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Quasilinear divergence form elliptic equations in rough domains. (English) Zbl 1190.35220

Summary: Existence and global Hölder continuity are proved for the weak solution to the Dirichlet problem
\[ \begin{cases} \text{div}(a^{ij}(x,u)D_ju+ a^i(x,u))= b(x,u,Du) &\text{in }\Omega\subset\mathbb R^n,\\ u=0 &\text{on }\partial\Omega,\end{cases} \]
over Reifenberg flat domains \(\Omega\). The principal coefficients \(a^{ij}(x,u)\) are discontinuous with respect to \(x\) with small BMO-norms and \(b(x,u,Du)\) grows as \(|Du|^r\) with \(r<1+2/n\).

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35J62 Quasilinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D30 Weak solutions to PDEs
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References:

[1] DOI: 10.1002/cpa.20037 · Zbl 1112.35053 · doi:10.1002/cpa.20037
[2] DOI: 10.1007/BF02547186 · Zbl 0099.08503 · doi:10.1007/BF02547186
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[6] DOI: 10.2307/2154833 · Zbl 0833.35048 · doi:10.2307/2154833
[7] DOI: 10.1002/3527600868 · doi:10.1002/3527600868
[8] DOI: 10.1007/s00208-006-0014-x · Zbl 1194.35157 · doi:10.1007/s00208-006-0014-x
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