The Dirichlet problem for nonlocal operators
The authors consider elliptic and parabolic Dirichlet problems for linear nonlocal operators with known boundary data on the complement of a given bounded set. Unique solvability is obtained in convenient spaces. The method is based on an appropriate combination of nonlocal versions of the Poincaré-Friedrichs inequality, a Gårding inequality, the Lax-Milgram lemma, the weak maximum principle for integro-differential operators in bounded domains, and the Fredholm alternative. Several concrete examples of the used kernels are included and discussed in the final section.
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