From 0d844cb23245f3df2dc8522053158ff2037428de Mon Sep 17 00:00:00 2001 From: Petr Velycko Date: Sat, 17 May 2025 06:31:51 +0200 Subject: [PATCH] proofreading --- jednoznacnost-reseni.tex | 15 ++++++++++++--- maximalni-reseni.tex | 4 ++-- skripta.pdf | Bin 227275 -> 227164 bytes 3 files changed, 14 insertions(+), 5 deletions(-) diff --git a/jednoznacnost-reseni.tex b/jednoznacnost-reseni.tex index e1822ce..f7f8cb1 100644 --- a/jednoznacnost-reseni.tex +++ b/jednoznacnost-reseni.tex @@ -47,9 +47,18 @@ Zavedeme značení $f \in C_x^1(\Omega)$, jestliže $\pdv{f}{x_i}$ existují a j Mějme $(x_0, t_0) \in \Omega$. Nechť $\delta > 0$ je takové, že množina $$M = \overline{U(x_0, \delta) \times (t_0 - \delta, t_0 + \delta)}$$ je podmnožinou $\Omega$. Z kompaktnosti $M$ máme, že parciální derivace $\pdv{f}{x_i}$ jsou omezené konstantou $K$. - Dále mějme dva body $(x, t),(y,t) \in M$. Potom $|f(x,t) - f(y, t)| = |f(x + 0(y - x), t) - f(x + 1(y - x)t)| = |\left[f(x+s(y - x), t)\right]^1_0| = |\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds|$. Pro derivaci $f$ platí $\odv*{f(x + s(y - x), t)}{s} = \sum_{i=1}^n \pdv{f}{x_i}(x + s(y - x), t) (y_i - x_i)$. Z toho máme, že - $$\left|\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds\right| \leq \int_0^1 \sum_{i=1}^n K|y_i - x_i| ds = \sum_{i=1}^n K \max_i |y_i - x_i| = $$ - $$n K \max |y_i - x_i| \leq nK | y - x |,$$ + Dále mějme dva body $(x, t),(y,t) \in M$. Potom + \begin{align*} + |f(x,t) - f(y, t)| &= |f(x + 0(y - x), t) - f(x + 1(y - x), t)| =\\ + &= |\left[f(x+s(y - x), t)\right]^1_0| = \left|\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds\right|. + \end{align*} + Pro derivaci $f$ platí + $$\odv*{f(x + s(y - x), t)}{s} = \sum_{i=1}^n \pdv{f}{x_i}(x + s(y - x), t) (y_i - x_i).$$ + Z toho máme, že + \begin{align*} + \left|\int_0^1 \odv*{f(x+s(y-x),t)}{s} ds\right| &\leq \int_0^1 \sum_{i=1}^n K|y_i - x_i| ds = \sum_{i=1}^n K \max_i |y_i - x_i| =\\ + &= n K \max |y_i - x_i| \leq nK | y - x |, + \end{align*} kde poslední nerovnost plyne z faktu, že $|y -x| = \sqrt{\sum_{i = 1}^n |y_i - x_i|^2}$. Tedy $f$ je lokálně lipschitzovská s konstantou $n \cdot K$. diff --git a/maximalni-reseni.tex b/maximalni-reseni.tex index b4d4421..ba3e1bf 100644 --- a/maximalni-reseni.tex +++ b/maximalni-reseni.tex @@ -14,9 +14,9 @@ V této kapitole se budeme věnovat otázce rozšíření řešení na co nejvě \begin{proof} Mějme řešení $(x, I)$ takové, že $I = (a, b)$. Budeme induktivně prodlužovat za bod $b$ (na druhou stranu se to pak udělá analogicky). Položme $x_0 = x$, $b_0 = b$, $I_0 = I$. V $n$-tém kroku dostaneme řešení $(x_n, I_n)$, kde $I_n = (a, b_n)$. Dále definujeme $\omega_n = \sup \{z > b_n; (x_n, I_n) \text{ lze prodloužit na } (a, z) \}$. Pokud příslušná množina je prázdná, jsme hotovi, neboť řešení již nejde prodloužit, tedy je maximální. - V opačném případě můžeme definovat $b_{n + 1} = \frac{b_n + \omega_n}{2}$ (pokud $\omega_n < \infty$), případně $b_{n + 1} = b_n + 1$. Tímto postupem získáme rostoucí posloupnost $b_n$, která musí mít limitu. Označme tuto limitu $\beta$. Dále položme $\tilde{I} = (a, \beta)$, $\tilde{x} = x_n(t)$, pro všechna $t \in \tilde{I}$ zvolím $n$ tak, aby $t \in I_n$. Na volbě $n$ nezávisí, neboť na příslušných intervalech jsou funkce $x_n$ stejné. + V opačném případě můžeme definovat $b_{n + 1} = \frac{b_n + \omega_n}{2}$ (pokud $\omega_n < \infty$), případně $b_{n + 1} = b_n + 1$. Tímto postupem získáme rostoucí posloupnost $b_n$, která musí mít limitu. Označme tuto limitu $\beta$. Dále položme $\tilde{I} = (a, \beta)$, $\tilde{x} = x_n(t)$, pro všechna $t \in \tilde{I}$ zvolím $n$ tak, aby $t \in I_n$. Na volbě $n$ nezáleží, neboť na příslušných intervalech jsou funkce $x_n$ stejné. - Dokážeme, že takto definované řešení $(\tilde{x}, \tilde{I})$ je maximální. Pro spor budeme předpokládat, že existuje rozšíření na $(a, \hat{\beta})$ takové, že $\hat{\beta}$. Okamžitě vidíme, že $\beta < \infty$. Vezmeme $n$ takové, aby $\beta - b_n < \hat{\beta} - \beta$ a $\beta - b_n < 1$ (existuje díky tomu, že $b_n$ konvergují k $\beta$). V tom případě $(x_n, I_n)$ má prodloužení až do $\hat{\beta}$, tedy $\omega_n \geq \hat{\beta}$. Pak ale (pokud $\omega_n = \infty$) $b_{n + 1} = b_n + 1 > \beta$, máme spor, případně pro $\omega_n$ konečné máme $b_{n + 1} = \frac{b_n + \omega_n}{2} > \frac{2\beta - \hat{\beta} + \hat{\beta}}{2} = \beta$, opět jsme došli ke sporu. + Dokážeme, že takto definované řešení $(\tilde{x}, \tilde{I})$ je maximální. Pro spor budeme předpokládat, že existuje rozšíření na $(a, \hat{\beta})$ takové, že $\hat{\beta} > \beta$. Okamžitě vidíme, že $\beta < \infty$. Vezmeme $n$ takové, aby $\beta - b_n < \hat{\beta} - \beta$ a $\beta - b_n < 1$ (existuje díky tomu, že $b_n$ konvergují k $\beta$). V tom případě $(x_n, I_n)$ má prodloužení až do $\hat{\beta}$, tedy $\omega_n \geq \hat{\beta}$. Pak ale (pokud $\omega_n = \infty$) $b_{n + 1} = b_n + 1 > \beta$, máme spor, případně pro $\omega_n$ konečné máme $b_{n + 1} = \frac{b_n + \omega_n}{2} > \frac{2\beta - \hat{\beta} + \hat{\beta}}{2} = \beta$, opět jsme došli ke sporu. \end{proof} V případě $f$ lipschitzovské se důkaz dá výrazně zjednodušit. 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