Liouville principles and a large-scale regularity theory for random\n elliptic operators on the half-space

2016/04/10 by Julian Fischer, Fischer, Julian, Claudia Raithel +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1604.02717

openalex publication_date 2016/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the large-scale regularity of solutions to second-order linear\nelliptic equations with random coefficient fields. In contrast to previous\nworks on regularity theory for random elliptic operators, our interest is in\nthe regularity at the boundary: We consider problems posed on the half-space\nwith homogeneous Dirichlet boundary conditions and derive an associated\nC^1,\α-type large-scale regularity theory in the form of a\ncorresponding decay estimate for the homogenization-adapted tilt-excess. This\nregularity theory entails an associated Liouville-type theorem. The results are\nbased on the existence of homogenization correctors adapted to the half-space\nsetting, which we construct - by an entirely deterministic argument - as a\nmodification of the homogenization corrector on the whole space. This adaption\nprocedure is carried out inductively on larger scales, crucially relying on the\nregularity theory already established on smaller scales.\n

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