Very degenerate elliptic equations under almost critical Sobolev regularity

Clop, Albert; Giova, Raffaella; Hatami, Farhad; Napoli, Antonia Passarelli di

doi:10.48550/arxiv.1803.08723

Cited by 1 publication

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References 28 publications

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“…The regularity of a solution to elliptic boundary value problems was studied in [31,32]. Moreover, the case, when the operator A has a potential and so the problem (1) can be 'equivalently' rewritten as the minimization problem to some variational integral, has recently attracted lot of attention and the regularity properties of solution/ minimizers has been closely investigated for A having non-standard growth but still being smooth with respect to spatial variable, see [6,8,22].…”

Section: Introductionmentioning

confidence: 99%

Homogenization of nonlinear elliptic systems in nonreflexive Musielak–Orlicz spaces

et al. 2019

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We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic N -function M , which may also depend on the spatial variable, i.e., the homogenization process will change the underlying function spaces and the nonlinear elliptic operator at each step. The problem of homogenization of nonlinear elliptic systems has been solved for the L p −setting with restrictions either on constant exponent or variable exponent that is assumed to be additionally log-Hölder continuous. These results correspond to a very particular case of N -functions satisfying both ∆ 2 and ∇ 2 -conditions. We show that for general M satisfying a condition of log-Hölder type continuity, one can provide a rather general theory without any assumption on the validity of neither ∆ 2 nor ∇ 2 -conditions.

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