The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

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Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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Natural question about harmonic functions

Let $U$ an open, bounded and convex set in $R^n$. Let $(u_k)_{k \in N}$ a sequence of functions defined in $\overline{U}$. Suposse that each $u_k \in C(\overline{U}) \cap C^{2}(U)$ and harmonic in ...
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An inequality concerning an harmonic function

Let $h$ be a positive harmonic function on $\Delta (0,\rho )=\lbrace z\in\mathbb{C} : |z|\leq \rho \rbrace$. I wish to show that $|\nabla h(z)|\leq \frac{2\rho}{\rho ^2-|z|^2}h(z)$. Since $h$ is ...
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Solving Laplace's Equation - weird boundary conditions?

The potential is given by: $$V = \sum_{n=0}^{\infty} \left[a_n r^n +b_nr^{-(n+1)}\right] P_n(cos \theta) $$ I want to find potential for $r \geq a$ using th definition $I_n = \int_0^1 P_n(x) \space ...
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45 views

Inequality for a harmonic function with gradient bounded from below

Consider $K \subset \mathbb R^n$ a compact set . Let $R > 0 $ such that $B(0,R) \supset K$ and $\partial B(0,R) \cap \partial K = \emptyset .$ Let $u : \overline{B(0,R)} \rightarrow \mathbb R$ a ...
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Questions about the Laplace's equation in polar coordinates

The Laplace's equation in polar coordinates at a cyclic disk: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}, \ \ \ 0 \leq r \leq a, \ \ \ 0 \leq \theta \leq 2 \pi$$ $$u(a,\theta)=h(\theta), \ ...
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49 views

resonance and collapsing of bridge [closed]

whenever there is lecture about vibration and resonance,lecturers sometimes give us example how can bridge be collapses if army soldiers will walk on it by regular steps,there is brief tutorial ...
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38 views

Laplacian transform of division by square root of t?

In this formula: $$f(t)=e^{-3t}t^{\frac{-1}2}$$ I saw examples on $t^n$ where $n>0$. But in above example $n<0$. I don't know how to deal with the $t^{\frac{-1}2}$. I know that ...
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56 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
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89 views

The harmonic measure on the unit disc is absolutely continuous with respect to length

I have read some pages from the book Conformally Invariant Processes in the Plane by Lawler, and found there the following definition of a harmonic measure: $$\text{hm}(z,D;V) = ...
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2answers
66 views

Laplace equation and minimum energy

A function $\Phi$ in a region $V$ satisfies given Dirichlet BC on the boundary $S$. How to show that $\int_V |\nabla\Phi|^2dV$ is minimum iff $\Phi$ satisfies the Laplace equation $\nabla^2 \Phi=0$ ...
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128 views

Positive harmonic function with harmonic reciprocal must be constant

Let $f(z)$ be a positive harmonic function on the unit disk such that $\frac{1}{f(z)}$ is also harmonic. Show $f(z)$ must be constant.
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1answer
53 views

Estimates for harmonic functions

Assume $u$ is harmonic in $U$. Then $$ |D^{\alpha}u(x_{0})|\leq \frac{C_{k}}{r^{n+k}}\|u\|_{L^{1}(B(x_{0},r))} $$ for each ball $B(x_{0},r)\subset U$ and each multi-index $\alpha$ of order $|\alpha| ...
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counterexample for ill posedness of the laplace equation

Consider the wave equation with initial data: $$u_{tt}(t,x) + u_{xx}(t,x) = 0$$ $$u(0,x) = u_0(x)$$ $$u_t(0,x) = u_1(x)$$ Hadamard showed that this problem is ill-posed: there exist large solutions ...
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21 views

Is $w =\text{ max}(v_1, v_2)$ subarmonic if $v_1$ and $v_2$ are?

I am studying Perron method to prove the existence of solution to \begin{equation} \Delta u = 0 \quad \text{in } \Omega \\ \ u = g \quad \text{in } \partial \Omega \end{equation} In the proof they ...
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117 views

Question about the proof of Harnack's inequality in Evans and Gilbarg's PDE book

I have some trouble in understanding the proof of Harnack's inequality. Since I have consulted two books, I explained my three questions one by one. In Evans' book Partial Differential Equations, 2nd ...
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37 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
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53 views

Vanishing Partial Derivatives of a Harmonic Function

If $u$ is a harmonic function such that all of its partial derivatives vanish at some point $z$, show that $u$ is constant.
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39 views

If $f$ and $g$ are holomorphic, then $\log(|f|+|g|)$ is subharmonic

Let $f$ and $g$ be two holomorphic functions on a plane domain, and let $u(z)=\log(|f(z)|+|g(z)|)$. Is it true in general that $u$ is subharmonic? I know it is true if $g=0$, but here I have some ...
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Subharmonic function and holomorphically parametrized integrals

Let $f_\lambda$ be a family of $L^1$ functions (say on $\mathbb{C}$) such that for all $z$ the map $\lambda \mapsto f_\lambda(z)$ is holomorphic. Consider the map $N(\lambda)=\log \int |f_\lambda(z)| ...
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38 views

Electromagnetic fields and Laplace equations along a square

I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$. I have ...
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36 views

Regularity of weakly harmonic map

Suppose $(M,g)$ is a smooth $n$-dimensional manifold with $C^k$-metric $g$ and let $U\subset M$ be an open subset. Does anyone have a reference for a statement about the regularity of a map ...
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Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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43 views

Mean Value Property and Harmonic functions: a simple exercise

Let $f:[a,b]\to\mathbb{R}$ such that for every $h>0$ such that $$(x-h,x+h)\subset[a,b]$$ we have $$f(x)=\frac{1}{2}(f(x-h)+f(x+h)).$$ How can I conclude that $f$ is harmonic in $[a,b]$? My idea ...
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Evaluating a Limit with Generalized Harmonic Numbers.

Using WolframAlpha, I could informally come up with the following result: $$ \lim_{n \rightarrow \infty} \frac{H_n^{(-\frac{1}{2})}}{n\sqrt{n}} = \frac{2}{3} $$ Allowing me to infer that ...
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40 views

Determining the Asymptotic Order of Growth of the Generalized Harmonic Function?

How should I proceed to determine the order of growth of the generalized harmonic numbers? $$ H_{n}^{(r)} \in \mathcal{O}(?) $$
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Question about the maximum principle for the Laplace's equation

Maximum principle for the Laplace's equation: $$\nabla^2 u= \Delta u=u_{xx}+u_{yy}$$ $$\Delta u=f(x,y) \text{ Poisson }$$ Problem with boundary values of the form Dirichlet: $$\left.\begin{matrix} ...
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Prove that a function is harmonic by the use of the mean property

Given $y\in\mathbb{R}^N$, define $v_y:\mathbb{R}^N\setminus\{y\}\to\mathbb{R}$ by $$v_y(x) = \frac{|y|^2 - |x|^2}{|x-y|^N}.$$ By denoting $u(x) = |y|^2 - |x|^2$ and $w(x) = |y-x|^{-N}$, one can ...
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30 views

example of harmonic function on sphere

Can anyone give me an example of a harmonic function on the sphere $S^{2}=\{(x,y,z):x^2+y^2+z^2=1,x,y,z\in{\mathbb{R}}\}$, which equals $1$ on the northern hemisphere and $-1$ on the southeren ...
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41 views

Proof that laplace's equation is rotationally invariant using chain rule

Suppose $(x, y)$ and $(p, q)$ are coordinates in the plane related by rotation around a fixed point $(a, b)$, as follows: $$\begin{bmatrix} p\\ q\end{bmatrix} = \begin{bmatrix} \cos(t) & -\sin(t) ...
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67 views

Harmonic conjugates on annulus slit

Let $D$ be an annulus slit with $$D= \{a<|z|< b \}$$ excluding $(-b,-a)$. Show that any harmonic function on $D$ has a harmonic conjugate on $D$. The hint says to fix $c$ between $a$ and ...
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59 views

What is a harmonic conjugate of $u=Arg(z)$?

Let $u=Arg(z)$ be a function maps $\mathbb{C}\setminus \{0\}$ to $ (-\pi,\pi]$. How do i find a harmonic conjugate of $u$ when $Arg(z)\in (-\pi,\pi)$?
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Extending bounded functions on $\mathbb{R}$ to $\mathbb{H}$ with the Poisson kernel.

Let $h(\phi):\mathbb{R}\to\mathbb{R}$ be a bounded piecewise continuous function on the real line. Define a function on the upper half-plane by the formula $\tilde{h}(s+it):=\int_{-\infty}^{\infty} ...
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2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
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Looking at Green's Function for the Dirichlet Problem, how do you calculate n(y)

If G(x,y) = (-1/4pi)[(|x-y|^-1)-(|x-r(y)|^-1)] where r(y)=(y1,y2,-y3) and x and y are vectors in R3. The question states 'Evaluate ∂G(x,y)/∂ny (n subscript y) for y ∈ ∂Ω. I know to do this you ...
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Analytic property of harmonic functions through mean value property.

I have this strong intuition that one should be able to at least prove that a harmonic function is infinitely differentiable, let alone analytic, using just the mean value property. Unfortunately I am ...
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1answer
77 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
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Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
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Intuitive significance of harmonicity

I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it ...
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Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
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Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
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Chain Rule (applied twice) for vector valued functions

I need to show that if $f: \mathbb{R}^2 \to \mathbb{R}$ is harmonic, i.e., $$\frac{\partial^2 f}{\partial x^2}(x,y) + \frac{\partial^2 f}{\partial y^2}(x,y) = 0 \quad \text{ for all } (x,y) \in ...
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Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
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44 views

Local regularity for harmonic functions (Laplace's equation)

I need a local Sobolev regularity result for a smooth solution $u$ of $$ -\Delta u=0 $$ with the equation satisfied in an open set $U$ (I have no boundary conditions). I know that such a smooth $u$ ...
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Second partial derivatives of harmonic functions

Given a twice-differentiable function $f:S\rightarrow \mathbb{R}$, where $S$ is a nonempty subset of $\mathbb{R}^n$, how does one prove that if f is harmonic then the second partials $D_jD_kf$ all ...
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An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
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Finding all harmonic functions satisfying initial conditions in the unit disk

Find all harmonic functions $\phi$ in the unit disk $D= \{\ z \in \mathbb{C} : |z|<1 \}\ $ that satisfy $\phi(\frac{1}{2})=4$ and $\phi(z)\ge 4$ for all $z \in D$. Through $\phi$ being harmonic, ...
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139 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
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Harmonic function vanishing on a set of positive measure.

I'm preparing for a qualifying exam, and came across a question I couldn't figure out: If $\Omega$ is a region and $h:\Omega\to \mathbb{R}$ is a harmonic function vanishing on a set of positive ...
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28 views

Derivatives of the Green's Function

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. ...