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Integrability for solutions to quasilinear elliptic systems. (English) Zbl 1224.35139
The authors give structural conditions on coefficients of quasilinear elliptic systems in divergence form \[ -\sum _{i=1}^n D_i\left (\sum _{j=1}^n \sum _{\beta =1}^N a_{ij}^{\alpha \beta }(x,u(x))D_j u^{\beta }(x)=0\right )\;\;\text{on \(\Omega \) for \(\alpha = 1,...,N\)} \] which guarantee the higher integrability of solution \(u\). The structural conditions require decay of non-diagonal coefficients for \(u \to \infty \) described by \[ g^{\gamma }(L)= \max _{i,j}\max _{\beta \neq \gamma }\sup _{| y^{\gamma }| >L} \sup _{x}| a_{ij}^{\gamma \beta }(x,y)| . \] With no extra assumptions \(g^{\gamma }\) is nonnegative, bounded and decreasing function. Then the authors prove the existence of a constant \(C\) such that \( | \{x \in \Omega; | u^{\gamma }(x)| > 2L\} | \leq C\big (g^{\gamma }(L)/L\big )^{2^{*}}. \) Here \(2^{*}= 2n/(n-2)\) is Sobolev’s embedding exponent. If non-diagonal coefficients decay polynomially, i.e., for a positive \(q\) \( | a_{ij}^{\gamma \beta }(x,y)| \leq c/| y^{\gamma }| ^{-q} \) and \(u^{\gamma }\) in bounded on \(\partial \Omega \) then \(u \in L^{2^*(1+q)}_{weak}(\Omega )\). Both results are obtained under weaker ellipticity condition: For given \(\gamma \) there are positive constants \(\nu , \theta \) so that \[ \theta ^{\gamma } \leq | y^{\gamma }| \Rightarrow \nu | \xi | ^2 \leq \sum _{i,j=1}^n a_{ij}^{\gamma \gamma }(x,y)\xi _i \xi _j. \]

35J62 Quasilinear elliptic equations
35J47 Second-order elliptic systems
35D30 Weak solutions to PDEs
35B45 A priori estimates in context of PDEs
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