Partial differential equations are a classical and very active field of research. One of its salient features is to break the rigid distinction between the evolution of the theory and the applications to real world phenomena, since the two are intimately intertwined in the harmonious development of such a fascinating and multifaceted topic of investigation.

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $s$-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.

Given a bounded open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $\mathrm{\Omega }$. We prove that the second eigenvalue ${\lambda }_2(\mathrm{\Omega })$ is always strictly larger than the first eigenvalue ${\lambda }_1(B)$ of a ball $B$ with volume half of that of $\mathrm{\Omega }$. This bound is proven to be sharp, by comparing to the limit case in which $\mathrm{\Omega }$ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.