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Volume 29, Issue 1
Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

Elsayed M. E. Zayed

DOI: 10.4208/jpde.v29.n1.6

J. Part. Diff. Eq., 29 (2016), pp. 59-70.

Published online: 2016-03

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  • Abstract

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator

\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}

for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and

\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}

is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.


  • Keywords

Separation biharmonic Laplace-Beltrami operator operator potential Hilbert space L_2(R^n H_1) coercive estimate existence and uniqueness Theorem

  • AMS Subject Headings

47F05, 58J99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

e.m.e.zayed@hotmail.com (Elsayed M. E. Zayed)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-29-59, author = {Zayed , Elsayed M. E.}, title = {Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem}, journal = {Journal of Partial Differential Equations}, year = {2016}, volume = {29}, number = {1}, pages = {59--70}, abstract = {

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator

\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}

for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and

\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}

is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.


}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v29.n1.6}, url = {http://global-sci.org/intro/article_detail/jpde/5079.html} }
TY - JOUR T1 - Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem AU - Zayed , Elsayed M. E. JO - Journal of Partial Differential Equations VL - 1 SP - 59 EP - 70 PY - 2016 DA - 2016/03 SN - 29 DO - http://doi.org/10.4208/jpde.v29.n1.6 UR - https://global-sci.org/intro/article_detail/jpde/5079.html KW - Separation biharmonic Laplace-Beltrami operator KW - operator potential KW - Hilbert space L_2(R^n KW - H_1) KW - coercive estimate KW - existence and uniqueness Theorem AB -

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator

\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}

for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and

\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}

is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach.


Zayed , Elsayed M. E.. (2016). Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem. Journal of Partial Differential Equations. 29 (1). 59-70. doi:10.4208/jpde.v29.n1.6
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