Unique continuation for \(|\Delta u| \leq V|\nabla u|\) and related problems (Q1175975)

The following theorem is proved: Theorem. Assume \(V\in L_{loc}^ r(\mathbb{R}^ d)\), \(u\in W_{loc}^{2,p}\), \(|\Delta u|\leq V|\nabla u|\) and \(\lim_{R\to 0}R^{-N}\int_{| x|\leq R}|\nabla u|^{p'}=0\) for all \(N\). Then \(u\) is a constant. As usual, \(1/p+1/p'=1\), \(1/p-1/p'=1/r\). This interesting result is a slight improvement of a theorem of Y. M. Kim. On the other hand, this paper is of interest from technical point of view as a variant of the Carleman method is proposed here and a partly new approach to proving the Carleman (i.e. weighted Sobolev) inequalities is developed.