Exceptional Domains in Higher Dimensions

Abstract We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> , $$N\ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . These domains arise as perturbations of complements of straight cylinders in $$\mathbb {R}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> , and by definition they support a positive harmonic function with vanishing Dirichlet boundary values and constant Neumann boundary values, the so-called roof function. While the domains have a similar shape as those constructed in the recent work [18] for the case $$N=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , there is a striking contrast with regard to the shape of corresponding roof functions which are bounded for $$N \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . Moreover, while the analysis in [18] does not extend to higher dimensions, the approach of the present paper depends heavily on the assumption $$N \ge 4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> .