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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) $\int\sb \Omega f(x,v(x),Dv(x))dx$ satisfies certain local boundedness conditions. Here $\Omega$ is a bounded open set in $\bbfR\sp n$ and $f$ is defined on $\Omega\times\bbfR\times \bbfR\sp n$. It is assumed that $f$ satisfies $$\sum \vert z\sb i\vert\sp{q\sb i}- c\sb 1\vert s\vert\sp r- g(x)\le f(x,s,z)\le c\sb 2\Bigl[\sum \vert z\sb i\vert\sp{q\sb i}+ \vert s\vert\sp r+ g(x)\Bigr],\tag2$$ where $1\le q\sb i\le \bar q\sp*$, $1\le r\le \bar q\sp*$, $\bar q= n(\sum q\sp{-1}\sb i)\sp{-1}$, $\bar q\sp*= n\bar q/(n-\bar q)$ if $\bar q< n$ or any $p$ if $\bar q\ge n$, all summations extend over $i= 1,2,\dots,n$, and $g(x)$ is a non-negative function in $L\sp p(\Omega)$, where $p>\max(1,n/\bar q)$. Let $W\sp{1,(q\sb i)}(\Omega)= \{u\in W\sp{1,1}(\Omega)$, $D\sb i u\in L\sp{q\sb i}\}$ then if $f$ satisfies (2) and if $u\in W\sp{1,(q\sb i)}(\Omega)$ is a local minimizer of (1) then $u\in L\sp \infty\sb{\text{loc}}(\Omega)$. The authors also consider the functional (3) $\int\sb \Omega f(Dv)dx$, where $f$ is now a $C\sp 2$ function from $\bbfR\sp n$ to $\bbfR$ satisfying $$\sum \vert z\sb i\vert\sp{q\sb i}\le f(z)\le c\sb 1\Bigl(1+ \sum\vert z\sb i\vert\sp{q\sb i}\Bigr),\tag4$$ $$c\sb 2\vert \xi\vert\sp 2\le \sum f\sb{z\sb i z\sb j}\xi\sb i \xi\sb j\le c\sb 2\Bigl(1+\sum \vert z\sb i\vert\sp{q\sb i-2}\vert \xi\vert\sp 2\Bigr)\tag5$$ for any $\xi\in \bbfR\sp n$, where $2\le q\sb i\le 2n/(n- 2)$ if $n>2$ ($q\sb i\ge 2$ if $n=2$) and $q\sb 1\le q\sb 2\le q\sb n$, $q\sb{n-1}< 2n/(n- 2)$. Then if $u\in W\sp{1,(q\sb i)}(\Omega)$ is a local minimizer of (3) and $f$ satisfies (4), (5) then $u\in W\sp{1,\infty}\sb{\text{loc}}(\Omega)\cap W\sp{2,2}\sb{\text{loc}}(\Omega)$.

49N60Regularity of solutions in calculus of variations
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