# Questions tagged [maximum-principle]

For questions about the use of maximum principle, either in the context of partial differential equations or complex analysis.

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### Maximum principle for elliptic equation in exterior domain

I have a question on maximum principle for elliptic equation in exterior domain. Suppose that $u$ is infinitely differentiable in $\mathbb{R}^n$ and bounded in $\mathbb{R}^n$. I want to prove that ...
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### Use Schwarz' Lemma and/or Maximum Modulus Principle to Prove This Proposition

Suppose the following conditions hold true: (1) The function $f$ is analytic and contractive in the open unit disk. (2) $f(0)=0$. (3) $\exists z_1 \ne z_2 \in B(0,1)$, $|z_1|=|z_2|$, $f(z_1)=f(z_2)$...
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### Maximum modulus principle problem

I have the function $f$ which is holomorphic on the open unit disc $|z|<1$ and satisfies $|f(z)|\leq cos(\frac{\pi |z|}{2})$. I am asked to find $f$. I know that I am supposed to use maximum ...
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### Dirichlet Problem with Isolated point has no solution

I have found this question on Fraenkel's Maximum Principles: Suppose I have a bounded domain with an isolated point on the boundary. I do not understand how there can be certain functions $g(x)$ for ...
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### If $f,g$ are holomorphic in an open connected set $U$ and for some $z_0\in U$ we have $|f|+|g|\leq|f(z_0)|+|g(z_0)|$ then $f$ and $g$ are constant [duplicate]

I want to prove the statement of the question. I know we should apply some form of the maximum modulus principle, however, I don't know which function to pick, because the absolute values are quite ...
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### Interior ball condition, outer unit normal vector

Suppose $U$ is bounded and open. If $U$ satisfies the interior ball condition at $x^0$, then there exists an open ball $B\subset U$ with $x^0\in \partial B$. Question: If $x^0\in\partial U$, is the ...
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### Show that $\max_{\bar \Omega} u=\max_{K_{T}} u$. Where $K_{T}$ is the parabolic boundary of $\Omega$.

Let $\Omega \subset \mathbb R^{n}$ be a bounded, smooth domain. Suppose that for some $T>0$, u is continuous on the closure $\Omega_{T}:= \Omega \times (0,T)$ and is a smooth, non-negative ...
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### prove you have found all such functions

find all possible entire functions f with the property that $|f(z)|\le2|z|+1$ for all $z\in C$. Prove that you have found all such functions. First of all I am self studying complex analysis so sorry ...
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### Maximum Principle of Laplace equations

$u$ is the $C^2$ solution of $$\begin{cases}\Delta u = 0 &\text{in }\mathbb{R}^d\backslash B_R\\ u = 0 & \text{on } \partial B_R\end{cases}$$ So the problem asks to show that ...
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While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ... 0answers 27 views ### Weak maximum principle for the Dirichlet problem In Brezis' functional analysis he goes through the weak maximum principle for the Dirichlet problem. Brezis assumes$u\in H^{1}(\Omega)\cap C(\overline{\Omega})$and then fixes a function$G\in C^{1}(\...
I'm interested in proving that the following parabolic problem has a unique solution: \begin{cases} \dfrac{\partial y}{\partial t} (t;x)=r\cdot y(t;x)+D\cdot\Delta y(t;x)-c(x;t)\cdot y(t;x),\ \...