Medea Tsaava
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