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.