Abstract
This is a survey of some recent contributions by the authors onglobal integrability properties of the gradient of solutions toboundary value problems for nonlinear elliptic equations indivergence form. Minimal assumptionson the regularity of the ground domain and of the prescribed dataare pursued.
Highlights
The present paper is mainly devoted to report on some contributions by the authors on integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form
We just refer to the the monographs [BF, Gia, GT, Gi, LU, Mo] and to the recent survey paper [Mi]
With this tool at disposal, global bounds for any rearrangement invariant norm of the gradient of solutions to either Dirichlet or Neumann boundary value problems are reduced to one-dimensional inequalities for Hardy type operators
Summary
The present paper is mainly devoted to report on some contributions by the authors on integrability properties of the gradient of solutions to boundary value problems for nonlinear elliptic equations in divergence form. Most of the relevant estimates are formulated in terms of pointwise inequalities for the distribution function of the length of the gradient, or, equivalently, for its decreasing rearrangement With this tool at disposal, global bounds for any rearrangement invariant norm of the gradient of solutions to either Dirichlet or Neumann boundary value problems are reduced to one-dimensional inequalities for Hardy type operators. The latter depend both on the class of elliptic differential operators under consideration, and on the regularity of the domain. We emphasize that the bound in question turns out to hold for systems
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