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

Abstract

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In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator<\/p>

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

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<\/p>

\\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*}<\/p>

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.<\/p>

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