Let $$\Delta^2 u-\lambda u =0$$ where $\lambda>0$ and $$\Delta^2 u = \frac{\partial ^4 u }{\partial x^4} + 2 \frac{\partial^4 u }{\partial x^2 \partial y^2} + \frac{\partial ^4 u }{\partial y^4}$$ Is there a maximum principle for this equation in a rectangular domain $[-a,a]\times[-b,b]$ or any other domain? Fact: it is known that there is a […]

Can please somebody tell me, how solve this problem ? We say that the uniformly elliptic operator $$Lu\ =\ -\sum_{i,j=1}^na^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$ satisfies the weak maximum principle if for all $u\in C^2(U)\cap C(\bar{U})$ $$\left\{\begin{array}{rl} Lu \leq 0 & \mbox{in } U\\ u \leq 0 & \mbox{on} \partial U \end{array}\right.$$ implies that $u\leq 0$ […]

We say that the uniformly elliptic operator $$Lu\ =\ -\sum_{i,j=1}^na^{ij}u_{x_ix_j}\ +\ \sum_{i=1}^nb^iu_{x_i}\ +\ cu$$ satisfies the weak maximum principle if for all $u\in C^2(U)\cap C(\bar{U})$ $$\left\{\begin{array}{rl} Lu \leq 0 & \mbox{in } U\\ u \leq 0 & \mbox{on } \partial U \end{array}\right.$$ implies that $u\leq 0$ in $U$. Suppose that there exists a function $v\in […]

Prove: for $f,g \in Hol(G\subset \mathbb{C})$ $\implies$ max $(|f|+|g|)$ is on the boundary of $G$. I don’t really have a direction for this. I know it’s got to do with the maximum modulus principle, but I’m not really sure how to get to it. Any guidance is welcome!

Let $f$ be non-constant analytic in a neighborhood of the closed unit disk such that $|f|=1$ on the unit circle. Show that $f$ has a zero inside the unit disk. It has been suggested to argue this by contradiction and using the maximum principle. However, is it possible to argue this question using properties of […]

Let $u(x,t)$ be the solution to the problem $u_{t} = 2u_{xx}-2u_{x}-u$, $0<x<1$ where the b.c.’s are $u(0,t)=4t+1$, $u(1,t)=\cos t$ and the intitial condition $u(x,0) = 1$. Let $a \leq u(x,t) \leq b$ for all $0 \leq x \leq 1$ and $t \geq 0$. I need to find the best estimates for the bounds $a$ and […]

I am trying to identify for which $\lambda$ so that I can obtain the Maximum principle, that is, $$\sup_{\overline{\Omega}}u\leq \sup_{\partial \Omega}u$$ for equation $-\triangle u = \lambda u$, where the domain $\Omega$ is open bounded. I tried to mimic the prove in Evans book. Indeed, by assuming that $\lambda\leq 0$, I have $$-\triangle u \leq […]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is constant or $f$ has a zero inside $\gamma$ Here is my take: if $f$ is constant, i dont see […]

Let $\Omega$ be the right half plane excluding the imgainary axis and $f\in H(\Omega)$ such that $|f(z)|<1$ for all $z\in\Omega$. If there exists $\alpha\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $$\lim_{r\rightarrow\infty}\frac{\log|f(re^{i\alpha})|}{r}=-\infty$$ prove that $f=0$. The hint is define $g_n(z)=f(z)e^{nz}$, then by previous exericise $|g_n|<1$ for all $z\in\Omega$.

Let $f$ be an analytic function on a disc $D$ whose center is the point $z_0$. Assume that $|f'(z)-f'(z_0 )|<|f'(z_0)| $ on D. Prove that $f$ is one-to-one on D.

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