Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Prove the constancy of a harmonic function with $\lim_{\vert x\vert\rightarrow\infty}\frac{\vert f(x)\vert}{\ln\vert x\vert}=0$.

Let $f : \mathbb R^2 \rightarrow \mathbb R$ be a harmonic function. Suppose $$\lim_{\vert x\vert\rightarrow\infty}\frac{\vert f(x)\vert}{\ln\vert x\vert}=0$$ Prove or disprove that $f$ is a constant....
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1 vote
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Conjugate of the discrete Laplacian Green's function on a square lattice

I have an engineering background and I am faced with the following problem. Green's function for the discrete Laplacian on a square lattice is well known and I think it is a discrete harmonic function....
1 vote
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Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$

Find harmonic $u : \mathbb{R}^2 \to \mathbb{R}$ such that $u(x, 0) = u_y(0,0) = 0.$ Without the last condition, we have $u = y.$ I'm trying to prove that if in addition $u>0$ on the upper half-...
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Finding the family of arclength parameterized curves and the first fundamental form of the real part of an analytic function

Define a surface as the real part of a holomorphic function which by definition is harmonic: $$f (x, y) = {Re} (f (x + i y))$$ How can we define the first fundamental ...
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Showing that probability of BM being in part of a boundary is harmonic

Let $D$ be a domain in $\mathbb{R}^d$ and let $A$ be a measurable subset of its boundary $\partial D$. For $x\in D$, define $$\phi(x)=\mathbb{P}(X_T\in A)$$ where $(X_t)_{t\geq0}$ is a Brownian motion ...
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Construct a solution of Laplace/Poisson problem with a non constant gradient jump

consider the square $[-1,1]^2$ and a ball of radius $R$ entered at the origin $B_R(0)$. The function $u(x,y)=- \frac{\ln(\max(r^2,R^2))}{2}$ solves the Laplace problem $-\Delta u=0$, and the jump of ...
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Prove for any $|f(z)|\leq M$ for all $z$ in the right half plane.

I am reviewing for the complex analysis but I am struck by the following question, Let $f(z)$ be a bounded analytic function on the right half plane. Suppose that $f(z)$ extends continuously to the ...
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Matching Coefficients of Fourier Series with Separation of Variables Solution for Discontinuous Boundary Conditions: 2D Slab Conduction

I am trying to find the temperature profile in a 2D domain with steady heat conduction. The non-dimensional domain is shown below. Domain dimensions, coordinate system, boundary conditions, and ...
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Poisson kernel for product of spheres

It is known that if $\mathbb{S}^3\subset\mathbb{R}^4$ is the 3-sphere, the Poisson kernel of $\mathbb{S}^3$ is $$P(x,\xi)=\dfrac{1-|x|^2}{|x-\xi|^4}.$$ My question is: is there an easy way to ...
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Definition of limsup when points tend to boundary of a region

I am reading Complex Analysis by Marshall,he said in the remark of maximal principle: The reader should verify the alternative form:if $u$ is continuous and subharmonic on a region $\Omega$ in the ...
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The Laplacian of the function $1/\|x\|$ in $\mathbb{R}^d$

Put $r_d(x)=\|x\|=(\sum_{i=1}^dx_i^2)^{1/2}$. The Laplacian of $1/r_d$ in $\mathbb{R}^d$ is given by $$\Delta(\frac{1}{r_d})=-\frac{d-3}{r_d^3}$$ as a direct calculation shows. Thus, $1/\|x\|$ is ...
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Question regarding harmonic functions having the MVP via proof in Gamelin complex analysis

The question I have is regarding the following part in Gamelin's Complex Analysis. This is regarding showing that harmonic functions have the MVP (page 85-86). The step in question is the last ...
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Mean value property for harmonic functions

Consider a bounded harmonic function $u:\mathbb{R}^p \to \mathbb{R}$ (i.e. $u$ is a $C^2$ function such that the Laplacian $\Delta u=0$). Prove, without using Liouville's theorem, the following ...
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Subharmonic functions, their critical points and values

Let $f \colon \mathbb{C} \to \mathbb{R}_{+} \cup \left\{ 0 \right\}$ be a $C^{\infty}$ subharmonic function. Be given a compact domain $K \subset \mathbb{C}$, we let $D_{K}(f)$ be the set of critical ...
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An example involving Poisson's Integral Formula on the circle

I would like a check on my answer to the following problem: Let $B$ be the unit circle in $\mathbb{R}^2$ and consider the Dirichlet problem $$\Delta u = 0 \text{ on }B \\u= g \text{ on }\partial B$$ ...
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How to solve a set of PDEs derived from elasticity?

The problem is discribed firstly, and a possoble strategy which might work (yet don't know how exactly) is suggested. How to gain an analytical solution? Suggestion on numerical method is equally ...
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Explicit form of Harnack's inequality

PDE text Evans defines Harnack's inequality for non-negative harmonic functions as $$\sup_{B_{R}(0)}u\leq c \inf_{B_{R}(0)}u$$ where $c$ is a constant that only depends on the dimension $n$ such that ...
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Discrete Harmonic functions converge to a harmonic function

I am having a problem about Jost's PDE book (3rd edition). Let $\Omega \subset \mathbb{R}^d$ be a domain (i.e., open and bounded). We consider the orthogonal grid of mesh size $h\mathbb{Z}$ ($h>0$),...
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I'm trying to prove 2.74 Theorem of Folland's book, Introduction to PDE. It says: If $u$ is harmonic on the complement of a bounded set in $\mathbb{R}^n$, the following are equivalent: a) $u$ is ...
Suppose $u\in C^1(\bar{U})\cap C^3(U)$, where $U$ is a bounded simply connected open set. If $\Delta u=0$ and $u(x)\neq0$ for all $x$ , show that \$\varphi=\frac{\vert Du\vert^2}{u^{\frac{2(n-1)}{n-2}}}...