Tagged Questions

1
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1answer
37 views

Characterize solutions to Laplace's and Poisson's equation in the unit square with periodic boundary conditions

So I am studying for a qualifying examination and there was this problem from an old exam. (a) Does the problem $\Delta u = 0$ in the unit square in the plane with u and and all of its partial ...
0
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1answer
35 views

Corollary to mean value property for harmonic functions?

For $\Omega \subset\mathbb{R}^n$ open, and $u_i:\Omega \to \mathbb{R}$ a sequence of harmonic functions which are uniformly bounded. Prove that for any multi-index $\alpha$ and for any $K \subset ...
1
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1answer
53 views

How to determine if the sums and products of harmonic functions is also harmonic?

Suppose I know that $u + iv$ is non-constant and analytic on a domain $D$, then I know that $u$ and $v$ are harmonic on $D$ and not both constant; but how do I then determine whether $3u^2v - v^3 + ...
2
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1answer
39 views

Mean Value Property

I'm currently studying the theory of PDEs and, in particular, harmonic functions. I've been given this question: Show that if $u:(a,b) \rightarrow \mathbb{R}$ is continuous, and satisfies the ...
2
votes
1answer
19 views

The extension of smooth function under the restriction of its Laplacian

$u$ is a smooth bounded function on $\Omega-\{0\}$ where $\Omega$ is an open neighborhood of $0$ in $\mathbb R^n$. If $\Delta u$ is a bounded function on $\Omega-\{0\}$, then can we extend $u$ to be a ...
7
votes
2answers
105 views

Mean Value Property of Harmonic Function on a Square

A friend of mine presented me the following problem a couple days ago: Let $S$ in $\mathbb{R}^2$ be a square and $u$ a continuous harmonic function on the closure of $S$. Show that the average of ...
1
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0answers
124 views

Green's function for the Laplace operator $\Delta$ in a rectangle (or square)?

What I'm looking for is an explicit form of the Green's function $G$ for the Laplace operator $\Delta$ in a rectangular (or square) domain $D\subset\mathbb R^2$, e.g. $D=[0,1]\times[0,1]$. Namely, I'm ...
1
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0answers
91 views

Biharmonic operator

Consider the problem: $$ \Delta^2 u = f$$ on the square domain $U=(0,1)\times(0,1)$ with boundary conditions: $$ u(x,y)=\Delta u(x,y) = 0$$ for $(x,y) \in \partial U.$ I try to solve it with the ...
0
votes
1answer
58 views

if $\Delta u \geq c$ for some $c>0$ then $u$ has a max on the boundary

Let $D=\{(x,y): \vert(x,y)\vert \leq 1\}$ and let $u:D\rightarrow \mathbb R$ be continuous function with three continous derivatives in the interior of $D$. Show that if there is a number $c>0$ ...
4
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1answer
158 views

Laplace equation Dirichlet problem on punctured unit ball.

Let $\Omega = \{ x \in \mathbb{R}^n: 0<|x|<1 \}$ and consider the Dirichlet problem \begin{align} \Delta u &= 0 \\ u(0) &= 1 \\ u &= 0 ~~~\text{if} ~~|x|=1 \end{align} By considering ...
1
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1answer
91 views

Discontinuity of double-layer potentials

I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...
1
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1answer
65 views

Harmonic function product, Knowing that one is Harmonic implies something about the other?

Hello I have been fighting with this problems for a couples of days and cant get anything better, I will thanks a lot for your help, for little hint, some clue or idea : Let $A$ be an Harmonic ...
1
vote
1answer
78 views

How do harmonic function approach boundaries?

Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$. Question ...
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0answers
54 views

Harmonic Function in $\Omega$ that is continuous in $\overline{\Omega}$ except at a point on the boundary

My problem is the following. Let $x_{0}\in\partial\Omega$ and $\Omega\subseteq\mathbb{R}^{2}$ open and connected domain. Suppose there exists $R\in\mathbb{R}$ such that $\Omega\subseteq B_{x_{0},R}$. ...
0
votes
1answer
75 views

How to prove that this is an Harmonic funtion?

Let $u$ be an Harmonic function in $B(0,a)$ in $R^3$ we define $I(x)=x\dfrac{a^2}{|x|^2} $ Let $w(x) = u(I(x))$. Is there a way to prove that $w$ is harmonic without making too much computation? ...
1
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1answer
84 views

Elliptic equations and harmonic functions

Hi, I need some help with the following problem: Let $u(x_0,y_0)$ be a point of the boundary of a domain $\Omega$ contained in a circle of radius $R$ with center at $(x_0,y_0)$. Let $u$ be an ...
1
vote
1answer
97 views

Existence of solutions for the Dirichlet problem in unbounded domains

Suppose we are trying to solve the Dirichlet problem in a possibly unbounded domain $\Omega \subseteq \mathbb R ^n$ with continuous prescribed boundary data $f$. When $\Omega$ is bounded, it is well ...
3
votes
1answer
137 views

What is the counter example?

The Liouville's theorem states that if $u$ is a non negative, subharmonic function, $L^\infty(\mathbf{R}^n)$, then $u$ is constant ($n\leq2$). Someone knows a counter example if $n>2$, or where can ...
2
votes
2answers
200 views

An inequality about the gradient of a harmonic function

Let $G$ a open and connected set. Consider a function $z=2R^{-\alpha}v-v^2$ with $R$ that will be chosen suitably small, where $v$ is a harmonic function in $G$, and satisfies $$|x|^\alpha\leqslant ...
8
votes
2answers
329 views

How to argue this consequence?

Suppose that $\Omega=\mathbf{R}^n_+$ and consider a function $0<u<\sup\limits_\Omega u=M<\infty$ such that: $$\Delta u+u-1=0 \ \ \text{in} \ \ \Omega,$$ $$u=0 \ \ \text{on} \ \ ...
3
votes
1answer
113 views

Harmonic function with condition on part of its boundary

Suppose $u$ is harmonic in the interior of the unit square $0 \leq x \leq 1$, $0\leq y\leq1$. Suppose furthermore that $u$ and its first derivatives continuously extend to the bottom side $0\leq x ...
0
votes
1answer
97 views

EDP problem in a ball

How to solve the exterior problem on a ball with radius $r$ in the 3d space? I have to found u such that: $\Delta u = 0$ in $B(0,r)^C$ Thanks!
2
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1answer
76 views

Removal of singularities for harmonic functions with finite energy

Denote by $B = B(0,1) \subset \mathbb{C}$ the open unit disc and by $B' = B \setminus \{ 0 \} \subset \mathbb{C}$ the punctured unit disc. Assume that $u : B' \rightarrow \mathbb{R}$ is a harmonic ...
1
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1answer
143 views

Mean Value Property of Harmonic Functions Proof Step

I'll only include the step that throws me off unless more info is requested, but this is from LC Evans PDEs book: $$ \displaystyle \lim_{t \to \, 0^+ } \left[ \frac{1}{n\,\alpha(n) \, t^{n-1}} \int_{ ...
1
vote
2answers
94 views

Mean-value formula for inhomogeneous harmonic functions

I ma working on Evans' PDE textbook problems, but I am stuck with the following problem about modification of the proof of the mean-value formula for harmonic functions. I cannot really see how to ...
0
votes
1answer
81 views

Interior gradient bound

I would like some help with the following problem (Gilbarg/Trudinger, Ex. 2.13): Let $u$ be harmonic in $\Omega \subset \mathbb R^n$. Use the argument leading to (2.31) to prove the interior ...
1
vote
1answer
371 views

Proving the mean value property of harmonic functions using distributions?

A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is ...
1
vote
0answers
109 views

Notation in Gilbarg/Trudinger? [Section 2.8]

Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
1
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0answers
155 views

Subharmonic/Superharmonic Inequality in Gilbarg/Trudinger [Section 2.8]

This is in Section 2.8 of Gilbarg and Trudinger. I believe there are some inaccuracies in the proof supplied, and in any case I think there is a more straightforward proof. Definition: A ...
1
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0answers
119 views

Poisson equation on half-space

Let $H$ be the open half-space of $\Bbb R^n$ defined by $x_n > 0$. Let $f : \overline H \to \Bbb R$ continuous and harmonic on $H$. Define the function $F : \Bbb R^n \to \Bbb R$ by $$ ...
1
vote
1answer
254 views

Maximum principle for harmonic functions in unbounded domains

We demonstrated the weak maximum principle for harmonic functions in bounded domains, proving it first considering the case u subharmonic, then approximating in this way: choose $v(x)=x_1^2-M$ so ...
5
votes
3answers
643 views

Composition of a harmonic function.

I noticed this question while reading several pdfs of lecture notes, and I'm having trouble figuring it out. Can anyone help? If $f$ is a harmonic function in a domain $D \subset \mathbb{C}$, and $g$ ...
2
votes
2answers
111 views

Limit involving the laplacian

I'm trying to prove that if $\Omega$ is an open subset of $\mathbb{R}^n$ and $u$ a $C^2$ function then $$\lim_{r\to 0}\frac{2n}{r^2}\left(u(x)-\frac{1}{|\partial B_r(x)|}\int_{\partial ...
13
votes
2answers
509 views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.