In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $$\Delta u = \lambda_{+\chi_{\{ u>0 \}}} - \lambda_{-\chi_{\{ u<0 \}}},\quad\lambda_{\pm}>0. $$ We prove that the free boundary touches the fixed boundary (uniformly) tangentially if the boundary data$f$and its first and second derivatives vanish at the touch-point.

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.

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Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on C3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity C2/Z2×C, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.

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