For questions regarding harmonic functions.

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339 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 ...
2
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1answer
40 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 : ...
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0answers
28 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 ...
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0answers
25 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 ...
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0answers
27 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 $$\alpha u(z_{0}) \leq u(z) \leq \beta u(z_{0})$$ ...
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1answer
53 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 ...
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1answer
65 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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25 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 ...
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1answer
47 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 ...
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1answer
57 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: ...
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1answer
122 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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1answer
239 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 ...
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1answer
27 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), ...
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2answers
98 views

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

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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1answer
70 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 \ ...
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1answer
168 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 ...
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1answer
48 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 ...
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1answer
50 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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1answer
118 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 ...
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1answer
70 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} \ ...
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2answers
79 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 ...
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1answer
115 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: $$ ...
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0answers
65 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
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1answer
33 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 ...
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3answers
76 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 ...
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1answer
62 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 ...
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2answers
118 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 ...
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0answers
45 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 ...
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30 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: $$u(z)+u(-z-2)=0\:\:\forall z\in\overline{\mathbb{D}}$$ ...
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1answer
144 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 $\phi$ ...
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1answer
66 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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0answers
50 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: $$ ...
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1answer
37 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 ...
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1answer
85 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 ...
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0answers
87 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 ...
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0answers
56 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: ...
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2answers
94 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 ...
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1answer
25 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 ...
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4answers
117 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 ...
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1answer
58 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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2answers
66 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), \ ...
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1answer
68 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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1answer
68 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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1answer
131 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) = ...
2
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2answers
86 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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3answers
192 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
85 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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1answer
98 views

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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1answer
28 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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1answer
240 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 ...