Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains

By Tadele Mengesha, N. Phuc. 2011

125 citations recorded by the source corpus.

Abstract Global weighted L p estimates are obtained for the gradient of solutions to nonlinear elliptic Dirichlet boundary value problems over a bounded nonsmooth domain. Morrey and Holder regularity of solutions are also established, as a consequence. These results generalize various existing estimates for nonlinear equations. The nonlinearities are of at most linear growth and assumed to have a uniform small mean oscillation. The boundary of the domain, on the other hand, may exhibit roughness but assumed to be sufficiently flat in the sense of Reifenberg. Our approach uses maximal function estimates and Vitali covering lemma, and also known regularity results of solutions to nonlinear homogeneous equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/489cafff4f0a7d492b58c0a4be70e6c0a9521d0f

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In Applied Mathematics

1. N. Phuc
2. Sun-Sig Byun
3. Lihe Wang
4. Seungjin Ryu
5. Tadele Mengesha
6. Yumi Cho
7. Giuseppe Mingione
8. Jihoon Ok

In Computational Theory and Mathematics

1. N. Phuc
2. Sun-Sig Byun
3. Lihe Wang
4. Seungjin Ryu
5. Tadele Mengesha
6. Yumi Cho
7. Jihoon Ok
8. Giuseppe Mingione

Citations in the corpus

• Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations (2013, methodology): “A proof of this lemma, which uses Lebesgue Differentiation Theorem and the standard Vitali covering lemma, can be found in [24].”
• Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains (2012, methodology): “The weighted version given below is proved in [ 29 ] following the arguments in [5,42].”
• Calderón–Zygmund theory for nonlinear elliptic problems with irregular obstacles (2012, methodology): “When it comes to a minimal geometric assumption we impose a Reifenberg flatness which turns out to be an appropriate one for nonlinear perturbation results, as in [6,21,27].”
• Global weighted estimates for the gradient of solutions to nonlinear elliptic equations (2013, background): “In the same spirit of that in Chapter 5 of the recent paper [20], we can find a finer regularity in Morrey spaces and Hölder spaces from Theorem 2.”