The traction boundary value problem for the elastostatic semi-infinite strip; existence of solution, and completeness of the Papkovich-Fadle eigenfunctions

The semi-infinite strip x≥0, −1≤y≤1 is in equilibrium under no body forces, with the sides y=±1, x>0 free of tractions, and on the end x=0% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaadIhacaWG4baabeaakiaacIcacaaIWaGaaiilaiaadMha% caGGPaGaeyypa0JaamOzaiaacIcacaWG5bGaaiykaiaacYcaaaa!4298!\[\sigma _{xx} (0,y) = f(y),\]% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS% baaSqaaiaadIhacaWG4baabeaakiaacIcacaaIWaGaaiilaiaadMha% caGGPaGaeyypa0Jaam4zaiaacIcacaWG5bGaaiykaiaacYcaaaa!4299!\[\sigma _{xx} (0,y) = g(y),\]where f(y), g(y) are independent, self-equilibrating tractions prescribed for y∈[−1, 1]. A rigorous proof is given that if f″, g″ are of bounded variation on [−1, 1], then this traction boundary value problem posesses a solution, and the stress field of this solution may be expanded, even onx=0, as a convergent series of the Papkovich-Fadle eigenfunctions. Thus these eigenfunctions are complete for the expansion of such data {f, g}.

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