Let Ω ⊆ n be open and bounded, with ∂Ω smooth. Sufficient conditions on f are given in order that for p > n, the equation

has a solution for every h ∈ Lp; L is a second order symmetric elliptic operator and B represents either the first, second or third boundary value problems. These conditions are in terms of the asymptotic behaviour of f, in its second variable, in relation to those λ for which there is a nontrivial solution of

For a handlebody H, we define two graphs, the augmented disk graph $\mathcal{ADG}(H)$ and the truncated augmented disk graph $\mathcal{TADG}(H)$, and we show they are hyperbolic in the sense of Gromov. In the process, we show they are quasi-isometric to two other disk graphs defined by U. Hamenstädt, the super conducting disk graph $\mathcal{SDG}(H)$ and the electrified disk graph $\mathcal{EDG}(H)$ respectively. So we reprove two theorems of Hamenstädt [12].

Our approach uses techniques from Masur–Schleimer's study on the hyperbolicity of the disk graph $\mathcal{DG}(H)$ [21].

Gross and Siebert developed a program for constructing in arbitrary dimension a mirror family to a log Calabi–Yau pair (X, D), consisting of a smooth projective variety X with a normal-crossing anti-canonical divisor D in X. In this paper, we provide an algorithm to practically compute explicit equations of the mirror family in the case when X is obtained as a blow-up of a toric variety along hypersurfaces in its toric boundary, and D is the strict transform of the toric boundary. The main ingredient is the heart of the canonical wall structure associated to such pairs (X, D), which is constructed purely combinatorially, following our previous work with Mark Gross. In the case when we blow up a single hypersurface we show that our results agree with previous results computed symplectically by Aroux–Abouzaid–Katzarkov. In the situation when the locus of blow-up is formed by more than a single hypersurface, due to infinitely many walls interacting, writing the equations becomes significantly more challenging. We provide the first examples of explicit equations for mirror families in such situations.