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

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F18%3A43951717" target="_blank" >RIV/49777513:23520/18:43951717 - isvavai.cz</a>
Result on the web
<a href="http://apcz.umk.pl/czasopisma/index.php/TMNA/article/view/TMNA.2017.055" target="_blank" >http://apcz.umk.pl/czasopisma/index.php/TMNA/article/view/TMNA.2017.055</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.12775/TMNA.2017.055" target="_blank" >10.12775/TMNA.2017.055</a>

Result language
angličtina
Original language name
On multiplicity of eigenvalues and symmetry of eigenfunctions of the p--Laplacian
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/LO1506" target="_blank" >LO1506: Sustainability support of the centre NTIS - New Technologies for the Information Society</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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