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On the regularity of minimizers to degenerate functionals

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  • In this paper, we prove a higher integrability result for the gradient of a minimizer of a functional of the type

    $I(\Omega , u)=\int_{\Omega}\sum_{i,j} a_{i,j} D_i u D_jv dx$

    whose coefficient matrix $A(x)= ^tA(x)$ satisfies the anisotropic bounds

    $\frac{|\xi |^2}{K(x)}\leq < A(x) \xi, \xi > \leq K(x) |\xi |^2\quad \forall \xi \in R^n,$ for a.e. $x\in \Omega,$

    where $ K:\Omega \subset R^n \rightarrow [1,+\infty),$ a locally integrable function in $\Omega$, belongs to $A_2 \cap G_n$ and has a majorant $Q(x)\geq K(x)$ of finite mean,

    limsup$_{R \rightarrow 0} \int_{B_R(x)} Q(y)dy < \infty $ at every point $x \in \Omega. $

    Mathematics Subject Classification: 42B25, 42B35.


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