A reflection result for harmonic functions which vanish on a cylindrical surface (Q298104)

Let \(N \geq 3\) and let \(B'\) denote the open unit ball in \(\mathbb{R}^{N-1}.\) In their previous paper [J. Math. Anal. Appl. 433, No. 2, 1870--1882 (2016; Zbl 1327.31012)] the authors show that any harmonic function in a finite cylinder \(B' \times (-a, a)\), \(a > 0\) which continuously vanishes on \(\partial B' \times (-a, a)\) has a harmonic extension to the strip \(\mathbb{R}^{N-1} \times (-a, a)\). In this paper the authors ask whether there is an analogous extension result when the given harmonic function is merely defined near the curved boundary and inside the cylinder. Their main result is the following theorem.NEWLINENEWLINETheorem. Let \(\phi : (-a, a) \rightarrow [0, 1)\) be upper semicontinuous. Then any harmonic function on NEWLINE\[NEWLINE\big\{ (x', x_N) : |x_N| < a, \phi (x_N) < ||x'|| < 1 \big\}NEWLINE\]NEWLINE which continuously vanishes on \(\partial B' \times (-a, a)\) has a harmonic extension to NEWLINE\[NEWLINE\{ (x', x_N) : |x_N| < a, \phi (x_N) < ||x'|| < 2 - \phi(x_N) \}.NEWLINE\]