Gradient estimates in anisotropic Lorentz spaces to general elliptic equations of \(p\)-growth. (English) Zbl 1493.35016
In the framework of the refined anisotropic Lorentz spaces the authors obtain useful global mixed norm gradient estimate for solutions of elliptic equations of \(p\)-growth having noncontinuous coefficients.
Is pointed out that Lorentz spaces, for some specific values of the exponents, are Lebesgue spaces so the study could be considered inserted in this context.
A key assumption for the highest order coefficients of the elliptic equations taken into consideration is that the boundary of the domain where are defined is locally presented as the graph of a Lipschitz continuous function with small Lipschitz constant.
The proof is based on an a priori pointwise estimate of the sharp functions for the spatial derivatives of weak solution, and a key ingredient is how to extend the Fefferman-Stein theorem of sharp functions to a new version with mixed-norm in the anisotropic Lorentz spaces.
In addition, the authors pointed out the use of the bootstrapping argument regarding the spatial dimension.
Is pointed out that Lorentz spaces, for some specific values of the exponents, are Lebesgue spaces so the study could be considered inserted in this context.
A key assumption for the highest order coefficients of the elliptic equations taken into consideration is that the boundary of the domain where are defined is locally presented as the graph of a Lipschitz continuous function with small Lipschitz constant.
The proof is based on an a priori pointwise estimate of the sharp functions for the spatial derivatives of weak solution, and a key ingredient is how to extend the Fefferman-Stein theorem of sharp functions to a new version with mixed-norm in the anisotropic Lorentz spaces.
In addition, the authors pointed out the use of the bootstrapping argument regarding the spatial dimension.
Reviewer: Maria Alessandra Ragusa (Catania)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35D30 | Weak solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
anisotropic Lorentz spaces; Calderón-Zygmund estimates; BMO discontinuous coefficients; \(C^{0, 1}\)-domain with small Lipschitz constantReferences:
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