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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Properties of harmonic functions

If $u:D \mapsto \mathbb{R}$ and $v:D \mapsto \mathbb{R}$ are harmonic functions, then also function $uv:D \mapsto \mathbb{R}$ is harmonic. Is the statement correct?
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Show that the following function is harmonic [Solved]

I am trying to show that the function: $$u(x)=|x|^{(2-n)}$$ is harmonic where $x$ is a vector in $\mathbb{R}^n\setminus\{0\}$ Here is what I tried: $\displaystyle u(x)=|x|^{(2-n)}$ ...
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Integral of harmonic function in a ball

Let $f\in C^2(\Omega)$ an harmonic function in $\Omega$, and: $$ \phi(r) = \frac{1}{2\alpha_2r} \int_{\partial B_r(x)} f(y) d \sigma(y) $$ Prove that $\phi '(r)=0$ by calculating the line integral. ...
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Improve Liouville's Theorem in Evans ' PDE

Here is Liouville's Theorem Suppose that $u \colon \mathbb{R}^n \to \mathbb{R}$ is harmonic and $u \geq 0$. Prove that $u$ is constant. (In this problem , instead of $u$ is bounded now $u \geq 0$ ...
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Calculating a harmonic conjugate

Is the following reasoning correct? Determine a harmonic conjugate to the function \begin{equation} f(x,y)=2y^{3}-6x^{2}y+4x^{2}-7xy-4y^{2}+3x+4y-4 \end{equation} We first of all check $f(x,y)$ ...
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What can you say about f if g is harmonic?

Suppose that f : R → R is such that, whenever g : $R^n$→ R is harmonic, so is f(g(x)). What can you say about f? This is my attempt , and I think f is a linear function.
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Mean value proof in Evans PDE

Here is the proof I don't really understand about the part beginning using Green's formula. How can Du(y) become du/dv . Is is using the directional derivative formula ? Aslo how can you get/pull ...
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Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$ I have concluded that $xu_x = yu_y$. Not sure how to proceed ...
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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 ...
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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) ...
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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}) ...
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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 ...
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1answer
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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 ...
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1answer
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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} ...
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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.
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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 ...
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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 ...
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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 : ...
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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 ...
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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 ...
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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 $$\alpha u(z_{0}) \leq u(z) \leq \beta u(z_{0})$$ ...
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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 ...
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51 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 ...
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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 ...
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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 ...
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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: ...
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42 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 ...
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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 ...
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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), ...
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In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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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 \ ...
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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 ...
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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 ...
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34 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 ...
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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 ...
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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} \ ...
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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 ...
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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: $$ ...
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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 ...
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1answer
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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 ...
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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 ...
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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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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 ...
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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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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 ...