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Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

Let ( X , ω ) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L p right hand side, p > 1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range ( X , ω ) of the complex Monge-Ampère operator acting on ω -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with L p -density belong to ( X , ω ) and proving that ( X , ω ) has the...

Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is 1 , 1 and the right hand side has a density in L p ( Ω ) for some p > 1, and prove the Hölder continuity of the solution.

Homogeneous Carnot groups related to sets of vector fields

Andrea Bonfiglioli (2004)

Bollettino dell'Unione Matematica Italiana

In this paper, we are concerned with the following problem: given a set of smooth vector fields X 1 , … , X m on R N , we ask whether there exists a homogeneous Carnot group G = ( R N , ∘ , δ λ ) such that ∑ i X i 2 is a sub-Laplacian on G . We find necessary and sufficient conditions on the given vector fields in order to give a positive answer to the question. Moreover, we explicitly construct the group law i as above, providing direct proofs. Our main tool is a suitable version of the Campbell-Hausdorff formula. Finally, we exhibit several...

Homogenization in polygonal domains

David Gérard-Varet, Nader Masmoudi (2011)

Journal of the European Mathematical Society

We consider the homogenization of elliptic systems with ε -periodic coefficients. Classical two-scale approximation yields an O ( ε ) error inside the domain. We discuss here the existence of higher order corrections, in the case of general polygonal domains. The corrector depends in a non-trivial way on the boundary. Our analysis substantially extends previous results obtained for polygonal domains with sides of rational slopes.

Homogenization of a monotone problem in a domain with oscillating boundary

Dominique Blanchard, Luciano Carbone, Antonio Gaudiello (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We study the asymptotic behaviour of the following nonlinear problem: { - div ( a ( D u h ) ) + | u h | p - 2 u h = f in Ω h , a ( D u h ) · ν = 0 on ∂ Ω h , . in a domain Ωh of ℝ n whose boundary ∂Ωh contains an oscillating part with respect to h when h tends to ∞. The oscillating boundary is defined by a set of cylinders with axis 0xn that are h-1-periodically distributed. We prove that the limit problem in the domain corresponding to the oscillating boundary identifies with a diffusion operator with respect to xn coupled with an algebraic problem for the limit fluxes.

Currently displaying 1981 – 2000 of 5493