Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in L^2. The nonlinear expression of the gradient under the divergence operator is shown to belong to W^{1,2}, and hence to enjoy the best possible degree of regularity. Moreover, its norm in W^{1,2} is proved to be equivalent to the norm of the right-hand side in L^2. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well.
Optimal second-order regularity for the p-Laplace system / Cianchi A.; Maz'ya V.G.. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - STAMPA. - 132:(2019), pp. 41-78. [10.1016/j.matpur.2019.02.015]
Optimal second-order regularity for the p-Laplace system
Cianchi A.
;
2019
Abstract
Second-order estimates are established for solutions to the $p$-Laplace system with right-hand side in L^2. The nonlinear expression of the gradient under the divergence operator is shown to belong to W^{1,2}, and hence to enjoy the best possible degree of regularity. Moreover, its norm in W^{1,2} is proved to be equivalent to the norm of the right-hand side in L^2. Our global results apply to solutions to both Dirichlet and Neumann problems, and entail minimal regularity of the boundary of the domain. In particular, our conclusions hold for arbitrary bounded convex domains. Local estimates for local solutions are provided as well.
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