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| −n−s |z|−n−s , with s∈(0,1) s∈(0,1) and n the dimension of the ambient space. The fractional Young’s law (contact angle condition) predicted by these models coincides, in the limit as s→1 − s→1− , with the classical Young’s 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 σ σ is negative, and larger if σ σ is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case s→0 + s→0+ 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.