The global, weak solutions for the semilinear problem (1) introduced in Ni-Sacks-Tavantzis (J. Differ. Eq. 54, 97-120 (1984)) are studied. Estimates on the Hausdorff dimension of their singular sets are found. As an application, it is shown that these solutions must blow up in finite time and become regular eventually when the nonlinearity is supercritical and the domain is convex. Mathematical Subject Classification (2000)

Upper bounds on the Hausdorff dimensions of the rupture set of a weak solution of the thin film equation in space-time and in space slices are derived. Finite time rupture is shown to occur for a class of thin films obeying the power law with power in (0, 1/2) under periodic boundary conditions.

Bernstein problem for affine maximal type equation (0.1)has been a core problem in affine geometry. A conjecture proposed firstly by Chern (Proc. Japan-United States Sem., Tokyo, 1977, 17-30) for entire graph and then extended by Trudinger-Wang (Invent. Math., 140, 2000, 399-422) to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C 4 -hypersurface in R N+1 must be an elliptic paraboloid. At the same time, this conjecture was solve completely by Trudinger-Wang for dimension N = 2 and θ = 3/4, and later extended by Jia-Li (Results Math., 56 2009, 109-139) to N = 2, θ ∈ (3/4, 1] (see also Zhou (Calc. Var. PDEs., 43 2012, 25-44) for a different proof). On the past twenty years, much efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N = 3. In this paper, we will construct nonquadratic affine maximal type hypersurfaces which are Euclidean compete for N ≥ 3, θ ∈ (1/2, (N − 1)/N). Contents

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In this paper, we will prove a result of nonexistence on harmonic diffeomorphisms between punctured spaces. In particular, we will given an elementary proof to the nonexistence of rotationally symmetric harmonic diffeomorphisms from the punctured Euclidean space onto the punctured hyperbolic space.