Extension of solutions of elliptic equations from discrete sets. (English) Zbl 0820.35020

Bukhgeim, A. L.

J. Inverse Ill-Posed Probl. 1, No. 1, 17-32 (1993).

The author studies the following inequalities \[ | \partial_{\overline z} u | \leq a | u |, \quad | \Delta u | \leq a \bigl( | u | + | \partial u | \bigr), \quad | \overline {\partial}_ A u | \leq a | u |, \] where \(z \in \mathbb{C}\), \(| z | \leq 1\), \(\partial_{\overline z}\) is Cauchy- Riemann operator, \(\overline {\partial}_ A = \partial_{\overline z} - A \partial_ z\) is the Beltrami differential operator with operator coefficient \(A\). If for a sequence of points \(z_ j\), \(| z_ j | < 1\) it is known that \(| u(z_ j) | \leq \varepsilon\), \(j = 1,2, \dots,n\), then the upper bound for the norm of the solution \(u\) is obtained. The case \(\varepsilon = 0\), \(n \to \infty\) gives uniqueness theorems. The proofs are based on weighted a priori estimates of Carleman type.

Reviewer: M.Perelmuter (Kiev)

MSC:

35B60	Continuation and prolongation of solutions to PDEs
35J15	Second-order elliptic equations
35B45	A priori estimates in context of PDEs

Keywords:

Cauchy-Riemann operator; Beltrami differential operator; a priori estimates of Carleman