The Dirichlet problem for the biharmonic equation in a Lipschitz domain

Björn E. J. Dahlberg; C. E. Kenig; G. C. Verchota

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 3, page 109-135
  • ISSN: 0373-0956

Abstract

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In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator Δ 2 , on an arbitrary bounded Lipschitz domain D in R n . We establish existence and uniqueness results when the boundary values have first derivatives in L 2 ( ∂ D ) , and the normal derivative is in L 2 ( ∂ D ) . The resulting solution u takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of ∇ u is shown to be in L 2 ( ∂ D ) .

How to cite

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Dahlberg, Björn E. J., Kenig, C. E., and Verchota, G. C.. "The Dirichlet problem for the biharmonic equation in a Lipschitz domain." Annales de l'institut Fourier 36.3 (1986): 109-135. <http://eudml.org/doc/74720>.

@article{Dahlberg1986,
abstract = {In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator $\Delta ^2$, on an arbitrary bounded Lipschitz domain $D$ in $\{\bf R\}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2(\partial D)$, and the normal derivative is in $L^2(\partial D)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2(\partial D)$.},
author = {Dahlberg, Björn E. J., Kenig, C. E., Verchota, G. C.},
journal = {Annales de l'institut Fourier},
keywords = {optimal estimates; Dirichlet problem; biharmonic operator; Lipschitz domain; non-tangential convergence},
language = {eng},
number = {3},
pages = {109-135},
publisher = {Association des Annales de l'Institut Fourier},
title = {The Dirichlet problem for the biharmonic equation in a Lipschitz domain},
url = {http://eudml.org/doc/74720},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Dahlberg, Björn E. J.
AU - Kenig, C. E.
AU - Verchota, G. C.
TI - The Dirichlet problem for the biharmonic equation in a Lipschitz domain
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 3
SP - 109
EP - 135
AB - In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator $\Delta ^2$, on an arbitrary bounded Lipschitz domain $D$ in ${\bf R}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2(\partial D)$, and the normal derivative is in $L^2(\partial D)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2(\partial D)$.
LA - eng
KW - optimal estimates; Dirichlet problem; biharmonic operator; Lipschitz domain; non-tangential convergence
UR - http://eudml.org/doc/74720
ER -

References

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