论文标题
针对一类非自治障碍问题的解决方案的可不同性结果具有较高的障碍性问题
Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
论文作者
论文摘要
我们建立了整数和分数顺序的一些更高的可不同性结果，以解决该形式的非自治障碍问题 \ begin {equation*} \ min \ left \ {\int_Ωf（x，dv（x））\，：\，v \ in \ Mathcal {K}_ψ（ω）\ right \}， \ end {equation*} 如果功能$ f $相对于梯度变量满足$ p-$增长条件，则$ 1 <p <2 $，而$ \ Mathcal {k}_ψ（ω）$是u_0+w^{1，p} _0（ω）$ v \ gege的$ v \ gege 6 $ a a in u_0+w^{1，p} in。 e。在$ω$中，其中$ u_0 \ in w^{1，p}（ω）$是固定的边界数据。 在这里，我们表明，如果部分地图$ x \ mapstod_ξf（x，ξ）$属于合适的sobolev或besov空间。这里的新颖之处在于，我们处理相对于梯度变量，i。 e。 $ f（x，ξ）\大约a（x）|ξ|^p $，$ 1 <p <2，$以及地图$ a $属于sobolev或besov-lipschitz空间。
We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf(x, Dv(x))\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions $v\in u_0+W^{1, p}_0(Ω)$ such that $v\geψ$ a. e. in $Ω$, where $u_0\in W^{1,p}(Ω)$ is a fixed boundary datum. Here we show that a Sobolev or Besov-Lipschitz regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i. e. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev or Besov-Lipschitz space.
