On removable sets of solutions of elliptic equations
The authors consider, in a bounded domain \(D\subset\mathbb R^n\), \(n\geq 3\), with the boundary \(\partial D\), the following elliptic equation \[ \mathcal L u=\sum^n_{i,j=1}a_{ij}(x)u_{ij}+\sum^n_{i=1}b_i(x)u_i+c(x)u=0,\tag{1} \] where \(\|a_{ij}\|_{i,j=1}^n\) is a real symmetric matrix. For a positive measurable function \(\omega(x)\), by \(\omega(E)\), for a measurable set \(E\) in \(\mathbb R^n\), is denoted \(\omega(E):=\int_E\omega(y)dy\). Denote by \(B_r^x\) the ball with center in \(x\) and radius \(r\). It is assumed that there exist constants \(\gamma\in (0,1]\) and \(b_0\geq 0\) such that \(\omega(B^x_R)\geq\gamma\omega(B^x_{2R})\) uniformly on \(R>0\), \[ \gamma|\xi|^2\omega(x)\leq\sum^n_{i,j=1}a_{ij}(x)\xi_i\xi_j \leq\gamma^{-1}\omega(x)|\xi|^2,\quad \xi\in\mathbb R^n,\;x\in D, \] and \[ |b_i(x)|\leq b_0,\;-b_0\leq c(x)\leq 0,\quad i=1,\dots,n,\;x\in D, \] where the coefficients \(b_i\) and \(c\) are measurable functions in \(D\) and \(a_{ij}\in C^1_{\omega}(\overline{D})\). Let \(\lambda\in (0,1)\). Denote by \(C^{\lambda}_{\omega}(D)\) a Banach space of functions \(u(x)\) defined on \(D\) with the finite norm \[ \|u\|_{C^{\lambda}_{\omega}(D)}:=\sup_{x\in D}\omega(x)|u(x)|+ \sup_{\underset{x\neq y}{x,y\in D}}\frac{\omega(x)|u(x)-u(y)|}{|x-y|^{\lambda}}. \] Finally, a compact set \(E\) in \(\overline D\) is called removable with respect to (1) if from \[ \mathcal Lu=0,\;x\in D\backslash E;\quad u|_{\partial D\backslash E}=0,\;u(x)\in C^{\lambda}_{\omega}(D), \] it follows that \(u(x)\equiv 0\) in \(D\). The aim of the paper is to find some sufficient conditions of removability of a compact set with respect to (1) in the space \(C^{\lambda}_{\omega}(D)\).
scientific article; zbMATH DE number 5964493