On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian

Kód výsledku v IS VaVaI

RIV/49777513:23520/18:43951717 - isvavai.cz

Výsledek na webu

http://apcz.umk.pl/czasopisma/index.php/TMNA/article/view/TMNA.2017.055

DOI - Digital Object Identifier

10.12775/TMNA.2017.055

Jazyk výsledku

angličtina

Název v původním jazyce

On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian

Popis výsledku v původním jazyce

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Omega subset R^N$. By means of topological arguments, we show how symmetries of $Omega$ help to construct subsets of $W_0^{1,p}(Omega)$ with suitably high Krasnosel'skiu{i} genus. In particular, if $Omega$ is a ball $B subset mathbb{R}^N$, we obtain the following chain of inequalities: $$ lambda_2(p;B) leq dots leq lambda_{N+1}(p;B) leq lambda_ominus(p;B). $$ Here $lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $lambda_ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $lambda_2(p;B)=lambda_ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=infty$ are also considered.

Název v anglickém jazyce

On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian

Popis výsledku anglicky

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Omega subset R^N$. By means of topological arguments, we show how symmetries of $Omega$ help to construct subsets of $W_0^{1,p}(Omega)$ with suitably high Krasnosel'skiu{i} genus. In particular, if $Omega$ is a ball $B subset mathbb{R}^N$, we obtain the following chain of inequalities: $$ lambda_2(p;B) leq dots leq lambda_{N+1}(p;B) leq lambda_ominus(p;B). $$ Here $lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $lambda_ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $lambda_2(p;B)=lambda_ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=infty$ are also considered.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

LO1506: Podpora udržitelnosti centra NTIS - Nové technologie pro informační společnost

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Topological Methods in Nonlinear Analysis

ISSN

1230-3429

e-ISSN

—

Svazek periodika

51

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

18

Strana od-do

565-582

Kód UT WoS článku

000441425700011

EID výsledku v databázi Scopus

—