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

learn more… | top users | synonyms (2)

0
votes
0answers
19 views

Reference request for properties of harmonic polynomials

I am reading this paper, and on page $4$ it takes a non-degenerate quadratic form $q$ on a finite dimensional complex vector space $V$, defines the laplacian assosciated to $q$, $\Delta$, acting on ...
0
votes
1answer
17 views

To Find the Transfer Function Z(s)/X(s) for the system…

Please, help me to answer the next problem: Objective: To find the Transfer Function $z(s)/x(s)$ for the system, using the next equations: "$a$", "$b$", "$c$" y "$k$" are constants $x(t) = a y(t) ...
2
votes
1answer
40 views

Solving the Laplace partial differential equation with particular boundary conditions [closed]

How this Laplace partial differential equation $$ u_{xx}+u_{yy} =0 $$ with initial conditions on $y=0 $ as $$ u(x,0)=0 $$ $$ u_{y}(x,0)=n^{−1} \sin{nx} $$ has solution $$u(x,y)=n^{−2} \sin({hny}) ...
1
vote
1answer
40 views

To show unique solution for the Laplace equation

The problem is in the top while its weak form at the end, source $\hskip 1in$ I know that the solution is unique because the boundary condition is Dirichlet. But I want to show this. How can you ...
2
votes
1answer
40 views

Fake proof using mean value property

Let $f : \Delta \rightarrow \mathbb{C}$ be a holomorphic function on the unit disk. Let $u = |f|$. Assume that $u(0)=f(0)=0$. Question : since $u$ is harmonic, the mean value property should imply ...
1
vote
1answer
22 views

Showing that a function is harmonic using a specific method

I'm trying to show that if $f$ is a harmonic function, then so is $\log|f|$. Moreover, I'm trying to do this using the following operator: $$ \Delta = 4\frac{\partial}{\partial z} ...
0
votes
2answers
16 views

Prove $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$

Prove that for some $C$, $F(r)=Cr^2\log(r)$ is a fundamental solution for $\Delta^2$ in $\mathbb R^2$. Recall that $$ \Delta^2u=u_{rr}+r^{-1}u_r+r^{-2}u_{\theta\theta}$$ My answer is below.
0
votes
1answer
37 views

Integral over compact boundary is finite in the context of potential function

Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to ...
1
vote
1answer
23 views

Finding the solution $u(x,y)$ to Laplace's equation in a rectangle.

I have the following problem which I basically understand, but I cannot understand how my professor did a substitution almost at the end of the problem. Thanks a lot in advance! Question: Find the ...
1
vote
1answer
28 views

When is a harmonic map biholomorphic?

Let $D_1$ and $D_2$ be simply-connected bounded open domains on $\mathbb{C}$. Riemann mappping theorem tells us that there exist biholomorphisms between them. On the other hand, let $\gamma : ...
3
votes
0answers
23 views

Harmonic except at a point [duplicate]

If $u$ is harmonic and bounded in $0<\lvert z \rvert< \rho$, show that $u$ becomes harmonic in $\lvert z \rvert< \rho$ when $u(0)$ is properly defined. What I was thinking so far is that ...
2
votes
0answers
23 views

Proof that function is real part of $\sec(z)$ [duplicate]

I'm working on the following problem: I've deduced that the key is to show that $u$ is the real part of $\sec(z)$. But, I'm getting stuck in the algebra and am hoping someone can point me in the ...
0
votes
0answers
16 views

Property about positive harmonic functions

Let $U$ be a region and $K$ a compact subset of $U$. Fix a $z_{0} \in U$. Why does there exist positive real numbers $\alpha$ and $\beta$ such that $$\alpha u(z_{0}) \leq u(z) \leq \beta u(z_{0})$$ ...
1
vote
1answer
38 views

Find an harmonic function in $\mathbb R^n$ which is a polynomial of degree 4 and is 1 at the origin

Find an harmonic function in $R^n$ which It is a polynomial of degree 4 and is =1 at the origin. It is a polynomial of degree 5 and its partial derivatives are both =0 at the origin. Important ...
0
votes
1answer
46 views

Comparison of the gradients of two harmonic functions near the boundary

Let $\Omega$ an open bounded domain in $R^n$. Let $u,v$ be nonconstant smooth functions in the interior of $\Omega$ and harmonic in $\Omega$. Suppose that $u,v \in C(\overline{\Omega})$ and $u \geq ...
0
votes
0answers
16 views

Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
1
vote
1answer
23 views

Potential equation in rectangle with boundary values

I'm running into problem with the boundary conditions for u(x). I get u(x) = sin((n*pi*x)/a) based on u(0,y)=0, but that doesn't agree with du/dx(a,y)=0 unless the whole function u(x)=0. Is that the ...
1
vote
1answer
33 views

Showing that two given functions are harmonic

I'm preparing for my complex analysis midterm on Thursday and our professor gave us the following as a practice problem: I'm a bit confused on how to approach part (a). Here's my train of thought: ...
0
votes
1answer
39 views

Showing that $u(x, \, y) = \ln(x^2 + y^2)$ is harmonic without computing partial derivatives

I'm trying to show that $u(x, \, y) = \ln(x^2 + y^2)$ is a harmonic function, without explicitly computing the partial derivatives and showing that $u_{xx} + u_{yy} = 0$. I believe that it would ...
2
votes
1answer
49 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
0
votes
1answer
24 views

Rewriting integrals over spheres involving $1/|x|$

The following derivation cames from calculations related to the Laplace equation and its fundamental solution. Let $g(x)$ be a test-function (meaning compact support and infinitely differentiable), ...
0
votes
2answers
83 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
1
vote
1answer
55 views

Computing a solution of the Laplace-Eigenvalueproblem with Neumann-b.c.

Good day! I was considering the Laplace-Eigenvalueproblem with Neumann b.c., i.e. find $u \in H^1(\Omega) \setminus \{0\}$ and $\lambda \in \mathbb{R}$, such that: \begin{eqnarray} -\Delta u \ ...
0
votes
1answer
38 views

Harmonic functions and polar differential forms

Given a harmonic function $u$, its differential and conjugate differential are $$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx ...
0
votes
1answer
20 views

Laplace Equation with non-const Dirichlet Boundary Conditions

I'm struggling to get a Laplace problem with inhomogeneous boundary conditions solved. My memories are very rusty, and it almost works out, but I've got my brain twisted in some way. So I'm kindly ...
0
votes
1answer
29 views

Harmonic function and Poincaré metric

Let $u$ be a harmonic function on the unit disk $\Delta$, taking values in $[0,1]$. Is it true that this implies that $u$ is Lipschitz for the Poincaré metric ? If not, what can be said about a ...
0
votes
1answer
47 views

Applications of PDE and laplace equation

The edge r = a of a circular plate is kept at temperature f(theta). The plate is insulated so that there is no loss of heat from either surface. Find the temperature distribution in steady state. I'm ...
0
votes
1answer
23 views

Dirichlet boundary value problem in convex domains with discontinuous boundary values

Consider $\Omega$ an open, bounded, and convex domain in $\mathbb{R}^n$. Let $g \in L^{2}(\partial \Omega)$ such that the problem $$ \left\{ \begin{array}{ccccccc} \Delta u = 0, \ \text{in} \ ...
3
votes
2answers
31 views

Harmonic non-surjective functions are constant

Let $u:\mathbb R^2 \to \mathbb R$ be a non-surjective harmonic function. $(i)$ Show that $u$ is bounded from below or from above. $(ii)$ Prove that $u$ is constant (and therefore any harmonic ...
8
votes
1answer
65 views

Green's identity contradicts Helmholtz theorem

Let F be a vector field on a bounded domain $V \subset \mathbb{R}^3$, which is twice differentiable, let $S := \partial V$. According to the Helmholtz Theorem, F can be decomposed, such that: $$ ...
0
votes
0answers
39 views

Newtonian potential is harmonic

I got this problem in page 321 from Advaced Calculus, written by Friedman: "Let $\sigma(x,y,z)$ be a continuous function, and let $S$ be a continuously differentiable surface. The Newtonian potential ...
2
votes
1answer
25 views

Potential Equation with Polar coordinates…

The problem: Consider Laplace's equation $$\nabla^2u=\frac{1}{r}(ru_r)_r + \frac{1}{r^2}u_{\theta\theta}=0$$ on the annulus ${(r,\theta)}: r \in (\frac{1}{2},2),\theta \in[0,2\pi]$. Find all ...
2
votes
3answers
51 views

Harmonic Maximum modulus

So, i am starting to solve some exercises of complex analysis, and i am a little rusty, so if anyone could help me with this exercise. I think that if i just can prove the mean value theorem for ...
0
votes
1answer
22 views

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 ...
4
votes
2answers
81 views

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 ...
1
vote
0answers
29 views

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 ...
0
votes
0answers
21 views

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: $$u(z)+u(-z-2)=0\:\:\forall z\in\overline{\mathbb{D}}$$ ...
0
votes
1answer
74 views

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 $\phi$ ...
2
votes
1answer
47 views

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 ...
0
votes
0answers
23 views

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: $$ ...
1
vote
1answer
33 views

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 ...
0
votes
1answer
28 views

Laplace's Equation - Why do we need to sum the solutions here?

I'm reading some notes on Legendre Polynomials and Laplace's Equation $\nabla^2 \psi = 0$ in $\mathbb{R}^3$ (so that $\psi : \mathbb{R}^3 \to \mathbb{R}$). Well, the notes suppose first that the ...
0
votes
1answer
28 views

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 ...
3
votes
0answers
60 views

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 ...
1
vote
0answers
43 views

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: ...
5
votes
2answers
67 views

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 ...
0
votes
1answer
19 views

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 ...
2
votes
4answers
88 views

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 ...
0
votes
1answer
34 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 ...
0
votes
2answers
44 views

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), \ ...