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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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##### References:
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