Asymptotic Expansions of the Contact Angle in Nonlocal Capillarity Problems

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels |z| 12n 12s |z| 12n 12s , with s 08(0,1) s 08(0,1) and n the dimension of the ambient space. The fractional Young\u2019s law (contact angle condition) predicted by these models coincides, in the limit as s\u21921 12 s\u21921 12 , with the classical Young\u2019s law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient \u3c3 \u3c3 is negative, and larger if \u3c3 \u3c3 is positive. In addition, we address the asymptotics of the fractional Young\u2019s law in the limit case s\u21920 + s\u21920+ of interaction kernels with heavy tails. Interestingly, near s=0 s=0 , the dependence of the contact angle from the relative adhesion coefficient becomes linear