Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms
We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in D⊂Rn and u=0 on ∂D, is not bounded generally if one imposes the norm ∫Du2p(x)dx=1, p=(1),2,3,…. The same holds with the norm ∫D|gradu|2pdx=1 when n>4p−1. On the other hand, if D⊂R2, re...
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| Language: | English |
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Wiley
1979-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171279000296 |
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| author | Yves Biollay |
| author_facet | Yves Biollay |
| author_sort | Yves Biollay |
| collection | DOAJ |
| description | We show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in D⊂Rn and u=0 on ∂D, is not bounded generally if one imposes the norm ∫Du2p(x)dx=1, p=(1),2,3,…. The same holds with the norm ∫D|gradu|2pdx=1 when n>4p−1. On the other hand, if D⊂R2, resp. R3 the norm ∫D|gradu|2dx=1 implies max|uk|→k→∞0, resp. max|uk|=0(1). |
| format | Article |
| id | doaj-art-3f2ef543e3ab488590e1861869db60ad |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1979-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3f2ef543e3ab488590e1861869db60ad2025-02-03T07:25:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251979-01-012233734010.1155/S0161171279000296Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various normsYves Biollay0Department of Mathematics, Cornell University, Ithaca 14850, New York, USAWe show in this paper that the sequence {max|uk|}, where the uk are the eigenfunctions of the problem Δu+λu=0 in D⊂Rn and u=0 on ∂D, is not bounded generally if one imposes the norm ∫Du2p(x)dx=1, p=(1),2,3,…. The same holds with the norm ∫D|gradu|2pdx=1 when n>4p−1. On the other hand, if D⊂R2, resp. R3 the norm ∫D|gradu|2dx=1 implies max|uk|→k→∞0, resp. max|uk|=0(1).http://dx.doi.org/10.1155/S0161171279000296elliptic equationseigenvalue problemseigenfuntions. |
| spellingShingle | Yves Biollay Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms International Journal of Mathematics and Mathematical Sciences elliptic equations eigenvalue problems eigenfuntions. |
| title | Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms |
| title_full | Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms |
| title_fullStr | Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms |
| title_full_unstemmed | Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms |
| title_short | Equation of the fixed membrane in the n-dimensional space: some remarks on the maxima of the eigenfunctions subjected to various norms |
| title_sort | equation of the fixed membrane in the n dimensional space some remarks on the maxima of the eigenfunctions subjected to various norms |
| topic | elliptic equations eigenvalue problems eigenfuntions. |
| url | http://dx.doi.org/10.1155/S0161171279000296 |
| work_keys_str_mv | AT yvesbiollay equationofthefixedmembraneinthendimensionalspacesomeremarksonthemaximaoftheeigenfunctionssubjectedtovariousnorms |