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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Prove that $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic

Suppose that $u,v$ are harmonic functions on doman $D$, and they are harmonic conjugate. Prove that function $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic on $D$. What I've done was to take the ...
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Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
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Differentiation in polar coordinates at $r=0$; question concerning Theorem 5.7 of Stein and Shakarchi Vol. I

This question arose while I was reading Theorem 5.7 of Stein and Shakarchi's Introduction to Fourier Analysis. It says (in part) Let $f$ be an integrable function defined on the unit circle. Then ...
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Can the measure of zeroes of a harmonic function be positive?

Let $u$ be a non-constant harmonic function of two variables defined, say, in the unit disk (or on the half plane for example). It is known that $u$ can vanish on some lines, as it discussed in here. ...
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Area growth of harmonic functions

Can one construct a harmonic function $f$ defined in unit disk with condition $f(0)\geq1$ such that area of $\{z\in\mathbb{D}: f(z)>0\}$ is small enough?
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23 views

Strange thing about Weak Maximum\Minimum Principle?

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline ...
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24 views

Proof using convolution?

there. I am a novice in graduate school. This is the first time I learn PDE in graduate level. I found it so hard. I am going to have a test next week and I am so worried about it. Since I always ...
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Why is the Laplace/Helmholtz equation only separable in a finite number of coordinate systems?

On MathWorld one finds that the Helmholtz equation $$(\nabla^2+k^2)\psi=0$$ is only separable in 11 coordinate systems. Similar statements can be found about the Laplace equation and maybe other ...
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1answer
21 views

Harmonic conjugate extend to boundary [duplicate]

Suppose u is a harmonic function in disc $|z|<1$, and u can be extended continuously to boundary, what about its harmonic conjugate v? Can it also be extended continuous to boundary? I know v can ...
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1answer
27 views

Harmonic functions in unit disk [closed]

Does there exist a harmonic function defined in the unit disk such that (1) $f(0)=1$ (2) area of $\{z\in\mathbb{D}:\,f(z)>0\}$ is equal to zero?
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1answer
59 views

Uniqueness in boundary value problem for the biharmonic functions

My attempt: I tried to use the Green's representation formula twice. The Green's reprensentation formula:$u(y)=\int_{\partial \Omega}(u(x)\frac{\partial G(x-y)}{\partial v}-G(x-y)\frac {\partial ...
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1answer
26 views

Help with biharmonic functions

I tried to use using Green's formula to prove $\triangle u=0$. But it turned out to be wrong because $u=0$ doesn't imply $\triangle u=0$ on the boundary. I am a novice in graduate pde, so I didn't ...
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1answer
31 views

Local barrier implies barrier?

there. This is part of the textbook of Gibarg's PDE: My question is that how to verify the part in red? How to know $\overline w$ is continous in $\overline \Omega$? Thanks so much! Your help ...
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38 views

Prove a function is subharmonic?__Gilbarg Exercise

My attempt: By a long time google search I found in a book about how to prove it in the case of harmonic function. Just construct another function $v$ which is harmonic in the $\Omega$, then prove ...
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1answer
34 views

Find a harmonic function on two concentric balls?

My attempt: I thought about using Poisson Integral formula since the area is two concentric balls. Then I get something like the following: $u(x)=\frac{1}{nw_nR}\int_{\partial ...
2
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1answer
35 views

Proof of reflection principle for harmonic functions

** My attempt: Hi, there! I have known how to prove the above statement when $u\in C^2(U)$, however, I have question about proving the above statement. Because it is $u\in C^2(U^{+}) \cap ...
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1answer
39 views

The solution of $\Delta u=u^3$ with zero boundary values is identically zero

My question: My attempt: I tried to use the Representation using Green's formula: Since $u=0$ on the boundary and $f(x)=x^3$, then the formula becomes: $$u(x)=\int_\Omega y^3G(x,y)dy \quad ...
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1answer
256 views

How to solve the two dimensional Laplace's equation for certain cases?

Had a doubt regarding Laplace's equation. In many textbooks, the general solution to the two dimensional Laplace equation is mentioned as: $$\Phi(\rho,\phi) = A_{0} + B_{0}\ln(\rho) + ...
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1answer
23 views

Harmonic functions that uniformly convergent?

Let $u_k$ be continuous on $\overline\Omega$, $u_k$ harmonic in $\Omega$. Suppose $u_k|\partial\Omega$ converge uniformly. Then $u_k$ converge uniformly in $\Omega$. The hint is using Maximum ...
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1answer
36 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
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2answers
65 views

A harmonic function bounded from below is constant

I am learning PDE on myself as a beginner. It takes me like several hours to finally think out this proof. However, I feel something not right about my proof, especially choosing "$R$" part, it ...
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1answer
40 views

Prove that $(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$

Knowing: $f(z)$ is analytical Prove: $$(\frac{\partial^2}{\partial^2 x} + \frac{\partial^2}{\partial^2 y})|f(z)|^2 = 4|f'(z)|^2$$ I have proved firstly that $\ln|f(z)|$ is harmonic function Let ...
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72 views

Maximum principle for harmonic functions on unbounded domain

I am learning PDE by myself now. I am considering converted the problem to bounded domain to use the strong maximum principle. My attempt: Using the $\lim u(x)=0$, then exists $\epsilon$ and $N$, ...
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1answer
33 views

Solution of Dirichlet problem in a ball?

If $u\in C^2(B_{R}(0))\cap C^0(B_R(0)))$ is harmonic, then $\large u(x)=\frac{R^2-|x|^2}{nW_nR}\int_{\partial B_R(0)}\frac{u(y)}{|x-y|^n}\,ds(y)=\int_{\partial B_R(0)}k(x,y)u(y)\,ds(y)$ by the ...
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29 views

How to do the “direct calculation”?

How I can get the equation (2.25)? What is the calculation? If $|x|$ is the length of the vector, I can't see how to do partial differential with respect to a length? Can someone help me? It is ...
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evaluate the sum of an alternating harmonic series with a fixed limit

Hi I stumbled across an alternating harmonic sum with a fixed limit on an practice exam and I've no idea how to calculate this sum. \begin{equation} \sum_{1}^{100}\frac{(-1)^{k-1}}{k}. \tag{1} ...
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Show $\phi \circ f$ is subharmonic.

Prove that if $f:G\rightarrow \Omega$ is a one-to-one holomorphic function and $\phi:\Omega \rightarrow \mathbb{R}$ is a smooth (twice continuously differentiable) subharmonic function, then $\phi ...
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1answer
25 views

Help with proof about sub harmonic function

I know how to prove it using strong maximum principle, but I need to show it using conditions for a relative maximum. Does this mean using second derivative test? I think if $p\in \Omega$ and ...
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1answer
38 views

A function satisfying the mean value property is harmonic

Here is the problem. I know that if $u$ is harmonic the equation holds, but I don't know how to prove it from the other direction.
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26 views

How to use second derivative test?

I read a proof in my book on pde which I find a bit strange. Let $\Omega$ be some bounded domain. For $f\in\mathscr C^2(\Omega)\cap\mathscr C^0(\Omega)$ satisfying $\Delta f\geq0$ it holds that ...
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31 views

Show that $\log \left| z \right|$ is harmonic and find its the conjugate harmonic function.

Is the form correct for the conjugate harmonic? Attempt: First, we are given \begin{align*} \log \left| z \right| &= u(x,y) + iv(x,y) = \log \sqrt{x^2 + y^2} + i \cdot 0 \\ u(x,y) &= \log ...
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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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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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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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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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1answer
25 views

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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1answer
46 views

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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1answer
69 views

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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1answer
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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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23 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 ...
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1answer
19 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) ...
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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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1answer
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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
44 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 ...
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1answer
23 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} ...
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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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76 views

Find upper and lower bound for $u(3/4)$ where $u$ is a positive harmonic function on unit disk

Let $u$ be positive harmonic function in the unit disk such that $u(0)=\alpha$. Find upper and lower bound for number $u(3/4)$. I tried to find an example, that is positive, harmonic( ...
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How to solve the exterior Dirichlet problem for Laplace's equation?

Problem Solve $\Delta u=0$ on $\Omega$, where $\Omega=\{x : \|x\|>1\}$. The conditions are $u=1$ on the boundary of $\Omega$, and $\lim_{x\to\infty}u(x)=0$. Context The domain here is the ...
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1answer
41 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 ...
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1answer
30 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 : ...