Exceptional Domains in Higher Dimensions

We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space RN R N , N≥ 4 N ≥ 4 . These domains arise as perturbations of complements of straight cylinders in RN R N , 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 $$N=3$$ N = 3 , there is a striking contrast with regard to the shape of corresponding roof functions which are bounded for N ≥ 4 N ≥ 4 . Moreover, while the analysis in [18] does not extend to higher dimensions, the approach of the present paper depends heavily on the assumption N ≥ 4 N ≥ 4 .