Recent work on sharp estimates in second-order elliptic unique continuation problems
This expository article concerns differential inequalities of the type \[ \left| \sum^ n_{i,j=1}a_{ij} \partial_ i \partial_ ju \right| \leq A | u |+B | \nabla u | \quad \text{in } \Omega, \tag{1} \] where \(\Omega\) is a domain in \(\mathbb{R}^ n\), \(A\) and \(B\) are functions on \(\Omega\), and \((a_{ij}(x))\) is a positive definite matrix which depends locally Lipschitz continuously on \(x \in \Omega\). Equation (1) is said to have the unique continuation property - UCP (resp. the strong unique continuation property -SUCP) if every solution \(u \in W^{2,2}_{\text{loc}}\) of (1) that vanishes on an open subset of \(\Omega\) (resp. that vanishes to infinite order at a point of \(\Omega)\) vanishes identically. The author shows: (1) has the UCP (resp. the SUCP) if \(A \in L^{n/2}_{\text{loc}}\), \(B\in L^ n_{\text{loc}}\) and \(n \geq 5\) (resp. if \(A\) and \(B\) are locally bounded). Moreover, for the case \(a_{ij}=\delta_{ij}\) (the Kronecker symbol) he proves more special results, and he discusses (by means of counterexamples) the question whether or not these results are sharp. The proofs of all ``positive'' results use Carleman inequalities.
scientific article; zbMATH DE number 513365
scientific article