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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.

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