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Let $\Omega\subset\RR^n$ ($n\ge 1$) be a bounded open set with a Lipschitzcontinuous boundary. In the first part of the paper, using the method ofbilinear forms, we give a rigorous characterization of the realization in$L^2(\Omega)$ of the fractional Laplace operator $(-\Delta)^s$ ($0<s<1$) withthe nonlocal Neumann and Robin exterior conditions. Contrarily to the classicallocal case $s=1$, it turns out that the nonlocal (Robin and Neumann) exteriorconditions are incorporated in the form domain. We show that each of the aboveoperators generates a strongly continuous submarkovian semigroup which is alsoultracontractive. In the second part, we show that the semigroup correspondingto the nonlocal Robin exterior condition is always sandwiched between thefractional Dirichlet semigroup and the fractional Neumann semigroup.

Realization of the fractional Laplacian with nonlocal exterior conditions via forms method