论文标题
在次级生长条件下的一类非同种元素椭圆问题的解决方案的较高的可不同性结果
Higher differentiability results for solutions to a class of non-homogeneouns elliptic problems under sub-quadratic growth conditions
论文作者
论文摘要
我们证明，对于$$ \ mathcal {f} \ left（w，ω\ right）= \int_Ω\ left [f \ left（x，dw（x）\ right）-f（x）-f（x）\ cdot w（x）dx $ fit nony-ny-autons concomply的$ c.梯度变量在$ p $ - 生长条件下，$ 1 <p <2 $。这里的主要新颖之处在于，假设部分地图$ x \ mapstod_ξf（x，ξ）$在某些lebesgue space $ l^q $中具有弱衍生物，并且假定基准$ f $属于合适的lebesgue space $ l^r $。 我们还证明，如果假定最小化器是先验的界限，则可以削弱基准$ f $和地图上的假设。
We prove a sharp higher differentiability result for local minimizers of functionals of the form $$\mathcal{F}\left(w,Ω\right)=\int_Ω\left[ F\left(x,Dw(x)\right)-f(x)\cdot w(x)\right]dx$$ with non-autonomous integrand $F(x,ξ)$ which is convex with respect to the gradient variable, under $p$-growth conditions, with $1<p<2$. The main novelty here is that the results are obtained assuming that the partial map $x\mapsto D_ξF(x,ξ)$ has weak derivatives in some Lebesgue space $L^q$ and the datum $f$ is assumed to belong to a suitable Lebesgue space $L^r$. We also prove that it is possible to weaken the assumption on the datum $f$ and on the map $x\mapsto D_ξF(x,ξ)$, if the minimizers are assumed to be a priori bounded.
