Local boundedness and continuity for weak solutions of \(-(\nabla-ib)^ 2u+Vu=0\)
The author considers weak solutions of the Schrödinger equation \[ -(\nabla- ib(x))^2u(x)+ V(x)u(x)= 0 \] in a domain \(\Omega\subset\mathbb{R}^n\), \(n\geq 3\), where \(b\) is a real-valued vector function and the potential \(V\) is complex-valued. For \(f\in L^1_{\text{loc}}(\mathbb{R}^n)\) let \[ \widetilde\eta(f; r):= \sup_{\mathbb{R}^n}\int_{B_r(x)}|f(y)||x-y|^{2-n}dy\quad\text{and} \quad \eta(f;r;G):=\widetilde\eta(f\chi_G; r), \] where \(\chi_G\) is the characteristic function of the domain \(G\) and \(B_r(x)\) is the ball with center \(x\) and radius \(r>0\). Then \(f\) is in the local Kato class \(K^{\text{loc}}_n(G)\) if \(\lim_{r\to 0}\eta(f;r;G')= 0\) for all bounded domains \(G'\subset\subset G\). The main results can be stated as follows: Assume \(|b|^2\), \(V\in K^{\text{loc}}_n(\Omega)\) and let \(u\in H^1_{\text{loc}}(\Omega)\) be a weak solution of the Schrödinger equation. Then \(u\in L^\infty_{\text{loc}}(\Omega)\). Moreover, if additionally \(\text{div }b\in K^{\text{loc}}_n(\Omega)\), then \(u\) is continuous in \(\Omega\) and \(|\nabla u|^2\in K^{\text{loc}}_n(\Omega)\). The local boundedness is obtained via Kato's inequality by arguments as in the paper of \textit{A. M. Hinz} and \textit{H. Kalf} [J. Reine Angew. Math. 404, 118-134 (1990; Zbl 0779.35026)]. The continuity result and the gradient regularity are formulated for solutions \(u\in H^1_{\text{loc}}(\Omega)\) to the more general equation \[ -\text{div}(A(x)\nabla u(x))+ c(x)\nabla u(x)+ V(x)u(x)=0, \] where the symmetric matrix \(A(x)\) is real-valued and uniformly elliptic and \(c\), \(V\) are complex-valued with \(|c|^2\), \(|V|\in K^{\text{loc}}_n(\Omega)\). The proof is based on the author's results in [Indiana Univ. Math. J. 43, No. 2, 411-440 (1994; Zbl 0805.35017)].
scientific article; zbMATH DE number 1028169
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