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Some remarks on the regularity of minima of anisotropic integrals. (English) Zbl 0795.49025

The authors state appropriate growth conditions to be imposed on the function f to ensure that a local minimizer u(x) of the functional (1) Ω f(x,v(x),Dv(x))dx satisfies certain local boundedness conditions. Here Ω is a bounded open set in n and f is defined on Ω×× n . It is assumed that f satisfies

|z i | q i -c 1 |s| r -g(x)f(x,s,z)c 2 | z i | q i + |s| r + g (x),(2)

where 1q i q ¯ * , 1rq ¯ * , q ¯=n(q i -1 ) -1 , q ¯ * =nq ¯/(n-q ¯) if q ¯<n or any p if q ¯n, all summations extend over i=1,2,,n, and g(x) is a non-negative function in L p (Ω), where p>max(1,n/q ¯). Let W 1,(q i ) (Ω)={uW 1,1 (Ω), D i uL q i } then if f satisfies (2) and if uW 1,(q i ) (Ω) is a local minimizer of (1) then uL loc (Ω). The authors also consider the functional (3) Ω f(Dv)dx, where f is now a C 2 function from n to satisfying

|z i | q i f(z)c 1 1+| z i | q i ,(4)
c 2 |ξ| 2 f z i z j ξ i ξ j c 2 1+| z i | q i -2 |ξ| 2 (5)

for any ξ n , where 2q i 2n/(n-2) if n>2 (q i 2 if n=2) and q 1 q 2 q n , q n-1 <2n/(n-2). Then if uW 1,(q i ) (Ω) is a local minimizer of (3) and f satisfies (4), (5) then uW loc 1, (Ω)W loc 2,2 (Ω).


MSC:
49N60Regularity of solutions in calculus of variations