We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in $L^q$ with $q>n$. Our approach is based on the scaling and comparison arguments of [13], and we show that all our assumptions are sharp.

As a consequence of our results, we deduce the ${\mathcal{C}}^{1,\alpha }$ regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

中文翻译：

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右手边的新边界Harnack不等式

对于右手边的方程，我们证明了Lipschitz域中新的边界Harnack不等式。我们的主要结果适用于可测系数有界的非散度形式算子和具有连续系数的散度形式算子，而右侧位于${\mathrm{大号}}^q$ 和 $q>ñ$。我们的方法基于[13]的比例和比较论证，并且我们证明了我们所有的假设都是正确的。

由于我们的结果，我们推断出 ${\mathcal{C}}^{\mathrm{1个}，\alpha }$ 完全非线性障碍问题和完全非线性薄障碍问题中自由边界的正则性。