zbMATH — the first resource for mathematics

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

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