# Questions tagged [maximum-principle]

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

371 questions
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### finding maximum value of BVP

Find the maximum value of $u$ in the following BVP \begin{align*} u_t &= u_{xx}, \; \; t,x \in (0,1) \\ u(0,t) &= 2t^2-t \\ u(1,t) &= \sin \pi t \\ u(x,0) &= x(...
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### Converse of the one-dimensional maximum principle

If a positive function $u:[0,1]\to \mathbb{R}^+$ is known to be continuous and twice differentiable, and to have no local maxima in $(0,1)$ (i.e. $u$ is maximised on the boundary $\{0,1\}$), is it ...
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### Maximum principle for discretized ODE

I discretized the following ODE using central finite differences for 1st and 2nd derivatives: $$u''-bu'=f(u), x\in (0,1)\\u(0)=1, u'(1)=0\\ b>0, f:\mathbb{R_{\ge 0}}\to \mathbb{R}_{\ge 0}$$ The ...
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### Maximum principle at the border

Let $Ω⊂⊂R^{2}$ a bounded regular domain and $u: \overline{Ω} → R$ a function such that $det(∇^{2}u)≤0$ in $Ω$. I'm trying to proof that you have reached your maximum and minimum in $∂Ω$. In my opinion,...
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### Maximum principle for first-order quasilinear pde

Let's suppose that $u\in C^{1}(K[0,1])$ solves $$a(x,y)u_{x}+b(x,y)u_{y}=-u$$ in $K[0,1]$. I need to show that if $$a(x,y)x+b(x,y)y>0$$ for all $(x,y)\in\partial K[0,1]$ then $u\equiv0$ in $K[0,1]$....
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### Uniqueness of Non-Linear Heat Equation

The following problem comes from an old exam: Consider \begin{cases} u_t - \Delta u + |u_{x_1}| = 0 \text{ in } \mathbb{R}^{n} \times (0,\infty) \\ u(x,0)=g(x) \text{ in } \mathbb{R}^n \end{cases} ...
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### Fastest structured way to get max(abc) if a+b+c=30

What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer.
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### Closed curve for an analityc function

If I have an analityc function $f(z)$ on a domain $B \subset \mathbb{C}$ and a simple and closed curve $C$ that encloses $B$, and if $|f|$ is constant over C, the $f$ is constant on $B$? I haven’t ...
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### Harmonic functions and Maximum Modulus Principle. (Real harmonic function implies holomorphic function?)

Theorem. (Maximum Modulus Principle) $\;$If $f$ is a non-constant holomorphic function in a region $\Omega$, then $f$ cannot attains a maximum in $\Omega$. Problem. Prove the maximum principle ...
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### Use maximum modules principle to prove that if $\sup_{|z| = R}|f(z)|\leq AR^{k} + B$ with $f$ entire, then $f$ is a polynomial

Use Cauchy inequality or maximum modules principle to prove that if $f$ is entire function that satisfies $$\sup_{|z| = R}|f(z)|\leq AR^{k} + B$$ for all $R >0$, for some $k \in \mathbb{Z}$ and ...
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### Weak Maximum Principle For Parabolic Equations With Neumann Boundary Condition

Consider the problem: \begin{equation} Lu\equiv \sum^n_{i, j=1} a_{ij}(x, t)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum^n_{i=1} b_i(x, t)\frac{\partial u}{\partial x_i}+c(x, t)-\frac{\partial u}...
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### Question on Laplace-Beltrami operator and the maximum principle

So I have this problem on my mind: Let $\Gamma$ be a $2d-$compact manifold and suppose that $u$ is a solution of: $-\Delta u=0 \;\forall x\in \Gamma$ ( $\Delta$ here denotes the Laplace-...
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### Proving $\rvert 1+\cos(z)+\cos(2z)+ \dotsb +\cos(nz)\lvert>n+1$

Let $n\in \Bbb N$. Show that for each $r>0$ there exists $z\in D(0,r)$ such that $\rvert 1+\cos(z)+\cos(2z)+ \dotsb +\cos(nz)\lvert>n+1$. I think it could work with the Maximum Principle ...