# 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 compact manifolds

Are the maximum/minimum principles on $\mathbb{R}^n$ available for both local or nonlocal operators can be modified for a compact Riemannian manifold? For instance, we have a nonlocal maximum ...
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### Maximum Principle Theorem

I am reading the mean curvature flow from Mantegazza. The maximum principle is stated as follows: Two questions: Why does $u_{\text{max}}:=\max u(p,t)$ exist? Why is there $T'$ instead of $T$ in ...
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### Estimation of heat flux at the boundary

So for heat equation $$T_t - k \Delta T = f(t)1_{r <= R}(r)$$ with initial condition $T(r,0) = 0$. where $r = ||(x,y,z)||$. $f(t)$ could be any positive function that it's integral over time is ...
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### Herman rings for polynomials

I am reading this link on complex dynamics and in Problem 12-1 it asks the reader to prove, using the Maximum Modulus Principle, that Herman rings cannot occur for polynomials. I have seen this ...
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### Cauchy problem for parabolic equation

Let $\mathcal{L}=a \partial_{xx}+b\partial_{x}+c$, where $a>0,c\in\mathbb{R},b\in \mathbb{R}$, I want to know if the following parabolic equation gets a smooth and unique solution \begin{align} &...
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### Maximum Principle for Poisson’s Equation

I understand how to prove the maximum principle for $u_{xx}+u_{yy}=0$, but how does this extend to a maximum principle for the equation $u_{xx}+u_{yy}=f$? I believe this is called Poisson’s equation. ...
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### Inequality for a harmonic equation

Disclaimer: I am attempting to make progress on the following problem that is part a homework assignment. Thus, I am hoping for hints/suggestions and not necessarily the full solution. Let the half-...
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### Assume its maximum value or minimum value in $\mathbb{C}$ or not?

Consider the following statements: (a) $|e^{\sin z}|$ does not assume its maximum value in $\mathbb{C}$. (b)$|\sin (e^z)|$ does not assume ist minimum value in $\mathbb{C}$, Then A. only A is true B. ...
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### Maximum of elliptic PDE

Let $\Omega =(0,1)^2$ and \begin{align} -div(p\nabla u)(x,y)=f(x,y) \text{ for } (x,y)\in \Omega\\ \end{align} for $p\in C^1$ and $u\in C^2(\Omega)\cap C(\overline{\Omega})$. If $f<0$ and $p\geq 0$ ...
I consider the very simple $p-$Laplacian equation with Dirichlet condition: $$- \Delta_p u = \alpha\ \mbox{in}\ \Omega \subset \mathbb{R}^N,\ N \geq 3,$$ $$u = 0\ \mbox{on}\ \partial \Omega,$$ ...
In the "textbook" theory of optimal control, the state variable $x(\cdot)$ is often assumed to be differentiable, or piece-wise differentiable. I am interested in a control problem in which \$...