Removability theorems for solutions of degenerate elliptic partial differential equations
Owing to a classical result by L. Carleson, a subset \(E\subset \Omega\) of an open subspace \(\Omega\subset \mathbb{R}^n\) is removable for a Hölder continuous harmonic function \(u\), with exponent \(0< \alpha< 1\), defined on \(\Omega\backslash E\) if and only if the \((n- 2+ \alpha)\)-dimensional Hausdorff measure of \(E\) is zero. The authors study an analogous problem for \({\mathcal A}\)-harmonic functions \(u\in C(\Omega)\cap W^{1, p}_{\text{loc}}(\Omega)\), weak solutions of the equation \(\text{div } {\mathcal A}(x, u(x))= 0\), by definition. Here \({\mathcal A}(x, \xi)\in \mathbb{R}^n\) is a function, measurable in \(x\) and continuous in \(\xi\), satisfying \[ {\mathcal A}(x, \xi)\geq \beta_1|\xi|^p,\;|{\mathcal A}(x, \xi)|\leq \beta_2 |\xi|^{p- 1},\;({\mathcal A}(x, \xi_1)- {\mathcal A}(x, \xi_2))\cdot (\xi_1- \xi_2)>0, \] \[ {\mathcal A}(x, \lambda\xi)= |\lambda|^{p- 2} \lambda {\mathcal A}(x, \xi) \] for appropriate \(p> 1\), \(0< \beta_1\leq \beta_2\) and all \(\lambda> 0\), \(\xi_1\neq \xi_2\). The crucial result is expressed in terms of an exhaustion \(K_1\subset K_2\subset\cdots \subset E= \bigcup K_i\) of \(E\): assuming Hölder-continuity, sufficient removability conditions are expressed by \(\int d(x, K_i)^{p(\alpha- 1)}dm(x)\) (integral over \(0< d(x, K_i)< 1\)), \(\liminf m(\{0< d(x, K_i)< r\})/r^b= 0\) \((r\to 0)\), where \(b= p- \alpha(p- 1)\). The case \(\alpha= 0\) of Hölder exponent is solved, too.
scientific article; zbMATH DE number 556416