  A new definition of conditional invariance for boundary value problems
involving a wide range of boundary conditions (including initial value problems
as a special case) is proposed. It is shown that other definitions worked out
in order to find Lie symmetries of boundary value problems with standard
boundary conditions, follow as particular cases from our definition. Simple
examples of direct applicability to the nonlinear problems arising in
applications are demonstrated. Moreover, the successful application of the
definition for the Lie and conditional symmetry classification of a class of
(1+2)-dimensional nonlinear boundary value problems governed by the nonlinear
diffusion equation in a semi-infinite domain is realised. In particular, it is
proved that there is a special exponent, $k=-2$, for the power diffusivity
$u^k$ when the problem in question with non-vanishing flux on the boundary
admits additional Lie symmetry operators compared to the case $k\not=-2$. In
order to demonstrate the applicability of the symmetries derived, they are used
for reducing the nonlinear problems with power diffusivity $u^k$ and a constant
non-zero flux on the boundary (such problems are common in applications and
describing a wide range of phenomena) to (1+1)-dimensional problems. The
structure and properties of the problems obtained are briefly analysed.
Finally, some results demonstrating how Lie invariance of the boundary value
problem in question depends on geometry of the domain are presented.
