In this paper we investigate the
Dirichlet problem
(1)
(2)
in a smooth domain Ω ⊂ ℝ2 for which ℝ2∖Ω is bounded. We sharpen previous
non-existence results for this exterior Dirichlet problem by showing that even the
smallness of the α- Hölder norm, 0 ≤ α < is not enough for the classical
solvability of (1) and (2), not imposing any asymptotical conditions at infinity upon
possible solutions. In particular, we explicitely exhibit smooth data f of arbitrary
small Cα-norm for which (1), (2) is not solvable in the space C0(Ω) ∩ C2(Ω).
The key idea of our proof is to replace the original problem (1), (2) on a
known domain but with unknown boundary conditions at infinity by the
corresponding problem on some unknown (bounded) domain, but with fixed
boundary data. By the same method we show the instability of the exterior
Dirichlet problem with respect to Cα-small perturbations of the boundary
data, 0 ≤ α <, provided that Ω is the complement of a strictly convex
set.
is not enough for the classical
solvability of (
, provided that 