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Estimates of the derivatives of minimizers of a special class of variational integrals

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  • The note concerns on some estimates in Morrey Spaces for the derivatives of local minimizers of variational integrals of the form $$\int_\Omega F (x,u,Du) dx $$ where the integrand has the following special form $$ F(x,u,Du)\, =\, A(x,u, g^{\alpha\beta}(x) h_{ij}(u) \frac{\partial u^i}{\partial x^\alpha} \frac{\partial u^i }{\partial x^\beta}), $$ where $(g^{\alpha\beta})$ and $(h_{ij})$ symmetric positive definite matrices. We are not assuming the continuity of $A$ and $g$ with respect to $x$. We suppose that $A(\cdot, u,t)/(1+t)$ and $g(\cdot)$ are in the class $L^\infty\cap VMO$.
    Mathematics Subject Classification: Primary: 49N60, 35J50, 35R05; Secondary: 46E30, 35B65.


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