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Dirichlet problem for integro-differential operators
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DOI：10.1016/j.spa.2026.104976.png)
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Abstract
En 中文
In this paper, we study the Dirichlet problem for the following integro-differential operator on bounded (not necessarily connected) open sets in & Ropf;(d) : Lu = 1 2 div(A(x)del(x)) + b(x) & sdot; del u(x) + 1/2 lim (epsilon -> 0) integral(d)({y is an element of & Ropf;) & ratio;|y-x|>epsilon} (u(y) - u(x))J(x, y)dy, where A(x) = (a(ij)(x))(1 <= i,j <= d) is a measurable d & times; d matrix-valued function on & Ropf;(d) that is uniformly elliptic and bounded, b is an & Ropf;(d) -valued function so that |b| (2) is in the Kato class Kd , and J(x, y) >= 0 is a measurable symmetric non-trivial kernel on & Ropf;(d) & times; & Ropf;(d) bounded above by c max{|x - y| -(d+alpha) , |x - y| -(d+beta)} for some 0 < beta <= alpha < 2 and c > 0. We show that there is a Feller process X on & Ropf;(d) having strong Feller property associated with the non-local operator L. We further show that for any bounded open set D in & Ropf;(d) that is regular with respect to the Feller process X and for every bounded function phi on D-c that is continuous on partial derivative D, the Dirichlet problem for L on D has a unique weak solution on & Ropf;(d) that is continuous on D. Moreover, the solution can be represented in terms of the associated Feller process
Keywords:
Diffusion with jumps
Dirichlet problem
Weak solution
Probabilistic representation
Girsanov transform
Dirichlet form
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105
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