Range of the gradient of a smooth bump function in finite dimensions

This paper proves the semi-closedness of the range of the gradient for sufficiently smooth bumps in the Euclidean space.

     

The infinity Laplacian in infinite dimensions

We study three properties of real-valued functions defined on a Banach space: The absolutely minimizing Lipschitz functions, the viscosity solutions of the infinity Laplacian partial differential equation, and the functions which satisfy comparison with cones. We prove that these notions are equivalent, and we show the existence of such functions. These results are new in the infinite-dimensional case.Received: 14 May 2003, Accepted: 4 August 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 49L25, 35J60, 54C20     

The steiner point in infinite dimensions

It is shown that the Steiner point cannot be extended continuously to all convex bodies in infinite dimensional Hilbert spaces. This follows as a corollary of a result on the local behavior of the point.     

The Hoffman-Wielandt inequality in infinite dimensions

The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.     

The Dolbeault complex in infinite dimensions II

We study the equation on a ball , and prove that it is solvable if is a Lipschitz continuous, closed -form.

     

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy-Riemann equations on affine spaces.

     

The convergence-extension theorem of Noguchi in infinite dimensions

In this note we give generalizations of Noguchi's convergence-extension theorem to the case of infinite dimension.

     

Semi-linear problems in infinite dimensions

We prove existence, smoothness and ergodicity results for semilinear parabolic problems on infinite dimensional spaces assuming the Logarithmic Sobolev inequality is satisfied. As a consequence we construct a class of nonlinear Markov semigroup which are hypercontractive.     

Sobolev embeddings in infinite dimensions

In this paper, we study Sobolev spaces in infinite dimensions and the corresponding embedding theorems. Our underlying spaces are ?^r for r ∈ [1, ∞), equipped with corresponding probability measures.For the weighted Sobolev space W_b^1,p(?^r, γ_a) with a weight a ∈ ?^r of the Gaussian measure γa and a gradient weight b ∈ ?^∞, we characterize the relation between the weights(a and b) and the continuous(resp. compact)log-Sobolev...     

On the continuity of Ornstein-Uhlenbeck processes in infinite dimensions

Summary Existence and continuity of Ornstein-Uhlenbeck processes in Banach and Hilbert spaces are investigated under various assumptions.This work was partly written when W. Smoleski visited the Mathematics Department in Angers     

Nonlinear stochastic differential equations in infinite dimensions

The stochastic variation of constants proved in[2] results to be an interesting tool to study properties of different classes of stochastic differential equations. In particular, we study the extension to the case of coefficients depending on the solution X _t. It turns out that the representation formula becomes a stochastic integral equation that has to be studied via anticipate calculas     

Coverings and dimensions in infinite profinite groups

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.     

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form. The authors are supported by the ‘VIDI subsidie’ 639.032.201 of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. Received: 28 June 2006 Revised: 5 January 2007