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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) \(\int_ \Omega f(x,v(x),Dv(x))dx\) satisfies certain local boundedness conditions. Here \(\Omega\) is a bounded open set in \(\mathbb{R}^ n\) and \(f\) is defined on \(\Omega\times\mathbb{R}\times \mathbb{R}^ n\). It is assumed that \(f\) satisfies \[ \sum | z_ i|^{q_ i}- c_ 1| s|^ r- g(x)\leq f(x,s,z)\leq c_ 2\Bigl[\sum | z_ i|^{q_ i}+ | s|^ r+ g(x)\Bigr],\tag{2} \] where \(1\leq q_ i\leq \bar q^*\), \(1\leq r\leq \bar q^*\), \(\bar q= n(\sum q^{-1}_ i)^{-1}\), \(\bar q^*= n\bar q/(n-\bar q)\) if \(\bar q< n\) or any \(p\) if \(\bar q\geq n\), all summations extend over \(i= 1,2,\dots,n\), and \(g(x)\) is a non-negative function in \(L^ p(\Omega)\), where \(p>\max(1,n/\bar q)\). Let \(W^{1,(q_ i)}(\Omega)= \{u\in W^{1,1}(\Omega)\), \(D_ i u\in L^{q_ i}\}\) then if \(f\) satisfies (2) and if \(u\in W^{1,(q_ i)}(\Omega)\) is a local minimizer of (1) then \(u\in L^ \infty_{\text{loc}}(\Omega)\). The authors also consider the functional (3) \(\int_ \Omega f(Dv)dx\), where \(f\) is now a \(C^ 2\) function from \(\mathbb{R}^ n\) to \(\mathbb{R}\) satisfying \[ \sum | z_ i|^{q_ i}\leq f(z)\leq c_ 1\Bigl(1+ \sum| z_ i|^{q_ i}\Bigr),\tag{4} \]
\[ c_ 2| \xi|^ 2\leq \sum f_{z_ i z_ j}\xi_ i \xi_ j\leq c_ 2\Bigl(1+\sum | z_ i|^{q_ i-2}| \xi|^ 2\Bigr)\tag{5} \] for any \(\xi\in \mathbb{R}^ n\), where \(2\leq q_ i\leq 2n/(n- 2)\) if \(n>2\) (\(q_ i\geq 2\) if \(n=2\)) and \(q_ 1\leq q_ 2\leq q_ n\), \(q_{n-1}< 2n/(n- 2)\). Then if \(u\in W^{1,(q_ i)}(\Omega)\) is a local minimizer of (3) and \(f\) satisfies (4), (5) then \(u\in W^{1,\infty}_{\text{loc}}(\Omega)\cap W^{2,2}_{\text{loc}}(\Omega)\).

MSC:
49N60 Regularity of solutions in optimal control
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