Quantitative uniqueness for elliptic equations with singular lower order terms
The authors consider the equation \[ Pu=Vu+W_1\cdot\nabla u + \nabla\cdot(W_2u),\tag{1} \] where \(P=\mathrm{div}(g\nabla u)\), \(g(x)=\{g^{ij}(x)\}^n_{i,j=1}\) is a real-valued symmetric matrix that satisfies the uniform ellipticity condition in \(\Omega\): \[ \lambda|\xi|^2\leq g(x)\xi\cdot\xi\leq\lambda^{-1}|\xi|^2,\quad x\in\Omega,\xi\in\mathbb R^n \] for some \(\lambda\in (0,1]\). It is also assumed the Lipschitz condition \[ |g(x)-g(y)|\leq\Lambda_0|x-y|,\qquad x,y\in\Omega. \] Finally, the lower order terms satisfy the following integrability conditions: \[ V\in L^{n/2}(\Omega)\quad \text{and}\quad W_1,W_2\in L^s(\Omega)\;\text{with}\;s>n, \] where \(W_1,W_2:\Omega\to\mathbb R^n\) and \(V:\Omega\to \mathbb R\). Then, the main aim of the work is to obtain quantitative propagation of smallness from sets of positive measure for solutions of (1). More precisely, let \(\Omega(\rho)=\{x\in\Omega\;:\)\ dist\((x,\partial\Omega)>4\rho\}\) for each \(\rho>0\). Due to the further restriction on \(\Omega\), the authors may consider only \(\rho<\rho^*\) such that \(\Omega(\rho)\) is connected. Then, one of the main results of the paper reads as follows. Let \(\Omega\) be a bounded domain with Lipschitz boundary, \(u\in W^1_2(\Omega)\) be a solution of (1), and the coefficients of the equation satisfy the above conditions. Further, let \(\rho<\rho^*\) and let \(E\) be a measurable subset of \(\Omega(\rho)\) of positive measure such that \(\|u\|_{L^2(\Omega)}\leq 1\), \(\|u\|_{L^2(E)}\leq\epsilon\). Then, \[ \|u\|_{L^2(\Omega(\rho))}\leq C|\log\epsilon|^{-c}, \] where \(C\) and \(c\) depend on \(\Omega\), \(\lambda\), \(\Lambda_0\), \(V\), \(W_1\), \(W_2\), \(|E|\), and \(\rho\) only. The second main theorem gives an estimate for \(\|u\|_{L^{\infty}(\Omega(\rho))}\).
scientific article; zbMATH DE number 6065232
scientific article