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March 12, 2008
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We obtain the classical global L p , 2 ≦ p < ∞, estimate for the spatial gradient of the weak solutions for a class of parabolic problems in a very general irregular domain whose model is a nonlinear parabolic equation in divergence form. We treat discontinuous nonlinearity of BMO type and δ-Reifenberg flat domains. These domains might have fractal boundaries.

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We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the Bott-Steenbrink-Danilov vanishing theorem.

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In this work, we describe the asymptotic behavior of complete metrics with prescribed Ricci curvature on open Kähler manifolds that can be compactified by the addition of a smooth and ample divisor. First, we construct an explicit sequence of Kähler metrics with special approximating properties. Using those metrics as starting point, we are able to work out the asymptotic behavior of the solutions given in Tian, Gang, Yau, Shing-Tung , Complete Kähler manifolds with zero Ricci curvature. I, J. Amer. Math. Soc. 3 (1990), no. 3, 579–609., in particular obtaining their full asymptotic expansion.

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We describe the Chow ring with rational coefficients of the moduli space of stable maps with marked points 0, m (ℙ n , d ) as the subring of invariants of a ring B*( 0, m (ℙ n , d ); ℚ), relative to the action of the group of symmetries S d . B*( 0, m (ℙ n , d ); ℚ) is computed by following a sequence of intermediate spaces for 0, m (ℙ n , d ) and relating them to substrata of 0,1 (ℙ n , d + m - 1). An additive basis for A*( 0, m (ℙ n , d ); ℚ) is given.

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This is the first in a series of papers in which we study the n -Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n . The methods we describe are practical in the case n = 3 for elliptic curves over the rationals, and have been implemented in MAGMA.

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In this paper, we prove the L p -L q maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov V. A. Solonnikov , On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvest. 31 (1988), 381–405., but he assumed that p = q > 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 < p , q < ∞ but also the infinite time interval case. Especially, we obtain the L p -L q maximal regularity theorem with exponential stability on the infinite time interval. Our method can be applied to any initial boundary value problem for the equation of parabolic type with suitable boundary condition which generates an analytic semigroup, for example the Stokes equation with non-slip, slip or Robin boundary conditions.

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We obtain results on the geometry of D -semianalytic and subanalytic sets over a complete, non-trivially valued non-Archimedean field K , which is not necessarily algebraically closed. Among the results are the Parameterized Smooth Stratification Theorem and several results concerning the dimension theory of D -semianalytic and subanalytic sets. Also, an extension of Bartenwerfer's definition of piece number for analytic K -varieties is provided for the D -semianalytic sets and the existence of a uniform bound for the piece number of the fibers of a D -semianalytic set is proved. There is a connection between the piece number and the complexity of a D -semianalytic set which is a subset of the affinoid line and therefore a simpler proof of the Complexity Theorem of Lipshitz and Robinson is made possible by these results. Finally we prove an analogue of a theorem by van den Dries, Haskell and Macpherson, which states that for each D -semianalytic X , there is a semialgebraic Y such that one dimensional fibers of X are among the one dimensional fibers of Y through an easy application of our earlier results.