Questions tagged [maximum-principle]
For questions about the use of maximum principle, either in the context of partial differential equations or complex analysis.
622
questions
1
vote
0
answers
22
views
$\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ has at most one solution
I want to show that the PDE $\Delta u = f(u,x)$ in $\Omega \subseteq \mathbb{R}^n$ with the boundary value $u = \varphi$ on $\partial \Omega$ has at most one solution under the following conditions:
$...
- 13
0
votes
0
answers
30
views
Transport equation to trace fluid particles
The method of characteristics finds curves along which the PDE becomes an ordinary differential equation.
How can I use the method (or any other methods) to rigorously relate the scalar quantity (lets ...
0
votes
1
answer
59
views
Prove that $u = |\nabla v|^2$ reaches its maximum in $\partial \Omega$, $v$ being a solution to the given Poisson pde
Let $\Omega \subset \mathbb{R}^2$ and $v: \mathbb{R}^2 \to \mathbb{R}$ be a solution to:
$$\begin{cases}
v_{xx}+v_{yy} = -2 &\text{in $\Omega$}\\
v = 0 &\text{in $\partial \Omega$}
\end{cases}$...
- 107
0
votes
0
answers
13
views
Polyharmonic system
In the problem, with $\Omega \subset \mathbb{R}^3$ a smooth bounded domain, and the given functions $h_1, h_2 \in C(\partial\Omega)$
\begin{equation}\nonumber%\label{eq:Pe}\tag{$P_{\varepsilon}$}
\...
- 493
2
votes
1
answer
43
views
Maximum entropy distribution with given mean, variance and skewness?
Is there a way to find the maximum entropy distribution with certain values for the three first moments, when the support is the set of real numbers? Or, without loss of generality, with mean zero, ...
- 541
0
votes
0
answers
30
views
Strong maximum principle for a local maximum
Suppose $L$ is a strictly elliptic operator in the non-divergence form with $c \equiv 0$, $u \in C^2(\Omega)\cap C(\Omega)$ and $Lu \le 0$ in $\Omega$. Prove that if $u$ attains a local maximum at an ...
- 57
1
vote
0
answers
21
views
Complex solutions of ordinary BVP with real/complex coefficients
The classic Protter and Weinberger: Maximum principles in Differential Equations states the following uniqueness theorem (theorem 8):
Let $u_1$, $u_2$ be solutions of the BVP $u''(x) + g(x) u'(x) + h(...
- 53
1
vote
0
answers
217
views
What hypothesis can I assume in order to obtain infinity solutions for this problem?
It is related to the classic theory of PDE.
Suppose that $U \subset \mathbb{R}^m$ is an open, connected and bounded set, $f : U \to \mathbb{R}$ and $g:\partial U\rightarrow \mathbb{R}$ and $c:\...
- 337
3
votes
1
answer
68
views
Showing uniqueness of solution to a non-linear Poisson problem
I'm trying to prove that a non-linear Poisson problem has a unique solution. The context is the following:
Let $\Omega \subset \mathbb{R}^n$ be a bounded open subset of class $C^2$. Consider the ...
- 6,679
1
vote
1
answer
48
views
A property about distance between open subsets and boundary of open set used in a proof of the strong maximum principle
I'm going through a proof of the following statement:
Let $\Omega \subset \mathbb{R}^n$ be a connected open set, $L$ be an uniformly elliptic operator in $\Omega$ with $c \equiv 0$ and $u \in C^2(\...
- 6,679
0
votes
0
answers
52
views
Solving this optimisation problem that contains integrals
Assume I have the following objective, which consists of choosing the bundle of goods that will maximise the intertemporal utility between date 0 and date T:
$$ \max_{C1(t), C2(t), C3(t)} U(C(t))= \...
- 15
1
vote
1
answer
78
views
Function cannot attain a minimum inside the domain
Question is that to prove if $$u'' + e^u = −x \text{ for } 0< x <1$$, then u cannot attain a minimum in $(0,1)$
Step I tried:
First, $u'' + e^u <0.$ I assume exists a $c \in (a,b)$ s.t. $u(c)=...
- 79
0
votes
0
answers
58
views
If $f(z)$ is analytic for $|z|<1$ and $f(z)≤\frac{1}{1－|z|}$,find the best estimate of $|f^{(n)}(0)|$ that Cauchy inequality will yield
There have a solution that Fix $n≥0$ and let $R＝\frac{n}{n＋1}$ so that $f(z)$ is analytic on$|z|＝R$. Then by Cauchy inequality
$$|f^{(n)}(0)|≤\frac{n!}{R^n} \max_{z＝R}|f(z)|＝\frac{n!}{R^n(1－R)}＝\frac{...
- 17
0
votes
1
answer
31
views
weak version of maximum principle for not-quite-subharmonic functions
For smooth functions $f(x,y)$ in a disk, if $f$ is subharmonic then it satisfies the maximum principle. What happens if we relax the subharmonic condition, by requiring only certain bounds on $\Delta ...
- 40.1k
0
votes
0
answers
34
views
Proving $u(x,t) \leq \alpha x(1-x) e^{-\beta t}$ for Heat Diffusion using the Maximum Principle
I have $u(x,t)$ to be defined as the solution to the following partial differential equation for heat diffusion over the domain $S = (0, 1) \times (0, \infty)$.
$$
\begin{cases}
u_t - u_{xx} &= 0 ...
- 1,124
1
vote
0
answers
75
views
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 ...
- 2,070
0
votes
0
answers
22
views
Variant of strong maximum principle
i have a question about strong maximum principle and mean value formula for heat equuation i saw two versions of these theorems one (In Evans Book) with hypothesis that $u \in C^{2,1}(U\times (0,T])...
- 31
0
votes
0
answers
43
views
How to prove the comparison principle of this parabolic partial differential equation?
Here $H$ is a positive uniformly convex function. I came across this comparison principle while studying the paper. Maybe the author thought it was easy to prove so he omitted the proof, but I failed ...
- 25
0
votes
1
answer
33
views
Limsup on boundary implies bound on modulus
I am working on the following problem:
Suppose $f$ is analytic on a bounded domain U and that
$$\limsup_{U\ni z\to w} |f(z)| \leq K \quad (\dagger)$$
for each $w \in \partial U$. Show that $|f(z)| \...
- 13
0
votes
0
answers
44
views
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 ...
- 33
0
votes
0
answers
58
views
Complex Analysis - Proving a function is constant
let:$\text{U}\subseteq \mathbb{C}$ as $U$ open and path connected.
$f$ analytic in $U$ and does:
$\forall z_0\in U,\forall \epsilon >0,\exists z_1,z_2\in Ball_{\epsilon }\left(z_0\right)st:f\left(...
- 179
0
votes
0
answers
56
views
Product of two convex combinations is a harmonic function
Let two convex combinations $\lambda = \sum_{i = 1}^N \lambda_i \theta_i$ and $\mu = \sum_{i = 1}^N \mu_i \theta_i$ of real constants $\lambda_i$ and $\mu_i$, and variables $\theta_i$ such that $\sum_{...
- 63
0
votes
0
answers
28
views
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 ...
- 651
0
votes
1
answer
61
views
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 ...
- 533
0
votes
0
answers
29
views
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}
&...
- 73
0
votes
0
answers
63
views
Minimum for a nonlinear heat equation
I'm trying to solve question 4.6 using maximum principle
Suppose that $u(x, t)>0$ solves $$ \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(u \frac{\partial u}{\partial x}\right)+u $...
- 1,452
0
votes
0
answers
55
views
Applying the Phragmén-Lindelöf principle to an entire function of finite order
Disclaimer: I posted recently to MSE a similar question that still remains unanswered. This question is not a duplicate and instead is about another approach suggested by a comment on the original ...
- 5,396
0
votes
0
answers
36
views
Question about Strong Minimum Principle
Denote the indicator function on a meausurable set $A$ as $\chi_A$.
On page 11 of the book Regularity of Free Boundaries in Obstacle-type Problems, the author states that:
Consider the problem (1)
$$\...
- 11k
0
votes
0
answers
50
views
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. ...
- 457
0
votes
1
answer
100
views
Perron's method for the Dirichlet problem: Necessity of subharmonic barrier
I am not seeing how to adapt my argument to produce a proof for the following statement:
Let $D \subset \mathbb{C}_{\infty}$ omit at least two points, and suppose that $\infty \notin \partial D$ and $\...
- 2,060
1
vote
0
answers
55
views
How to prove the uniqueness and stability of solution to Poisson equation using the maximum principle?
I came across this problem in an exam. Given a Poisson equation on a bounded region $\Omega$,
$$\begin{cases}
-\Delta u=f& u\in\Omega\\
\alpha u+\beta\dfrac{\partial u}{\partial n}=g&u\in\...
- 329
0
votes
1
answer
93
views
Comparison principle for p-harmonic functions.
The proof begins by considering the open set $D_{\varepsilon} = \{ x \in \Omega : u(x) > v(x) + \varepsilon \}$ for some $\varepsilon > 0. $ Then it says that $D_{\varepsilon} = \emptyset$ or $...
- 103
1
vote
0
answers
93
views
Maximum principle or comparison principle
the equation
$\varphi_t + D_p H(x, Du ^\varepsilon)\cdot D \varphi = \varepsilon\Delta \varphi$ is a linear parabolic equation. Thus, by the comparison principle for parabolic equations, we have for ...
- 33
1
vote
1
answer
59
views
Usual maximum principle
We assume that
$max_{\mathbb{R}^n \times[0,T]} f(x,t) = f(x_0,t_0)\quad\text{for some }(x_0,t_0)\in\mathbb{R}^n \times[0,T]$
Then, my book says that If $t_0>0$, then, by the usual maximum principle,...
- 33
1
vote
1
answer
53
views
Missing step in the proof of the maximum modulus principle
During class, we proved the maximum modulus principle using the fact that:
$$f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(z+ \rho e^{i\theta})dz$$
Where we are considering a curve centered in z and with radius $...
- 215
0
votes
0
answers
24
views
Finding $M_f(r)$ - maximum modulus of a function $e^{\sin z}$ as a function of $r$ (modulus of a complex-valued number $z$)
I'm not asking for a full solution or answer. Please give me a direction in which to get the right answer.
I found in a university book that if a complex-valued function $f$ is holomorphic, then (as $...
- 1
3
votes
0
answers
78
views
Dirichlet Problem and Comparison Principle - Estimation with Supremum
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ and $u, v$ be functions of the class $C^2(\Omega) \cap C(\bar{\Omega})$.
Prove that if
$$
\left\{\begin{array}{l}
\Delta u=f \text { in } \Omega ...
- 2,098
1
vote
1
answer
124
views
Help with a step on a proof of the maximum principle for harmonic functions over a connected region
I know how to prove that if $f:A \rightarrow \mathbb{C}$ is a harmonic function (here $A$ is an open connected region in $\mathbb{C}$) such that $|f|$ admits a local maximum on a point $z_0\in A$, ...
- 1,734
1
vote
0
answers
60
views
Maximum Value Principle: if $u$ is harmonic on $\{x+iy:0 <x<1\}, u(x+iy) \leq A cosh(ky)$, then $u \leq (1-x)A_0 + xA_1$ where $A_t = \sup\{u(t+iy)\}$
Let $\Omega = \{x+iy : 0 < x < 1 \text{ and } y \in \mathbb R\}$. Suppose $u: \bar\Omega \to \mathbb R$ is continuous, harmonic on $\Omega$ and satisfies $u(z) \leq A \cosh(k \text{ Im}(z))$ ...
- 3,331
1
vote
0
answers
53
views
Papa Rudin $4.22$ Theorem
There is the theorem:
Every orthonormal set $B$ in a Hilbert space $H$ is contained in a maximal orthonormal set in $H$.
There is the proof:
Let $\mathscr P$ be the class of all orthonormal sets in $H$...
- 626
2
votes
1
answer
189
views
Maximum Principle for nonlinear elliptic PDE
I've followed a course about elliptic equations. I've studied the maximum principle for linear elliptic pde's in its various forms. I was wondering if I can use them to deduce something useful for ...
- 670
1
vote
0
answers
32
views
A maximum principle in $\mathbb{R}^N$
Let $\delta > 0$ and define
$$
H_\delta(x) = \prod_{j=1}^{N} \cosh(\delta x_j), \quad \forall x \in \mathbb{R}^N.
$$
By straightforward calculations we get $\Delta H_{\delta} (x) = \delta^2 H_\...
- 891
1
vote
1
answer
27
views
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-...
- 5,396
1
vote
1
answer
100
views
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. ...
- 1,127
1
vote
1
answer
143
views
Weak maximum principle for elliptic operators without classical smoothness assumptions
I am currently reading chapter 8 in Gilbarg-Trudinger's Elliptic PDE of the Second order.
Let $L$ denote a strictly elliptic operator (with some additional assumptions on the coefficients). $\Omega\...
- 3,026
1
vote
1
answer
116
views
Pontryagin's Maximum Principle expaination
I am having problems understanding the Pontryagin's Maximum principle.
I really dont understand the necessary conditions for minimization problem. On every website that I checked I have the impression ...
- 732
2
votes
0
answers
102
views
Fixed endpoint and free end time problem
So I have this problem but I dont know if my solution is right. Can someone help me?
We consider the following fixed endpoint and free end time problem
$\min \int_{0}^{t_f}1 dt \\
s.t\\
x_{1}{'}(t) = ...
- 732
2
votes
1
answer
64
views
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 $p\geq 0$ ...
- 345
0
votes
0
answers
25
views
strong maximum p-Laplacian equation
I consider the very simple $ p-$Laplacian equation with Dirichlet condition: $$ - \Delta_p u = \alpha\ \mbox{in}\ \Omega \subset \mathbb{R}^N,\ N \geq 3, $$ $$ u = 0\ \mbox{on}\ \partial \Omega, $$ ...
- 169
0
votes
0
answers
17
views
Optimal Control With Absolutely Continuous State Variable (not necessarily differentiable).
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 $...
- 31