FELLER SEMIGROUPS WITH BOUNDARY CONDITIONS 3
1) The operator Q is a second-order degenerate elliptic differential operator on
dD with non-positive principal symbol. In other words, the a , J are the components
of a C°° symmetric contravariant tensor of type (0) on dD satisfying
N-l N-l
£ *'*(*%$ ° *' dD, £' = £ tjdx T*x,{dD).
« J = I i=i
Here T*,(dD) is the cotangent space of dD at #'.
2) Q l =
7
e C°°(dD) and 7 0 on dD.
3) // C°°(dD) and // 0 on 9D.
4) « C°°{dD) and S 0 on dD.
5) n = ( n i , . . . , njsi) is the unit interior normal to the boundary dD.
6) The integral kernel r(x',y') is the distribution kernel of a pseudo-differen-
tial operator R L\~QKl(dD), K,\ 0, and r(x',yf) 0 off the diagonal AQD =
{(x',x')\x' dD} in dD x dD. The density dy' is a strictly positive density on
dD.
7) The integral kernel t(x, y) is the distribution kernel of a properly supported,
pseudo-differential operator T Ll~QK2(RN), «2 0, which has the transmis-
sion property with respect to the boundary dD, and t(x, y) 0 off the diagonal
{{x,x);xeKN}inIiN xRN.
8) The function r(x, y) is a C°° function on D x D, with compact support in a
neighborhood of the diagonal AQ£, such that, at each point x' of dD, r(x'', ?/) = 1
for ?/ in a neighborhood of z' in 5 . The function r(x, y) depends on the shape of
the boundary dD.
9) The operator T is a boundary condition of order 2 min(/ci, is?), and satisfies
the condition
ri(*') = ,(*') + / r(x', y')[l - r(x', y')]dy'
JdD
+ [ i(x',y)[l-T(x',y)]dy0 on dD.
JD
The boundary condition L is called a second-order Ventcel' boundary condition
(cf. [17]). The terms of L
N-l N-l
3
i—\
»\;=i
dx{
du.
, r / JV-1
J^t(x',y){u(y)-T(x',y)[u(x')+J2(yj-xi)-£:(*')
dy',
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