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Elliptic operators with nonstandard growth condition: some results and open problems. (English) Zbl 1439.35173
Kuchment, Peter (ed.) et al., Differential equations, mathematical physics, and applications. Selim Grigorievich Krein centennial. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 734, 277-292 (2019).
Summary: This is a short survey of certain results and open problems on the Dirichlet problem for nonlinear monotone elliptic operators with nonstandard growth condition. In this paper we pay attention to the case when the so-called Lavrentiev phenomenon may occur. Furthermore, as a rule the growth condition accepted in the paper is more general than the well-known $$p(x)$$ growth condition. Also we discuss the homogenization result obtained by V. Zhikov and S. Pastukhova in the case of $$p(x)$$ growth and its potential extensions.
For the entire collection see [Zbl 1420.35009].

MSC:
 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 47H05 Monotone operators and generalizations
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References:
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