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

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12 views

Diffusion/Heat equation, weak maximum principle

Consider the problem: $$u_t -div(A(x) \nabla u) +a(x) u = f $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) $$ and its variational formulation $$ \lt \dot{u(t)},v \gt_* + B(u(t),v;t) =(f,v)_{L^2} \quad \forall ...
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1answer
23 views

Counterexample to Maximum modulus principle [on hold]

How can I show a counterexample to the maximum modulus principle? The Maximum modulus principle states, suppose $f$ is holomorphic and nonconstant in a region $G$. Then $|f|$ does not attain a weak ...
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9 views

Strong maximum principle for heat equation

Let $M$ be a closed Riemannian manifold. If $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ is a weak solution of $$u_t - \Delta u = f$$ $$u(0) =u_0$$ where $f \in L^2(0,T;L^2)$ with $f(t,x) \geq 0$ a.e. and ...
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1answer
13 views

Sign of coefficient in the weak maximum principle

( Weak maximum principle) Let $u \in C^{2}(\Omega)\cap C(\bar{\Omega})$ satisfies $Lu \geq 0$ with $c(x) \leq 0$ Em $\Omega$. Then $u$ attains in $\partial \Omega$ its nonnegative maximum in ...
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1answer
51 views

On the curve $y =\frac{ 2x - 1} {x - 1}$ , $x > 1$ drag tangent such that area of right triangle bounded by this tangent and the axes is minimal .

On the curve $y =\frac{ 2x - 1} {x - 1}$ , $x > 1$ drag tangent such that area of right triangle bounded by this tangent and the axes is minimal . What is the minimum value of this area ? My ...
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43 views

Dirichlet problem of second-order elliptic PDE on unbounded open set

Set $\Omega$ an unbounded open set in $\mathbb{R^n}$, $L$ is the elliptic operator $Lu=-a_{ij}D_{ij}u+b_i D_i u+cu$ which satisfies the uniform elliptic condition. Also $c \ge 0$, $u \in C^2(\Omega) ...
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1answer
31 views

$L^\infty$ bound on solution of $u_t - \Delta u + cu = f$ and dependence on $c > 0$

Let $u$ be the weak solution to $$u_t - \Delta u + cu = f$$ $$u(0) = 0$$ with zero Neumann data, on a bounded domain and time interval $[0,T]$. Here $c > 0$. If $f$ is in $L^\infty$ in time and ...
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1answer
28 views

Why $\frac{\partial u}{\partial \nu} \ge 0$ at the maximum point?

In Evans' book "Partial Differential Equations", on page 348, it is mentioned that the outer normal derivative at the maximum point (if it exists) $\displaystyle \frac{\partial u}{\partial \nu}(x_0) ...
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1answer
14 views

Analytic function on the annulus $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ s.t. $C_1 \to C_1$, $C_4 \to C_2$?

Question: Does there exist an analytic function mapping $A = \{z : 1 \le |z| \le 4\}$ onto $B = \{z : 1 \le |z| \le 2\}$ and taking $C_1 \to C_1$, $C_4 \to C_2$, where $C_r$ is the circle of radius ...
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51 views

(without Phragmén-Lindelöf) $f$ is of exponential type and bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}$

Question: Without using Phragmén-Lindelöf, show that if $f$ is of exponential type and uniformly bounded on the real axis, then there exists $C>0$ such that $|f(x+iy)|\leq Ce^{\tau |y|}.$ ...
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1answer
38 views

How can we compute the maximum of $|\zeta(z)|$ on the square with vertices $2,3,3+i$ and $2+i$?

By the maximum modulus principle, since $\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^z}$, with $\Re z>1$, is analytic inside the square Q of vertices $2,3,3+i$ and $2+i$, and continuous on Q, ...
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0answers
20 views

Inequality among Hessians

I know that this could be a stupid question but I would like to be sure on this point. I need to study the Maximum Principle on Petersen's Riemannian Geometry. This is the first Lemma. Let $ ...
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0answers
40 views

Maximum Principle for Nonlinear Heat Equation

For a function $f(x,t)$ I'm aware of a fairly strong result about the PDE: $\sigma^2 f_{xx} + f_t = 0,$ $f(x,0) = h(x)$, which guarantees that any local maxima of $f$ or any of its derivatives in ...
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1answer
44 views

Inhomogeneous Diffusion Equation - Maximum Principle

Let $u$ be a solution of the inhomogeneous diffusion equation $u_{t} = ku_{xx} + f$ on the rectangle {${0 < x < l, 0 < t < T}$} with initial data $\phi$ and boundary data $g$. Prove that ...
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0answers
35 views

Maximum Principle of the Diffusion Equation

Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {$0\leq x\leq l,0\leq t<\infty$} a) Let $ M(t)$ = the maximum of $u(x,t)$ in the closed rectangle {$0\leq x\leq l,0\leq ...
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0answers
44 views

How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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1answer
47 views

Harmonic function zeros on open subset

Let $h$ be an harmonic function on $\Omega\subset\mathbb{C}$. Let $A\neq\emptyset$ an open subset of $\Omega$ such that $h\mid_A\equiv 0$. Prove $h\mid_\Omega\equiv 0$. I thought on taking a ...
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0answers
30 views

Infinite number of points with modulus 1

Let $f:D(0,1)\rightarrow \mathbb{C}$ be analytic such that $f(0)=1$. Prove that there are an infinite number of distinct points $z_{n}$ in $D(0,1)$ such that $|f(z_{n})|=1$ Remark: $D(0,1)$ is the ...
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0answers
25 views

maximum principle in a harmonic and positive function

$\begin{equation}u=\frac{1-(x^2+y^2)}{(1-x^2)+y^2}\end{equation}$ is harmonic and positive in $x^2+y^2<1$ because $u=0$ in $x^2+y^2=1$ except in $(1,0)$. Is the maximum principle valid for $u$?. ...
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0answers
14 views

Maximum principle for heat equation with Neumann boundary condition

please help me with this problem. Let $u\in C^2_1([0,l)\times (0,T))\cap C^0([0,l]\times [0,T])$ be a solution of the heat equation $u_t-u_{xx}=0$ such that $u_x(0,t)=0$. Prove that the maximum of $u$ ...
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3answers
106 views

Prove that $|f|\leq 1$ whenever $|x|\leq 1$.

Let $f :\mathbb{R}^2\rightarrow \mathbb{R}^2 $ be everywhere differentiable such that the Jacobian is not singular at any point in $\mathbb{R}^2$. Assume $|f|\leq 1$ whenever $|x|=1$. Prove that ...
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1answer
38 views

Maximum modulus pronciple

The problem I had was: If $f$ is holomorphic on $U \subseteq \mathbb{C}$ and $\exists z_0$ in $D(0,1)$ (which is fully contained in $U$) such that $|f|$ has a local maximum at $z_0$ then $f$ is ...
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1answer
46 views

Some basic questions regarding the Maximum Principle for Harmonic Functions,

I've seen a uniqueness argument come up a few times but I don't really understand it. The argument is that if two harmonic functions $f$ and $g$ agree on the boundary of some domain, then $f-g$ or ...
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1answer
28 views

Why does $v(z)=\text{Im}\left[\left(\frac{1+z}{1-z}\right)^2\right]$ not contradict maximum principle?

Since $\left(\frac{1+z}{1-z}\right)^2$ is holomorphic in $\mathbb{D}$, its imaginary part is harmonic, and we have $$\underset{r \uparrow 1}{\lim}v(re^{i\theta})=0 \quad \forall ~ \theta \in [0, ...
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4answers
59 views

Showing $f(z)$ is constant when $Im(f^2(z)-f(z))=0$

I don't have the full question but I'm assuming $f(z)$ must be entire for this to occur. Also note that $u>1$ where $u$ is the real part of $f(z)$. If we solve the given eqn then we are left with ...
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2answers
68 views

An entire function such that $if(z)= \bar f(z)$ for all $z$ is constant

Let $f$ be an entire function such that $if(z)= \bar f(z)$ for all $z\in\Bbb C$. Show that $f$ is the constant function. I didn't understand it but I took some notes which seem like gibberish to me ...
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1answer
33 views

Prove $v$ is harmonic and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$

Prove that if $v(z) = \mathrm{Im}[(\frac{1+z}{1-z})^2]$, then $v$ is harmonic on the unit disc and $\lim_{r \uparrow 1} v(re^{i\theta}) = 0$ for all $\theta \in [0,2\pi)$. Explain why this does not ...
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0answers
38 views

Prove or disprove the following:$\{p_n\}$ converges uniformly to $f$ on $\{z\in \mathbb{C}: |z|=a\}$

Let $A=\{z\in \mathbb{C}: a<|z|<b\}$. Suppose $f$ is continuous on the boundary of $A$. Suppose there is a sequence of holomorphic polynomials $\{p_n\}$ that converges uniformly to $f$ on ...
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2answers
50 views

If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial B$, then $f$ must have a zero in $B$ [duplicate]

The question goes like this: Let $B=\{z\in C: |z|<R\}$ and $\partial B$ is the circle of radiu $R$. If $f$ is a non-constant analytic function on $B$ such that $|f|$ is a constant on $\partial ...
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1answer
51 views

Uniqueness for a mixed Dirichlet/Neumann problem

Let $L$ be a uniformly elliptic operator with $c \equiv 0$ on a bounded domain $U \subset \mathbb{R}^n$ with $C^2$-boundary, and let $\partial U = S_1 \cup S_2$. Suppose $u \in C^2 (\bar{U})$ ...
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1answer
26 views

Hyperplane problem by Lagrange multiplier method

Solve the problem using Lagrange multiplier method. Find the point that belongs to both hyperplanes xT c = β and xT d = γ which is closest to the origin.
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1answer
19 views

Holomorphic function vanishes on the boundary

Let $f \in \mathcal{O}(D) \cap \mathcal{C}(\overline{D})$, $D$ - a bounded region in $\mathbb{C}$. Suppose that $f(z)=0$ for $z \in \partial D$. Prove that $f(z)=0$ for all $z \in D$. At first I ...
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2answers
34 views

Maximum minimum of $x^2$ on the open interval $-1<x<1$

The question goes like this: Does the function $f(x)=x^2$ have a maximum on the open interval $(-1,1)$ ? And a minimum? Explain Not exactly sure how to give a proper answer really. For me it makes ...
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1answer
48 views

Expected value of a maximum of two draws compared to expected value of each

I am no mathematician, so I apologise in advance for not explaining myself properly, and for asking something that is probably utterly obvious for most of you. The question has to do with the ...
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0answers
76 views

How can I conclude that f(z) is constant? [duplicate]

I'm struggling a bit to arrive at the conclusion that $f(z)$ is a constant. Suppose $f(z)$ is holomorphic on and inside the unit disk, and that it has no zeroes on the interior. Also, assume that ...
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1answer
32 views

Prove that if $K$ is polynomially convex, then $\mathbb{C}-K$ is connected.

Prove that if $K$ is polynomially convex, then $\mathbb{C}-K$ is connected. My Try: Suppose $K$ is polynomially convex. Then $K=\hat{K}$. So, given $z\in \mathbb{C}-K$, there is a polynomial ...
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1answer
28 views

Weak maximum principle for the Telegrapher's equation (?)

I'm struggling with the following: Let a bounded connected domain $U \subset \mathbb{R}^2$ be given. Prove the weak maximum principle for the equation $$u_{xx} + 2u_{yy} + u_{y} =0 \qquad (*)$$on ...
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2answers
52 views

$f$ is a monotone increasing, but not necessarily continuous, on $\mathbb{R}^n$, $A$ is compact. Is $f$ always has a maximum on $A$?

Call a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ nondecreasing if $x,y \in \mathbb{R}^n$ with $x \geq y$ implies $f(x) \geq f(y)$. Suppose $f$ is a nondecreasing, but not necessarily ...
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1answer
24 views

Inequality between Laplacians at a point of local maximum

I never before worked with maximum principle. Can someone tell me how I proceed with this: Let $v\in C^{\infty}$ $u$ is continuous and $\Omega\in \mathbf{R}^n$ and suppose that $u − v$ has a ...
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1answer
57 views

Prove $\max_{\bar \Omega} |\nabla u|^2=\max_{\partial \Omega} |\nabla u|^2$.

Let $\Omega \in \mathbb R^n$, $n>1$, be a bounded domain with smooth boundary. Let $u \in C^1 (\bar \Omega)$ be harmonic in $\Omega$. (a) Prove $\max_{\bar \Omega} |\nabla u|^2=\max_{\partial ...
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1answer
38 views

If $\Delta u - u \geq 0$ in $D$, show that $u\leq \max\{\ \max\limits_{\partial D}(u), \ 0 \ \}$

Let $D$ be a bounded domain, with boundary $\partial D$. Suppose $u \in C^{2}(D)\cap C^{0}(D\cup\partial D)$ satisfies $\Delta u - u \geq 0$ in all of $D$. I am supposed to prove then that $u\leq ...
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0answers
44 views

Is this just a direct consequence of the Maximum-Modulus Theorem?

Let $D$ be a bounded region and $f$ is an analytic function on $D$. Show that if there is a constant $c ≥ 0$ such that $|f(z)| = c$ for all $z$ in the boundary of $D$ then either $f$ is a constant ...
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0answers
34 views

Discrete maximum principle for finite difference to the heat equation

Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - ...
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0answers
26 views

If two holomorphic function whose maximal on compact set is the same, then are they equal?

I tried to prove that if equality holds for Hadamard's three circle theorem, then the function is form of $f(z) =Cz^{\lambda}$ where $C \in \mathbb{C}$, $\lambda \in \mathbb{R}$. I know one of the ...
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2answers
57 views

Does there exist a holomorphic function with the following property?

Does there exist a holomorphic function $f$ defined over $D = \{ z : |z| < 1 \}$ such that $|f| \rightarrow \infty$ when $|z| \rightarrow 1$? My approach: If such an $f$ exists, then for a given ...
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0answers
54 views

Prove solution to heat equation is bounded if initial data is bounded

Heat equation: $$\begin{cases} u_t-ku_{xx}=0, x \in \mathbb{R}, t \gt 0 \\ u(x,0) = u_0(x), x \in \mathbb{R} \\ \end{cases} $$ I'm given that for constants $M, N \in \mathbb{R}$, we have $N \leq ...
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2answers
35 views

Finding the maximum of a function with two variables

I started with f(x,y)=20x-x^2+30y-y^2+xy I need to find the maximum values for both x and y. I have differentiated it using implicit differentiation and have gotten to an answer of: ...
2
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4answers
90 views

Is there a Maximum Principle for Biharmonic eigenvalue problem?

Let $$\Delta^2 u-\lambda u =0$$ where $\lambda>0$ and $$\Delta^2 u = \frac{\partial ^4 u }{\partial x^4} + 2 \frac{\partial^4 u }{\partial x^2 \partial y^2} + \frac{\partial ^4 u }{\partial y^4}$$ ...
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0answers
36 views

Maximum principle for heat equation with dissipation

Suppose that $u$ satisfies $$u_t - ku_{xx} + cu=0$$ $$u(x,0) = \phi(x)$$ for $t >0 $ and $x \in [a,b]$. Show that u satisfies the maximum principle My attempt If $v =e^{ct}u$, then $v$ satisfies ...
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1answer
44 views

Maximum Modulus principle on Rudin's theorem.

I have studied rudin's Real and complex analysis, and have a question on the proof that why the level set $E$ is compact. Could you give me some hint? 11.13 Theorem If a continuous function $u$ ...