On a frequency function approach to the unique continuation principle (Q435161)

The paper deals with the unique continuation principle for the equations NEWLINE\[NEWLINE \Delta u =0 \qquad \text{in } G \, , \qquad \qquad \Delta u = b(x) \cdot \nabla u \qquad \text{in } G \, . NEWLINE\]NEWLINE The authors prove: Let \(G\) be an open and connected subset of \({\mathbb R}^n\), \(n \geqslant 2\). If \(u\) is a solution, belonging to \(C^2(G)\) of one of the equations above and \(u = 0\) in \(D \subset G\), then \(u = 0\) in \(G\).NEWLINENEWLINEThese results are already known, also for more general equations, but the goal of the authors is to provide a more straightforward treatment for the classical proof. The technique makes use of the so called Almgren's frequency function or of some of its modifications. In the last section the authors show the same result for the \(p\)-Laplace equation under some assumptions about a possible extension of the frequency function to the nonlinear case. They suppose this frequency function to be locally bounded, but local boundedness for this function seems to be still not known.