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Regularity of degenerate weak \((L_1,L_2)\)-BLD mappings. (Chinese. English summary) Zbl 1115.30018

Summary: we give the definition of degenerate weak \((L_1,L_2)\)-BLD mappings in space, and by using the technique of Hodge decomposition and weakly reverse Hölder inequality we prove the following regularity result of degenerate weak \((L_1,L_2)\)-BLD mappings: For every \(q_{1}\) such that \( 0<{L_2^l}n^\frac{l}{2} l^{2}2^{n+l}\times100^{n^{2}}[2^{\frac{3l}{2}}( 2^{4l+n}+1)](l-q_{1})<1 \) there exists an integrable exponent \(p_{1}=p_{1}(n,l,q_{1}, L_{1},L_{2})>l\), such that every degenerate weak \((L_1,L_2)\)-BLD mapping \(f \in W_{\text{loc}}^{1,q_{1}}(\Omega,\mathbb R^n)\) belongs to \(W_{\text{loc}}^{1,p_{1}}(\Omega,\mathbb R^n)\), that is, \(f\) is a degenerate \((L_1,L_2)\)-BLD mapping in the usual sense.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J60 Nonlinear elliptic equations