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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Function $u(x,y) \in C^2$ that admits harmonic conjugate

Problem 1) Prove that if the real and imaginary part of a holomorphic function are of class $C^2$, then they are harmonic. 2)Deduce from 1) that if $u(x,y) \in C^2$ is a function that admits a ...
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limit of harmonic functions Gilbarg and Trudinger page 27

Here is a passage from Gilbarg and Trudinger page 27 Let $\Sigma$ be a bounded domain in $\mathbb{R}^n$ ($n\geq 3$) with smooth boundary $\partial\Omega$, and let $u$ be the harmonic function (often ...
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Difference of two subharmonic functions and signed measures

One of the reasons subharmonic functions are interesting is that if you take their laplacian, you get a measure (and conversely any finite Radon measure with compact support can be obtained in this ...
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A functional equation for harmonic functions

Does there exist a non zero function $u\in C(\mathbb{C})$, harmonic in $\mathbb{C}\setminus\mathbb{T}$ that satisfies the following equation: $$u(z)+u(-z-2)=0\:\:\forall z\in\overline{\mathbb{D}}$$ ...
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Dirichlet problem on a disk with polynomial boundary values

Suppose that $\phi$ is a real valued harmonic function on the unit disc that is continuous up to the boundary such that $\phi$ agree with a real valued polynomial on the unit circle. Then $\phi$ ...
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Uniqueness of harmonic function

Let $u\in C(2\overline{\mathbb{D}})$ be harmonic in $\mathbb{D}$, and also harmonic inside the annulus $\{1<z<2\}$. Suppose $v\in C(2\overline{\mathbb{D}})$ is another function that is harmonic ...
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Harmonic function in circle - exercise from Partial Differential Equations book by Y. Pinchover

Could I please ask about help with the following exercise: Let $u(x, y)$ be the harmonic function in $D = \{ (x, y) : x^2 + y^2 < 36\}$ which satisfes on $D$ the Dirichlet boundary condition: $$ ...
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The Laplacian and a nice PDE

Given the Laplacian: $$\Delta u= \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} $$ I had to show that by using this $$v(r,\theta ):=u(r\cos \theta ,r\sin \theta ) $$ I can ...
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The limit function of decreasing sequence of subharmonic is also subharmonic

Let $u(z)$ be a continuous function on a domain $D \subset \mathbb{C}$ to $[−\infty, \infty)$. Suppose $u_n(z)$ is a decreasing sequence of subharmonic functions on $D$ such that $u_n(z) \to u(z)$ for ...
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Inequalities for Laplacian operator eigenvalues

Do the Dirichlet and Neumann eigenvalues of the minus Laplace-Beltrami operator on a compact surface w/boundary interlace? There're known inequalities of the form $$\mu_{k+N}\le\lambda_k,$$ where ...
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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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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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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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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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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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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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56 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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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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137 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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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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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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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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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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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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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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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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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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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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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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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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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 $u(z)>4x^2+9y^2+1$.