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Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations

Guy Barles, Emmanuel Chasseigne, Cyril Imbert (2011)

Journal of the European Mathematical Society

This paper is concerned with the Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth...

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.

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