An end point Orlicz type estimate for nonlinear elliptic equations. (December 2018) - Jang, Yunsoo; Kim, Youchan
Abstract: We study nonlinear elliptic equations of p -Laplacian type in divergence form to establish a natural Calderón–Zygmund type theory of an Orlicz space type, where the Lebesgue space is the special case with Φ ( t ) = t q p : Φ ( | F | p ) ∈ L 1 ( Q 2 R ) ⟹ Φ ( | D u | p ) ∈ L 1 ( Q R ) . In the previous results, the estimates obtained were strictly above the natural exponents, and the function such as Φ ( t ) = t log ( 1 + t ) was ruled out for the candidate of Φ . But with our approach, Φ can be selected up to the end point case of the estimates, and the functions Φ ( t ) = t and Φ ( t ) = t log ( 1 + t ) can be additionally selected for Φ .