Poisson equations and Morrey spaces
Let \(\Omega\) be an open bounded subset of \(\mathbb{R}^ n\) (\(n\geq 3\)) and there exists \(A>0\) such that \(| \Omega\cap B(x,r)|\geq A| B(x,r)|\) for \(x\in\Omega\) and \(0<r<\text{diam }\Omega\); \(0<\lambda<n\), \(f\in L^{1,\lambda}(\Omega)\), \(u\in L^ 1(\Omega)\) such that \[ -\int_ \Omega u \sum^ n_{i,j=1} \partial_ j (a_{ij} \partial_ i\varphi) dx=\int_ \Omega \varphi f dx \quad \text{for every} \quad \varphi\in H^ 1_ 0(\Omega)\cap C^ 0(\overline {\Omega}) \] such that \(\sum^ n_{i,j=1} \partial_ j(a_{ij} \partial_ i \varphi)\in C^ 0(\overline {\Omega})\), where \(a_{ij}\in L^ \infty(\Omega)\), \(a_{ij}=a_{ji}\) for \(i,j=1,\dots,n\) and there exists \(\nu>0\) such that \((1/\nu)|\xi|^ 2 \leq \sum^ n_{i,j=1} a_{ij} \xi_ i \xi_ j\leq \nu|\xi|^ 2\) for every \(\xi\in\mathbb{R}^ n\). Theorem 1. If \(\lambda\in]0,n-z[\) then \(u\) is the weak \(L^{p_ \lambda}(\Omega)\), where \(1/p_ \lambda= 1-2/(n-\lambda)\); if \(\lambda=n-2\) then \(u\) is locally in \(\text{BMO}(\Omega)\); if \(f\in \widetilde{S}\) then \(u\) is bounded, if \(f\in S\) then \(u\) is continuous where \(\widetilde{S}\) and \(S\) are variants of Stummel-Kato classes; if \(\lambda\in ]n-2,n[\) then \(u\) is locally Hölder continuous. Theorem 2. Let \(\Omega\) be satisfying a uniformly exterior sphere condition and \(a_{ij}\) Dini continuous for \(i,j=1,\dots,n\); if \(\lambda\in]n-2,n-1[\) then \(\nabla u\) is in the weak \(L^{p_ \lambda}(\Omega)\), where \(p_ \lambda=(n- \lambda)/(n- \lambda-1)\); let \(a_{ij}\) be Hölder continuous for \(i,j=1,\dots,n\), too: if \(\lambda=n-1\) then \(\nabla u\) is locally in \(\text{BMO}(\Omega)\); if \(\lambda\in]n-1,n[\) then \(\nabla u\) is locally Hölder continuous.
second-order uniformly elliptic operator
scientific article; zbMATH DE number 62873