  
PAOLO ROSELLI ◽  
MICHEL WILLEM
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
First Eigenvalue ◽  
Nodal Solutions ◽  
The First Eigenvalue ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Sign Changing Solutions ◽  
Nirenberg Problem ◽  
Least Energy

We prove the existence of (a pair of) least energy sign changing solutions of [Formula: see text] when Ω is a bounded domain in ℝN, N = 5 and λ is slightly smaller than λ1, the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions on Ω.

scholarly journals On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Olha P. Kupenko ◽  
Rosanna Manzo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Direct Method ◽  
Dirichlet Boundary Conditions ◽  
Elliptic Variational Inequalities ◽  
The Matrix ◽  
Degenerate Elliptic ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Linear Degenerate

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.

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