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

摘要

Besides other things we prove that if u is an element of L-loc(infinity)(Omega; R-M), Omega subset of R-n, locally minimizes the energy integral(Omega) [a(vertical bar(del) over tildeu vertical bar) + b(vertical bar partial derivative(n)u vertical bar)] dx, (del) over tilde :=(partial derivative(1),..., partial derivative(n-1)), with N-functions a <= b having the Delta(2)-property, then vertical bar partial derivative(n)u vertical bar(2)b(vertical bar partial derivative(n)u vertical bar) is an element of L-loc(1)(Omega). Moreover, the condition b(t) <= const t(2)a(t(2)) (*) for all large values of t implies |(del) over tildeu vertical bar(2)a(vertical bar(del) over tildeu vertical bar is an element of L-loc(1)(Omega). If n = 2, then these results can be improved up to vertical bar del u vertical bar is an element of L-loc(s) (Omega) for all s < infinity without the hypothesis (*). If n >= 3 together with M = 1, then higher integrability for any exponent holds under more restrictive assumptions than (*).