Computational complexity of two-dimensional domains whose boundaries are polynomial-time computable Jordan curves with polynomial inverse moduli of continuity is studied. It is shown that the membership problem of such a domain can be solved in
, i.e., in polynomial time relative to an
oracle in
, in contrast to the higher upper bound
for domains without the
property of polynomial inverse modulus of continuity. On the other hand, the lower bound of
for the membership problem still holds for domains with polynomial inverse moduli of continuity. It is also shown that the shortest path problem of such a domain can be solved in
PSPACE, close to its known lower bound, while no fixed upper bound was known for domains without this
property.