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I have been trying to make sense of the proof of Lemma 3.1 in the article Stabilizing inverse problems by internal data by P. Kuchment and D. Steinhauer. The lemma is on page 6 and the proof is found on page 17. Most of it makes sense to me, however, what I don't understand is how they obtain equation (60).

They state that they apply Taylor's theorem to $ F $ to obtain the inequality, but I have a few problems in following exactly how they are doing this. Here are my thoughts.

$ F(y,z,w) $ is assumed to be smooth for $ y,z \in \mathbb{R} $ and $ w > 0 $, thus a Taylor explansion is fine: $$ F(x+\delta x) = F(x) + \nabla F(x)\cdot\delta x + \delta x^T H_{F(\xi(x))}\delta x, $$ where $ x = (y,z,w) $ and $ \delta x = (\delta y, \delta z, \delta w) $. However, I have a problem with the parameter $ |\mathbf{h}_0 + \mathbf{h}| $ in their expression for $ E $. Somehow it seems to me like the expand $ F(f_0+f,g_0+g,|\mathbf{h}_0+\mathbf{h}|) $ using the above Taylor expansion, by putting $ w = |\mathbf{h}_0| $ and $ \delta w = \frac{\mathbf{h}_0\cdot\mathbf{h}}{|\mathbf{h_0}|}, $ which I don't understand how they can do, as $ |\mathbf{h}_0+\mathbf{h}| \neq |\mathbf{h}_0| + \frac{\mathbf{h}_0\cdot\mathbf{h}}{|\mathbf{h}_0|} $ in general. And since $ \mathbf{h} $ is an $ L^2(\Omega)^n $-function I feel like there is even more complexity than this simple vector relation to consider.

Can anyone explain what it is I am missing here?

1 Answers 1

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Let $ \delta w $ satisfy $ |h_0 +h| = |h_0| + \delta w $, then $$ \delta w = |h_0 +h | - |h_0| = \frac{|h|^2 + 2(h_0,h)}{|h_0+h|+|h_0|}. $$

Let $ E $ be as in the article, $$ \begin{aligned} E &= F(f_0+f,g_0+g,\underbrace{|h_0+h|}_{|h_0|+\delta w})-F(f_0,g_0,|h_0|) -\nabla F(f_0,g_0,|h_0|)\cdot\left(f,g,\frac{(h_0,h)}{|h_0|}\right). \end{aligned} $$ Using the Taylor expansion, $ x = (f_0,g_0,|h_0|) $, $ \delta x = (f,g,\delta w) $, $$ F(x+\delta x) = F(x) + \nabla F(x)\cdot\delta x + \delta x^T H_{F(\xi)}\delta x, $$ with $ \xi = \xi(\delta x) $, we obtain $$ \begin{aligned} E &= \partial_wF(f_0,g_0,|h_0|)\left(\delta w - \frac{(h_0,h)}{|h_0|}\right) + (f,g,\delta w)^T H_{F(\xi)}(f,g,\delta w). \end{aligned} $$ We have $$ \begin{aligned} \delta w - \frac{(h_0,h)}{|h_0|} &= \frac{|h|^2}{|h_0+h|+|h_0|} + \frac{2(h_0,h)}{|h_0+h|+|h_0|} - \frac{(h_0,h)}{|h_0|} \\ &=\frac{|h|^2}{|h_0+h|+|h_0|} + \left(\frac{h_0}{|h_0|}\left(\frac{|h_0|-|h_0+h|}{|h_0+h|+|h_0|}\right),h\right) \\ &=\frac{|h|^2}{|h_0+h|+|h_0|} - \left(\frac{h_0}{|h_0|}\left(\frac{|h|^2+2(h_0,h)}{(|h_0+h|+|h_0|)^2}\right),h\right) \\ &=\frac{|h|^2}{|h_0+h|+|h_0|} - \frac{|h|^2(h_0,h)}{|h_0|(|h_0+h|+|h_0|)^2} - \frac{2(h_0,h)^2}{|h_0|(|h_0+h|+|h_0|)^2}. \\ \left|\delta w - \frac{(h_0,h)}{|h_0|}\right|&\leq \frac{|h|^2}{|h_0+h|+|h_0|} + \frac{|h|^3}{(|h_0+h|+|h_0|)^2} + \frac{2|h_0|\cdot|h|^2}{(|h_0+h|+|h_0|)^2} \\ &= |h|^2\left(\frac{1}{|h_0+h|+|h_0|} + \frac{|h|+2|h_0|}{(|h_0+h|+|h_0|)^2}\right) \leq C_1|h|^2. \end{aligned} $$ The constant comes from $ |h(x)| < m $ on $ \Omega\backslash U $.

We note that since $ D^\alpha F $ is continuous and all of $ f_0,f,g_0,g,|h_0|,|h| $ are bounded on $ \Omega\backslash U $, $ H_{F(\xi)} $ is bounded, i.e. there is a constant $ |H_{F(\xi)}v| \leq C_2|v| $.

Finally we consider $ \delta w^2 $, $$ \begin{aligned} |\delta w| &\leq \left|\delta w - \frac{(h_0,h)}{|h_0|}\right| + \left|\frac{(h_0,h)}{|h_0|}\right| \leq C_1|h|^2 + |h| \\ |\delta w|^2 & \leq (C_1^2m^2 + 1 + 2C_1m)|h|^2 = C_3|h|^2. \end{aligned} $$ With all of this in place, let $ C $ denote an arbitrary positive constant, not necessarily the same every time $$ \begin{aligned} |E| &= \left|\partial_wF(f_0,g_0,|h_0|)\left(\delta w - \frac{(h_0,h)}{|h_0|}\right) + (f,g,\delta w)^T H_{F(\xi)}(f,g,\delta w)\right| \\ &\leq \left|\partial_wF(f_0,g_0,|h_0|)\right|\cdot\left|\delta w - \frac{(h_0,h)}{|h_0|}\right| + \left|(f,g,\delta w)^T H_{F(\xi)}(f,g,\delta w)\right| \\ &\leq \left|\partial_wF(f_0,g_0,|h_0|)\right|C_1|h|^2 + \left|(f,g,\delta w)\right|\cdot\left|H_{F(\xi)}(f,g,\delta w)\right| \\ &\leq C|h|^2 + C_2|(f,g,\delta w)|^2 \leq C|h|^2 + C_2(|f|^2+|g|^2+|\delta w|^2) \\ &\leq C|h|^2 + C_2(|f|^2+|g|^2+C_3|h|^2) \leq C(|f|^2+|g|^2+|h|^2). \end{aligned} $$