Medea Tsaava

The Boundary Value Problems for the Bi-Laplace-Beltrami Equation

abstract:
The purpose of the present paper is to investigate the boundary value problems for the bi-Laplace-Beltrami equation $\Delta^2_{\mathcal{C}}{\varphi}=f$ on a smooth hypersurface $\mathcal{C}$ with the boundary $\Gamma={\partial}\mathcal{C}$. The unique solvability of the BVP is proved on the basis of Green's formula and Lax-Milgram Lemma.

We also prove the invertibility of the perturbed operator in the Bessel potential spaces $\Delta^2_{\mathcal{C}}+\mathcal{H}\,I:\mathbb{H}^{s+2}_p(\mathcal{S})\to \mathbb{H}^{s-2}_p(\mathcal{S})$ for a smooth closed hypersurface $\mathcal{S}$ without boundary for arbitrary $1<p<\infty$ and $-\infty<s<\infty$, provided $\mathcal{H}$ is a smooth function, has non-negative real part $\RRe\mathcal{H}(t)\geqslant0$ for all $t\in\mathcal{S}$ and non-trivial support $\mes\supp\RRe\mathcal{H}\neq 0$.

Mathematics Subject Classification: 35J40, 35M12

Key words and phrases: Bi-Laplace-Beltrami equation, Günter's tangential derivatives, boundary value problems, mixed boundary condition, Bessel potential spaces