The semi-infinite strip x≥0, −1≤y≤1; completeness of the Papkovich-Fadle eigenfunctions when Φ_xx(0,y), Φ_yy(0,y) are prescribed

Abstract

For the problem of bending of a semi-infinite strip x≥0, −1≤y≤1, with the sides y=±1 clamped, we give a proof that the end-data

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaarmWu51MyVXgaiuGacqWFgpGzdaWgaaWcbaGaaeiEaiaabIha% aeqaaGqbaOGae4hiaaIaaiikaiaaicdacaGGSaGae4hiaaIaamyEai% aacMcacqGFGaaicqGH9aqpcqGFGaaicaWGMbGaaiikaiaadMhacaGG% PaGaaiilaaqaaiab-z8aMnaaBaaaleaacaqG5bGaaeyEaaqabaGccq% GFGaaicaGGOaGaaGimaiaacYcacqGFGaaicaWG5bGaaiykaiab+bca% Giabg2da9iab+bcaGiaadAgacaGGOaGaamyEaiaacMcacaGGSaaaaa% a!5D6D!\[\begin{array}{l} \phi _{{\rm{xx}}} (0, y) = f(y), \\ \phi _{{\rm{yy}}} (0, y) = f(y), \\ \end{array}\] where f(y), g(y) are ‘arbitrary’ independent functions prescribed on (−1,1), may be expanded as a series of the bi-orthogonal Papkovich-Fadle eigenfunctions for the strip. This represents an advance on the standard work of R. T. C. Smith [6], who proved such an expansion, but under conditions which are often not satisfied in practice. In particular we are able to solve this bi-harmonic boundary value problem when f, g do not satisfy the side conditions

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaacaWGMbGaaiikaiabgglaXkaaigdacaGGPaqedmvETj2BSbac% faGae8hiaaIaeyypa0Jae8hiaaIaamOzamaaCaaaleqabaGaai4jaa% aakiab-bcaGiaacIcacqGHXcqScaaIXaGaaiykaiab-bcaGiabg2da% 9iab-bcaGiaaicdacaGGSaaabaGaam4zaiaacIcacqGHXcqScaaIXa% Gaaiykaiab-bcaGiabg2da9iab-bcaGiaadEgadaahaaWcbeqaaiaa% cEcaaaGccqWFGaaicaGGOaGaeyySaeRaaGymaiaacMcacqWFGaaicq% GH9aqpcqWFGaaicaaIWaGaaiilaaaaaa!6222!\[\begin{array}{l} f( \pm 1) = f^' ( \pm 1) = 0, \\ g( \pm 1) = g^' ( \pm 1) = 0, \\ \end{array}\]and when the ‘conditions of consistency’

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qmaeaacaWGNbGaaiikaiaadMhacaGGPaqedmvETj2BSbacfaGa% e8hiaaIaamizaiaadMhacqWFGaaicqWF9aqpcqWFGaaidaWdXaqaai% aadMhacaWGNbGaaiikaiaadMhacaGGPaGae8hiaaIaamizaiaadMha% cqWFGaaicqGH9aqpcqWFGaaicaaIWaaaleaacqWFsislcqWFXaqmae% aacqWFXaqma0Gaey4kIipaaSqaaiabgkHiTiaaigdaaeaacaaIXaaa% niabgUIiYdaaaa!5A1B!\[\int_{ - 1}^1 {g(y) dy = \int_{ - 1}^1 {yg(y) dy = 0} } \]are not satisfied.

The present completeness proof thus answers questions raised recently (in the mathematically equivalent context of Stokes flow) by Joseph [3], and Joseph and Sturges [5], who showed that if the side conditions (A), (B) are relaxed then the corresponding eigenfunction series may still converge; but they left open the more difficult question of whether these series still converge to the data.

The method of proof used here also succeeds in proving a corresponding completeness theorem for the Williams eigenfunctions for the wedge with the data.

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaadaabciqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaa% daqadiqaamaalaaabaGaaGymaaqaaiaadkhaaaqedmvETj2BSbacfi% Gae8NXdygacaGLOaGaayzkaaaacaGLiWoadaWgaaWcbaGaamOCaiab% g2da9iaaigdaaeqaaGqbaOGae4hiaaIaeyypa0Jae4hiaaIaamOzai% aacIcacqaH4oqCcaGGPaGaaiilaaqaamaaeiGabaWaaSaaaeaacqGH% ciITdaahaaWcbeqaaiaaikdaaaGccqaHgpGzaeaacqGHciITcqaH4o% qCdaahaaWcbeqaaiaaikdaaaaaaOWaaeWaceaadaWcaaqaaiaaigda% aeaacaWGYbaaaiab-z8aMbGaayjkaiaawMcaaaGaayjcSdWaaSbaaS% qaaiaadkhacqGH9aqpcaaIXaaabeaakiab+bcaGiabg2da9iab+bca% GiaadEgacaGGOaGaeqiUdeNaaiykaiaacYcaaaaa!6B9C!\[\begin{array}{l} \left. {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = f(\theta ), \\ \left. {\frac{{\partial ^2 \phi }}{{\partial \theta ^2 }}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = g(\theta ), \\ \end{array}\]prescribed on −α<θ<α, (where 2α is the wedge angle).