Questions tagged [harmonic-functions]

For questions regarding harmonic functions.

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Prove the uniqueness of $u\in H_0^1(\Omega)$ with $\Delta u=\vert u\vert^{q-1}u+f$ in $\Omega$ with $\Omega$ as a bounded domain with smooth boundary.

Problem: Let $\Omega\subset\mathbb R^2$ be a bounded domain with smooth boundary. Prove that, for all $p>1$ and $1\le q<\infty$, for all $f \in L^p(\Omega)$, there exists a unique $u\in H_0^1(\...
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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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  • 804
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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....
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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: \begin{equation} f (x, y) = {Re} (f (x + i y)) \end{equation} 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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2 votes
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23 views

Laplace equation on a rectangle with inhomogeneous boundary conditions

I am trying to solve Laplace equation in cartesian coordinates, on a rectangle defined by $x_1<x<x_2$ and $-y_0<y<y_0$: $$\nabla^2 g=0$$ with $$g(x,y=\pm y_0)=f(x)\\ g(x_1,y)=f(x_1) \\ \...
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Spherical harmonics orthogonality

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'} (\theta,\phi) \mathrm d\theta \mathrm d\phi $$ I've tried to use the ...
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  • 149
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Applying multivariate chain rule

If I have functions $g$ and $f=u+iv$ such that $$h(x,y) = g(u(x,y),v(x,y))$$ is reasonably definded, what would $\partial^2h/\partial x \partial x$ and $\partial^2h/\partial y \partial y$ look like? ...
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A boundary value problem of a Harmonic potential

A 2D electrostatic (i.e. harmonic potential) boundary value problem is shown in the figure. The solid lines are conductors (all are parallel), the two conductors with potential $V$ are infinitely long,...
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Nonnegative subharmonic function with harmonic majorant

If $u\geq 0$ is subharmonic, and $u$ has a harmonic majorant, is it true that $u=|h|$ for some harmonic function $h$? This is for $u$ subharmonic in $\mathbb{D}\subset \mathbb{C}$
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let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$

the question I'm struggling a bit with is: let h be a harmonic function in a simply connected space $D$, prove that there exists an analytic $f=u+iv$ in $D$ s.t $u+v=h$ I tried to solve by placing ...
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If $u$ is continuous on $\bar D$, harmonic on $D$ and vanishes on an open arc in $\partial D$, then is $u=0$?

Greene and Krantz, Chapter 7 ex. 11. Where $D$ is the unit disc. I have made some progess. I will list my results. If we assume that the answer is "no", I was able to deduce that any nonzero ...
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If $u$ is complex valued and harmonic on a connected open set, and if $u^2$ is harmonic, then either $u$ is holomorphic or $\bar{u}$ is

I don't even know where to start, so some starting points would be appreciated. Thanks! (Greene and Krantz Function Theory of one complex variable, Chapter 7 ex. 13)
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A similar problem about the unique continuation for harmonic function

The unique continuation conjecture states that Let $\Omega$ be a Lipschitz domain with boundary $\partial \Omega$ and let $\Sigma \subset \partial \Omega$ be an open subset. Let $u$ be a harmonic ...
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  • 875
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Decomposition into spherical harmonics

I'm trying to follow a text I found online. The author decomposes EM fields such $$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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4 votes
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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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1 answer
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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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2 answers
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How does Laplace's equation ensure that the condition of saddle (i.e. the Hessian have eigenvalues of mixed sign) is satisfied?

For a function $f(x,y,z)$ that satisfies the Laplace's equation, $$\nabla^2f=0,$$ every point must be a saddle point (because it allows no local maxima or minima). Therefore, the condition $\nabla^2f=...
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Query about Laplace Eqution and Harmonic functions

In the text below, I am unable to understand how Eq 2.5 represents the Laplace Equation because I don't see any partial derivatives etc. Does Eq 2.5 somehow reduce to $\nabla^2 u=0$? The snippet is ...
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Expansion for Harmonic Functions

Take $u\in W_\delta^{2,p}$(weighted Sobolev space), $\delta\notin \mathbb{Z}.$ If we have $$ \Delta u=0,\quad |x|>R, $$ where $R$ is a constant. It's said that "the classical expansion for ...
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Mean value formula for harmonic functions on product of spheres

Is there a mean value formula for harmonic functions over product of spheres? I mean, let us consider $\mathbb{R}^4$ with coordinates $\alpha$, $\beta$, $\gamma$ and $\delta$ and the laplacian $\Delta:...
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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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0 votes
1 answer
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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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Laplace's equation with the inner unclosed edge

Hello I'm Owen and I'm new here. Recently I was working on a boundary element problem about the Laplace equation. The basic Laplace equation is as follows: $$ \left\{\begin{array}{ll} \Delta \varphi=0 ...
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5 votes
1 answer
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Given $u(x,y) = e^x (x \cos y - y \sin y)$ for analytic $f(z) = u + iv$, find $v(x,y)$

Given $u(x,y) = e^x (x \cos y - y \sin y)$ for analytic $f(z) = u + iv$, find $v(x,y)$. I know that there is an answer to this question in For an analytic function $f(z)=u+iv$, if $u=e^x(x \cos y-y \...
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  • 491
2 votes
0 answers
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Does a bounded harmonic function on a manifold with negative curvature have bounded Hessian

I would like to find an example of a manifold $(M,g)$ along with a non-constant harmonic function having bounded gradient and bounded Hessian. What came to my mind was that it is known (by the work of ...
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1 answer
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Solving the Laplace equation in polar coodinates

So, I have the Laplace equation to solve in polar coordinates. I have the following details: \begin{equation} \Delta u=0 \end{equation} with boundary conditions \begin{cases} u(r,0)=u(r,\pi/2)=0\\ u(1,...
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Component wise convergence of a sequence of complex harmonic functions

It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...
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Rudin Real and Complex Analysis Exercise 14.9

Let $U=\{z\in\mathbb C:|z|<1\}$ and $\Omega=\{z\in\mathbb C:-1<\Re z<1\}$. The first part asks to find an explicit biholomorphism $f:\Omega\to U$ with $f'(0)>0$, which is given by $$f(z)=\...
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2 answers
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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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1 vote
0 answers
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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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0 answers
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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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2 votes
0 answers
38 views

Fractional Sobolev space norm given by Laplace Beltrami operator

I'm currently reading Lions and Magenes Non-Homogeneous Boundary Value Problems and Applications and I'm stuck at one point. Let $\Gamma$ be the smooth boundary of an open bounded subset $\Omega\...
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Level curves of harmonic functions are analytic curves?

In the paper, of Flatto, Newman, Shapiro about level curves of harmonic functions, namely curves $\Gamma$ for which there exists a harmonic function $u(x,y)$ vanishing on $\Gamma$ but not identically,...
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Green's Function Computation

I want to calculate Green's Function to solve $\triangle u = f(x,\ y)$, using Laplace Transforms. My plan was to tailor boundary conditions to the problem which simplify the computation. Because the ...
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1 vote
1 answer
42 views

bounding the rate of the harmonic function can decay

Let $u$ be the harmonic function on $\Bbb{R}^n$, we can build the following estimate using the inverse Poincare inequality: $$\int_{B_{2r}}u^2 \ge (1+c(n)) \int_{B_r}u^2 \tag{*}$$ Where $c(n)$ is a ...
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Constructing meromorphic differential over compact Riemann surface

On a compact Riemann surface, one can construct meromorphic differential having only simple poles by using dipole Green function by means of Perron method. (the construction is here and having such ...
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  • 151
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1 answer
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how does a non linear term affect Simple harmonic motion

Hey guys i had a question on simple harmonic motion. so im dealing with a trivial SHM equation which has a non linear term alpha included to the spring force. $$\frac{d^2x}{dt^2} = -{\omega_0^2}(1 + \...
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0 answers
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Sufficient condition for existence of Neumann problem for Laplace equation.

First, which of the following, concerning the solution of the Neumann problem for the Laplace equation, $$\Delta u=0~ \text{on}~~\Omega, \frac{\partial u}{\partial n}=f(x,y),\text{on}~\partial\Omega$$ ...
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  • 5,278
0 votes
1 answer
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Necessary Condition for existence of solution for Laplace equation.

If the Laplace equation $$u_{xx}+u_{yy}=0, 1<x<2,1<y<2$$ with the boundary conditions $$u_x(1,y)=y,u_x(2,y)=5,u_y(x,1)=a\frac{x^2}{7}, u_y(x,2)=x$$ has a solution, then find the constant $...
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  • 139
1 vote
0 answers
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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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2 votes
0 answers
18 views

Question about harmonic functions on the complement of a bounded set.

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 ...
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1 vote
1 answer
38 views

Harmonic Function and Subharmonic function

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}}}...
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3 votes
0 answers
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Necessary and sufficient conditions for existence of solutions to $\Delta \phi = f$ on torus

Let $\mathcal{T}$ be a torus with Riemannian metric. Consider the sourced Laplace equation on $\mathcal{T}$: \begin{align} \tag{1} \Delta \phi = f. \end{align} I'd like to know necessary and ...
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