arXiv subjects
Publications and source records attributed to Adriano Prade.
We study a variational problem modeling equilibrium configurations of charged liquid droplets resting on a surface under a convexity constraint. In the two-dimensional case with Coulomb interactions, we establish the validity of Young's law for the contact angle for small enough charges.
We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has order $2s$ then the solution is $C^{s-\varepsilon}$ regular for all $\varepsilon>0$ provided the flatness parameter is small enough. The proof relies on an induction argument and its main ingredients are the construction of a suitable barrier and the comparison principle.
For $0<s<1$, we consider the nonlocal equation $(-\Delta)^s u = f$ over a Reifenberg flat domain $\Omega$ with $f \in C({\overline{\Omega}})$ and null Dirichlet exterior condition. Given $\alpha \in (0,s)$, we prove that weak solutions are $\alpha$-H\"older continuous up to the boundary when the flatness parameter is small enough. The main ingredients of the proof are an iterative argument and a nonlocal version of the ABP maximum principle.