Variational integrals of splitting-type : higher integrability under general growth conditions

Abstract

Besides other things we prove that if u\in L_{loc}^{\infty}(\Omega;\mathbb{R}^{M}),\Omega\subset\mathbb{R}^{n}, locally minimizes the energy \int_{\Omega}\left[a(\left|\tilde{\nabla}u\right|)+b(\left|\partial_{n}u\right|)\right]dx, \tilde{\nabla}:=(\partial_{1},...,\partial_{n-1}), with N-functions a\leq b having the \Delta_{2}-property, then \left|\partial_{n}u\right|^{2}b(\left|\partial_{n}u\right|)\in L_{loc}^{1}(\Omega). Moreover, the condition b(t)\leq constt^{2}a(t^{2}) (*) for all large values of t implies \left|\tilde{\nabla}u\right|^{2}a(\left|\tilde{\nabla}u\right|)\in L_{loc}^{1}(\Omega). If n = 2, then these results can be improved up to \left|\nabla u\right|\in L_{loc}^{s}(\Omega) for all s<\infty without the hypothesis (*). If n\geq3 together with M = 1, then higher integrability for any exponent holds under more restrictive assumptions than (*).