0
$\begingroup$

Let $U_1, U_2\subset\mathbb R^3$ open and homeomorphic to the open unit ball, with sufficiently smooth boundary, such that $\overline{U}_1\subset U_2$.

Is is true that every biharmonic function $u$ in $U_1$ (i.e., $\Delta^2 u=0$), which extends continuously to $\overline{U}_1$, can be approximated by linear combinations of the form $$ c_0+\sum_{j=1}^n c_j |x-x_j|, $$ where $n\in\mathbb N$, $c_j\in\mathbb R$, and $x_j\in\partial U_2$. In particular, I am interested in approximation with respect to the norms of the spaces $C^\ell\big(\overline{U}_1\big)$.

Note that $\varphi(x)=-\frac{1}{8\pi}|x|$ is biharmonic function in $\mathbb R^3\smallsetminus\{0\}$ and a fundamental solution of $\Delta^2$, i.e., $\Delta^2\varphi=\delta$, in the sense of distributions.

NOTE. There is a correction: $c_0$ is added in the linear combination.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ The answer is "No" even for 2 balls. The easiest way to see it is to notice that both the value at the center and the integral over a sphere allow one to recover $\sum c_j$ but they are not related. The underlying reason is obvious: it is pretty hard to recover the normal derivative by the boundary difference quotients, which go tangentially. $\endgroup$
    – fedja
    Dec 19, 2013 at 14:26
  • $\begingroup$ See NOTE in the edited version. $\endgroup$
    – smyrlis
    Dec 19, 2013 at 18:58

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.