We construct Green’s function for second order elliptic operators of the form (Formula presented.) in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients (Formula presented.) is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition (Formula presented.). In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients (Formula presented.) and b satisfy the Dini conditions and the domain is (Formula presented.).
Bibliographical noteFunding Information:
S. Kim is partially supported by NRF Grant No. NRF-2016R1D1A1B03931680 and No. NRF-20151009350.G. Sakellaris has received funding from the European Union’s Horizon 2020 research and innovation program under Marie Skłodowska-Curie grant agreement No. 665919, and is partially supported by MTM-2016-77635-P (MICINN, Spain) and 2017 SGR 395 (Generalitat de Catalunya).
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All Science Journal Classification (ASJC) codes
- Applied Mathematics