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arXiv 1903.007512019-03-02math.AP

Fully anisotropic elliptic problems with minimally integrable data

Angela Alberico, Iwona Chlebicka, Andrea Cianchi, Anna Zatorska-Goldstein

英文摘要：

We investigate nonlinear elliptic Dirichlet problems whose growth is driven by a general anisotropic $N$-function, which is not necessarily of power type and need not satisfy the $Δ_2$ nor the $\nabla _2$-condition. Fully anisotropic, non-reflexive Orlicz-Sobolev spaces provide a natural functional framework associated with these problems. Minimal integrability assumptions are detected on the datum on the right-hand side of the equation ensuring existence and uniqueness of weak solutions. When merely integrable, or even measure, data are allowed, existence of suitably further generalized solutions - in the approximable sense - is established. Their maximal regularity in Marcinkiewicz-type spaces is exhibited as well. Uniqueness of approximable solutions is also proved in case of $L^1$-data.

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