FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 5
Intuitively, condition (A) implies that one of the absorption, reflection, viscosity
phenomena occurs at each point of the boundary dD.
Furthermore, we assume that:
(H) There exists a second-order VentceP boundary condition Lv such that
Lu(x') = ii{x')L¥u(x') + l{x')u{x') - 6{x')Wu{x'\ x' dD,
where the boundary condition Lv is given in local coordinates ( « i , - - , z;v_i) by
the formula
»*,i=i
dX{
N-l
du , _ ,v , ,v . v-^ »/ /\
®u
( i
+ J ?(*'.y') [u(y') - r(x',y') (u(x) + £ ( » - *i)^W)
^ ?(*', y) [u(y)-T(x',y) («(*') + £(% - * )^-(x')^dy,
+
and satisfies the condition
fj(x') + / f(x\ y')[l " r(x', y')]dy' + / (*', y)[l - r(z', y)]dy 0 on dD.
JdD
JD
We remark that the boundary condition L is not transversal on dD, while the
boundary condition Lv is transversal on 3D, since //(#') = 1 on 3D.
Intuitively, condition (H) implies that the diffusion along the boundary, the
inward jump phenomenon from the boundary and the jump phenomenon on the
boundary are "dominated" by the reflection phenomenon.
Now we introduce a subspace of C(D) which is associated with the boundary
condition L.
We let
M = {xf dD; n(xf) = S(x') = 0, / t(x\ y)dy oo}.
JD
Then, by condition (H), we find that
M = {x'e dD; fjL(x') = S(x') = 0}.
Further, in view of condition (^4), it follows that the boundary condition
Lu = fiL,,u -f 7 (U\SD) - 5 (Wu\dD) = 0 on dD
includes the condition
u = 0 on M.
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