WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 24, 2025
On Separability of Non-Linear Tri-Harmonic Operators with Matrix Potentials
Authors: , ,
Abstract: In this research, we explore the separability of the non-linear tri-harmonic form operator: in the case of $$A[u]=\Delta^{3} u(x)+V(x, u(x)) u(x), x \in \mathbb{R}^{\mathrm{n}}$$, in the space $$L_{2}\left(\mathbb{R}^{\mathrm{n}}\right)^{l}$$ with the operator potential $$V(x, u(x)) \in$$ $$L_{2}\left(\mathbb{R}^{\mathrm{n}}\right)^{l}$$ for every $$x \in \mathbb{R}^{\mathrm{n}}$$, where $$\Delta=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}{ }^{2}}$$ is the Laplace operator in $$\mathbb{R}^{\mathrm{n}}$$. That is the coercive inequality $$\left\|\Delta^{3} u\right\|+\|V u\|+\left\|V^{\frac{1}{2}} \Delta^{2} u\right\|+\sum_{i=1}^{n}\left\|V^{\frac{1}{2}} \frac{\partial^{2} u}{\partial x_{i}^{2}}\right\| \leq N\|A[u]\|$$, holds true.
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Keywords: Tri-harmonic operators, Coercive inequalities, Separability, Non-linearity, Matrix potentials
Pages: 44-50
DOI: 10.37394/23206.2025.24.6