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In this paper we study the existence assertion of the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = \Delta e^{-\Delta u}$. This problem arises in the mathematical description of the evolution of crystal surfaces. Our investigations reveal that the exponent in the equation can have a singular part in the sense of the Lebesgue decomposition theorem, and the exponential nonlinearity somehow “cancels” it out. The net result is that we obtain a solution $u$ that satisfies the equation and the initial boundary conditions in the almost everywhere (a.e.) sense.

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths, and is close to a convection CFL condition for large mean free paths. Ou r analysis is based on very simple energy estimates.

In this project we study the energy functional on the set of Lagrangian tori in CP2. The energy functional has been introduced in [2] as integral of the potential of 2D periodic Schrödinger operator associated to Lagrangian torus. It has been conjectured in [2] that the Clifford torus is the unique global minimum of energy functional (the statement is later referred to as the energy conjecture). Due to geometric interpretation of energy functional as linear combination of the volume andWillmore functionals, this conjecture can be seen as the CP2 analogue of the well-known Willmore conjecture for tori in R3, recently proved in [18]. The energy conjecture has been verified for two families of Hamiltonian-minimal Lagrangian tori in [2]. Results of [5] and [23] imply the conjecture for minimal Lagrangian tori of sufficiently high spectral genus and non-embedded minimal Lagrangian tori, respectively. In the present work we prove the energy conjecture for a family of Hamiltonian-minimal Lagrangian tori in CP2 constructed in [4]. In sharp distinction with cases considered in [2], the value of the energy functional for these tori can not be calculated exactly. The proof relies on analytic bounds for certain elliptic integrals arising from the induced metric of tori. Possible directions of further work are: 1. Consider local behaviour of the energy functional. Are the critical points of the energy functional governed by an integrable PDE, akin to Tzizeica equation describing minimal Lagrangian tori? The same questions for critical points under Hamiltonian variations. 2. Is there an analogue of the energy conjecture for other Kähler-Einstein surfaces? The case of K3 surface is of special interest as minimal Lagrangian tori in K3 can be related to elliptic fibrations (for instance [20]) making the conjecture amenable to algebrogeometric analysis. 3. Examples of monotone Lagrangian tori with trivial Floer cohomology were constructed in [21]. Do there exist critical points of the energy functional with trivial Floer cohomology?

For a one-parameter family (V, {i}pg i=1) of general type hypersurfaces with bases of holomorphic n-forms, we construct open covers V = Spg i=1 Ui using tropical geometry. We show that after normalization, each i is approximately supported on a unique Ui and such a pair approximates a Calabi-Yau hypersurface together with its holomorphic n-form as the parameter becomes large. We also show that the Lagrangian fibers in the fibration constructed by Mikhalkin [9] are asymptotically special Lagrangian. As the holomorphic n-form plays an important role in mirror symmetry for Calabi-Yau manifolds, our results is a step toward understanding mirror symmetry for general type manifolds.