论文标题
针对一类非自治障碍问题的解决方案的规律性结果
Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
论文作者
论文摘要
我们为解决形式的非自治障碍问题的解决方案建立了一些更高的可不同性结果 \ begin {equation*} \ min \ left \ {\int_Ωf\ left（x，dv（x）\ right）dx \，：\，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）$是一类可允许的功能。在这里，我们表明，如果障碍物$ψ$有限，则Sobolev的规律性假设对障碍物$ψ$转移到解决方案梯度的梯度，前提是部分地图$ x \ x \ mapstod_ξf（x，x，ξ）$属于sobolev space，$ w^^{1，p+2} $。这里的新颖之处在于，我们处理相对于梯度变量的次级生长条件，即$ f（x，ξ）\ of a（x）|ξ|^p $，$ 1 <p <2，$以及地图$ a $属于Sobolev空间。
We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf\left(x, Dv(x)\right)dx\,:\, 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. Here we show that, if the obstacle $ψ$ is bounded, then a Sobolev 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 Sobolev space, $W^{1, p+2}$. 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 space.
