# Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle

Nachman Aronszajn; R. D. Brown; R. S. Butcher

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 3, page 49-89
- ISSN: 0373-0956

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topAronszajn, Nachman, Brown, R. D., and Butcher, R. S.. "Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle." Annales de l'institut Fourier 23.3 (1973): 49-89. <http://eudml.org/doc/74142>.

@article{Aronszajn1973,

abstract = {A technique is developed for constructing the solution of $\Delta ^2u=F$ in $R=\lbrace (x,y):\vert x\vert < a,\;\vert y\vert < b\rbrace $, subject to boundary conditions $u=\varphi $, $\{\partial u\over \partial n\}=\psi $ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2(R)$ onto the subspace $\{\bf H\}$ of square integrable functions harmonic in $\{\bf R\}$. This problem is solved by decomposition $\{\bf H\}$ into the closed direct (not orthogonal) sum of two subspaces $\{\bf H\}^\{(1)\},\{\bf H\}^\{(2)\}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^\{(1)\}$, $P^\{(2)\}$ of $L^2(R)$ onto $\{\bf H\}^\{(1)\}$, $\{\bf H\}^\{(2)\}$ respectively. The resulting construction yields an approximation technique with both a priori and a posteriori error bounds (the latter very precise). In a short appendix the numerical results are given of the application of the technique in some specific examples and the a posteriori error evaluated.},

author = {Aronszajn, Nachman, Brown, R. D., Butcher, R. S.},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {3},

pages = {49-89},

publisher = {Association des Annales de l'Institut Fourier},

title = {Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle},

url = {http://eudml.org/doc/74142},

volume = {23},

year = {1973},

}

TY - JOUR

AU - Aronszajn, Nachman

AU - Brown, R. D.

AU - Butcher, R. S.

TI - Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle

JO - Annales de l'institut Fourier

PY - 1973

PB - Association des Annales de l'Institut Fourier

VL - 23

IS - 3

SP - 49

EP - 89

AB - A technique is developed for constructing the solution of $\Delta ^2u=F$ in $R=\lbrace (x,y):\vert x\vert < a,\;\vert y\vert < b\rbrace $, subject to boundary conditions $u=\varphi $, ${\partial u\over \partial n}=\psi $ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2(R)$ onto the subspace ${\bf H}$ of square integrable functions harmonic in ${\bf R}$. This problem is solved by decomposition ${\bf H}$ into the closed direct (not orthogonal) sum of two subspaces ${\bf H}^{(1)},{\bf H}^{(2)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{(1)}$, $P^{(2)}$ of $L^2(R)$ onto ${\bf H}^{(1)}$, ${\bf H}^{(2)}$ respectively. The resulting construction yields an approximation technique with both a priori and a posteriori error bounds (the latter very precise). In a short appendix the numerical results are given of the application of the technique in some specific examples and the a posteriori error evaluated.

LA - eng

UR - http://eudml.org/doc/74142

ER -

## References

top- [1] R. ADAMS, N. ARONSZAJN, M. HANNA, Theory of Bessel Potentials, Part III, Ann. Inst. Fourier Grenoble, v. 19, 1969, 281-338. Zbl0176.09902MR54 #915
- [2] R. ADAMS, N. ARONSZAJN, and K.T. SMITH, Theory of Bessel Potentials, Part II, Ann. Inst. Fourier Grenoble, v. 17, 1967, 1-135. Zbl0185.19703MR37 #4281
- [3] N. ARONSZAJN, Recherches sur les fonctions harmoniques dans un carré, Journ. de Math. Pures. et Appl., Neuvième série, 27 (1948), 87-175. Zbl0033.18302MR10,116a
- [4] N. ARONSZAJN, Theory of reproducing kernels, Trans. Am. Math. Soc., 68 (1950), 337-404. Zbl0037.20701MR14,479c
- [5] N. ARONSZAJN, Some integral inequalities, Proceedings of the Symposium on Inequalities at Colorado Springs, 1967. Zbl0226.26018
- [6] N. ARONSZAJN and W.F. DONOGHUE, Variational approximation methods applied to the eigenvalues of a clamped rectangular plate, Part I, Technical Report 12, University of Kansas, 1954. Zbl0058.32902
- [7] N. ARONSZAJN and G.H. HARDY, Properties of a class of double integrals, Ann. of Math., 46 (1945), 220-241. Errata to this paper, Ann. of Math., 47 (1946), p. 166. Zbl0060.14202MR7,116b
- [8] N. ARONSZAJN and K.T. SMITH, Theory of Bessel Potentials, Part I, Ann. Inst. Fourier Grenoble, v. 11, 1961, 385-475. Zbl0102.32401MR26 #1485
- [9] N. ARONSZAJN and P. SZEPTYCKI, Theory of Bessel Potentials, Part IV, to appear as a technical report. Zbl0121.09604
- [10] S. TIMOSHENKO, Theory of Plates and Shells, McGraw-Hill, New York, 1959. JFM66.1049.02
- [11] S. ZAREMBA, Le problème biharmonique restreint, Annales Ec. Norm. Sup., 26, 1909. JFM40.0842.01

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