We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $\Omega \subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1 (\Omega)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.

中文翻译：

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具有梯度非线性的半线性椭圆型方程的部分数据反问题。

我们表明，在有界的闭合固有子集上消失的有界光滑域$ \ Omega \ subset \ mathbb {R} ^ n $上，谐波函数梯度的标量集合的线性跨度在$ L中是密集的^ 1（\ Omega）$。我们将此密度结果用于解决一类具有二次梯度项的半线性椭圆PDE的部分数据逆边界问题。