0
votes
0answers
25 views

Caccioppoli inequality

Assume we have established the following version of Caccioppoli inequality $$\int |\nabla u|^2 \psi^2 dA\leq C \int u^2 |\nabla \psi| ^2 dA$$ for $C^2(\mathbb C)$- smooth functions $u\geq 0$ with ...
1
vote
2answers
34 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
0
votes
0answers
11 views

What is a 'general proper solution' of a partial differential equation?

I would like to know what a 'general proper solution' of a partial differential equation is. The term came reading the abstract of the paper 'Quasiregular mapping by branched circle packing', but I ...
0
votes
1answer
34 views

Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
3
votes
1answer
70 views

Fourier series for e^x

I'm trying to teach myself partial differential equations from Strauss' book. I have run into a very bizarre problem - I cannot figure out what is the Fourier series of e^x! And not even Google has ...
3
votes
2answers
54 views

Elliptic Operators and Continuity

I am reading a book on Hodge theory (Ref: http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/hodge-smf.pdf) or for english ...
1
vote
1answer
47 views

Integration by parts with complex numbers

Suppose $u$ is a complex-valued wave function $u(x,y,z,t)$. Also, suppose you have the integral $\int(u\overline{u}_{t}+u_{t}\overline{u})dx$. I need to get ...
1
vote
1answer
53 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
1
vote
1answer
115 views

Contour Integration (Choice of Contour)

Let $ \alpha \le 0 $ and $\sigma > 0$ . I want to choose a contour, including $ [\sigma - iR, \sigma+iR] $ , such that i can apply Cauchy's Residue theorem and evaluate: $$ \lim_{R \rightarrow ...
1
vote
1answer
96 views

Prove $\sum_{m \geq 1} {\frac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \frac{t^m}{(1-x)^{2m-1}}} $is divergent

How do I show that the following power series is divergent? $$ u(t,x) = \sum_{m \geq 1} {\dfrac{(2m-2)!}{(1-\rho)\cdots(m-\rho)} \dfrac{t^m}{(1-x)^{2m-1}}} $$ where $t$ is complex 1-dimensional, $x$ ...
4
votes
0answers
146 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
2
votes
0answers
43 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
0
votes
1answer
144 views

Analytic function, $\text{ Im} f(e^{i\theta}) = \sin\theta $

Suppose that $ f(z) $ is analytic in $ |z| < 10 $ and $ \text{ Im}f(e^{i\theta})=\sin\theta $, for $ \theta \in \mathbb{R} $. Find $ f $ in $ |z| < 10 $ and justify your answer. I denoted ...
3
votes
1answer
219 views

Laplace equation upper plane using conformal mapping

Find a solution $u(x,y)$ of Laplace’s equation on the domain $-\infty< x < \infty$ and $0 <y<$ $\infty$ for which $u(x,0)=x^{1/2}$ for $0<x< \infty$. What is $u(x,0)$ for $-\infty ...
1
vote
2answers
252 views

The harmonic conjugate of $\Im e^{z^2}$?

It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it? However, the solutions manual I'm consulting gives the answer as $\Im ...
4
votes
2answers
195 views

Harmonic Function bounded by a linear function

Let $u$ be a harmonic function on $\mathbb C$. Suppose that for each $\epsilon > 0$, there is a constant $C_\epsilon$ such that $$u(z) \leq C_\epsilon + \epsilon |z| .$$ I am trying to show that ...
1
vote
1answer
149 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
votes
0answers
46 views

Riemann mapping between arbitrary triangles

Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? Comment---I look for the conformal equivalence of interiors promised ...
2
votes
1answer
206 views

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ ...
1
vote
1answer
36 views

Laplacian $\Delta u$ in spherical coordinates

The Laplacian $\Delta u$ in spherical coordinates is $$\Delta u=\frac{\partial^2u}{\partial\rho^2}+\frac{2}{\rho}\frac{\partial ...
2
votes
0answers
69 views

Closed form formula for a double series related to wave equation

Does anyone have a closed form formula for the double series $$\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(-1)^{m+n}}{(2m+1)^3(2n+1)^3}\cos\left(\pi t \sqrt{(2m+1)^2+(2n+1)^2}\right)?$$ This is related ...
9
votes
1answer
202 views

Does holomorphic a.e. and continuous imply holomorphic everywhere?

Suppose $D$ is a domain in $\mathbb{C}$, $f:D\rightarrow \mathbb{C}$ is a continuous function. Suppose $f$ is holomorphic outside the zero set $f^{-1}(0)$, and $f^{-1}(0)$ has Lebesgue measure zero. ...
4
votes
1answer
112 views

Weak holomorphicity implies smooth and holomorphic.

This is an extension of a previously asked question: A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm ...
0
votes
1answer
56 views

Weak Holomorphicity: Notation clarification.

A function $f\in L^2(D)$ is weakly holomorphic if, for every $\phi\in \mathcal{C}^{\infty}_c(D)$, $$\int_D f\partial_{\bar{z}}\phi = 0.$$ I'm trying to show that each such $f$ is smooth on the ...
4
votes
1answer
219 views

Weighted $L^2$ Estimates for Domains $\Omega\subseteq\mathbb{C}$

EDIT: After mrf's comment below and some discussion with my instructor for the course it was decided that the below was not really an issue. Namely, I went into reading this lecture with the notion ...
1
vote
1answer
76 views

Question about derivatives of complex-valued functions

For $ z \in \mathbb{C}, t \in \mathbb{R}, \\f : \mathbb{C} \times \mathbb{R} \to \mathbb{C}, \\a : \mathbb{C} \times \mathbb{R} \to \mathbb{R}, \\b : \mathbb{C} \times \mathbb{R} \to \mathbb{R}$ And ...
3
votes
1answer
140 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
1
vote
2answers
342 views

The solution of Cauchy-Riemann equation

Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?
1
vote
2answers
226 views

Integrability of the Poisson kernel

Let $$P_t(x)=\frac{2t}{\omega_{n+1} (|x|^2+t^2)^\frac{n+1}{2}}$$ denote to the Poisson kernel on the upper half space $\mathbb{R}^n\times \mathbb{R}_+$. How do I see $P_t \in L^1 \cap L^\infty$ and ...
5
votes
3answers
1k 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
0answers
215 views

Question on harmonic functions

By the maximum principle, every harmonic function on a bounded domain is uniquely determined by its boundary values. However, for unbounded domains, we can have infinitely many harmonic functions with ...
2
votes
0answers
287 views

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open subset of the complex plane and suppose we have a differential operator of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and only ...
97
votes
5answers
6k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
1
vote
2answers
310 views

What is a reference for the ( classical and well-known ) proof of Weyl's lemma?

What is a reference for the (classical and well-known) proof of Weyl's lemma that states: Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U ...