Elliptic patching of harmonic functions (Q935553)

Let \(T\) denote the \(1\)-dimensional torus, and let \(\alpha>0\). Let \(u_-\) and \(u_+\) be two harmonic functions on \(]-\alpha,\alpha[\times T\) such that \(u_-(0,y)=u_+(0,y)\) for all \(y\in T\). The pair \((u_-,u_+)\) is called a \textit{gluable pair} if there is a positive constant \(\epsilon_0\) with the following property. For any positive number \(\epsilon<\epsilon_0\), there exists a function \(u_{\epsilon}\in C^{1,1}(]-\alpha,\alpha[\times T)\) such that: (i) \(u_{\epsilon}\) is a solution on \(]-\alpha,\alpha[\times T\) of some linear uniformly elliptic equation with bounded measurable coefficients and no zero order term; (ii) \(u_{\epsilon}=u_-\) on \(]-\alpha,-\epsilon]\times T\), and \(u_{\epsilon}=u_+\) on \([\epsilon,\alpha[\times T\); (iii) there is a constant \(M\) such that \(|u_{\epsilon,x}|<M\) on \([-\epsilon,\epsilon]\times T\). The author gives a condition necessary for a pair \((u_-,u_+)\) to be gluable, and a different condition sufficient for \((u_-,u_+)\) to be gluable.