0
votes
0answers
35 views

TT* Duality argument

I am trying to obtain $L^p$ estimate for a system of nonlinear PDE. I do not have $L^2$ to work with for my problem, I heard $TT^*$ argument is useful tool if one wishes to mapp from $L^p$ to $L^p$. ...
1
vote
2answers
44 views

Why are all Laplacian eigenfunctions on the square obtained by separation of variables?

Let our domain $\Omega = (0,2\pi)\times(0,2\pi)$ be a square with sides parallel to the axes. Consider the Dirichlet eigenvalue problem $\Delta u+\lambda u=0$ with Dirichlet boundary condition $u=0$ ...
2
votes
0answers
65 views

Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
0
votes
1answer
31 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
1
vote
1answer
31 views

Estimative with kernels of the Riesz transform

Let $x=(x_1,x_2)$ and $k(x)=\frac{x_1}{|x|^3}$, $|x|>0$. Condider $x,y\in\mathbb{R^2},\xi=|x-y|, \tilde{x}=\frac{x+y}{2}$ then $|k(x-t)-k(y-t)|\leqslant C\frac{|x-y|}{|\tilde{x}-t|^3}$ when ...
1
vote
1answer
29 views

Bound on the inverse laplacian

Let's consider the equation $$ \Delta u(x,y)=0\;\in\;\mathbb{R}\times\mathbb{R}^+,\;u(x,0)=g(x), $$ where $g$ is an integrable, smooth enough function. Let me write $P_y$ for the Poisson kernel for ...
0
votes
0answers
14 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
1
vote
1answer
64 views

Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
1
vote
0answers
228 views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
1
vote
0answers
37 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
1
vote
1answer
51 views

Sobolev spaces vs. Hardy Spaces

I have seen sobolev spaces (the ones with the p norms of the derivatives of a multivariable function) and Hardy spaces (the objects investigated in harmonic analysis when one asks about tangential and ...
1
vote
1answer
43 views

Convolution of distribution and Poisson kernel

I know that for a general tempered distribution (see here) $f$ the convolution $f\star P_t$ is not meaningful. Where $P_t$ is the Poisson kernel (see here) which is given by ...
4
votes
1answer
76 views

Taylor expansions of harmonic functions.

Let $D \subset \mathbb{R}^2$ be the unit open disc. Note that any harmonic function on $D$ is real analytic. How can one prove that there exits a constant $C>0$ such that the Taylor expansion of ...
1
vote
2answers
115 views

(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
0
votes
1answer
38 views

Show that $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$

Elias M. Stein said that by an application of Green's theorem the following equality holds $\int_{S^{n-1}}(\Delta _{S}f) gd\sigma=\int_{S^{n-1}}f( \Delta _{S}g)d\sigma$ where $\Delta _{S}$ is a ...
1
vote
1answer
34 views

Existence and uniqueness of solution for an eliptic problem.

Let $\Omega \subset R^n$ be an open and bounded set with smooth boundary and $f \in C^2(\Omega)\cap C(\overline{\Omega})$. Let $ a \geq 0$ be a constant. Consider the following problem: $$ ...
1
vote
1answer
38 views

distance function is p- superharmonic?

In this article in page 3: http://arxiv.org/pdf/0904.1332.pdf says: If I consider $\Omega$ a open bounded and convex set, then the function $\delta (x) = \displaystyle\min_{y \in \partial \Omega } ...
2
votes
0answers
39 views

sequence of p-harmonic functions

Consider $\Omega$ a bounded open set in $R^n$ with $ \partial \Omega$ smooth and $u_n \in C^{1,\alpha}(\Omega)$ a bounded sequence in $C^{1,\alpha}(\Omega)$. Suppose that each $u_n$ is p - harmonic ...
2
votes
2answers
344 views

strong maximum principle - harmonic function

Consider the following the theorem in the classical PDE book of Evans( chapter 2 ) : (part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u \in C^2 (U) \cap C(\overline{U})$, with ...
3
votes
0answers
84 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
-3
votes
1answer
52 views

How to prove $\mathrm{supp} ~ R(t)\in \{|x|<t\}$

Let $R(t)$ define as $$R(t):=F^{-1} \left(\frac{\sin |\xi|t}{|\xi|} \right),$$ how to show that $$\mathrm{supp} ~ R(t)\in \{|x|<t\},$$ where $F$ is the Fourier transform.
4
votes
1answer
195 views

Proof of weak maximum principle.

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 ...
8
votes
2answers
320 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 ...
5
votes
0answers
91 views

Can we do some scaling argument in the presence of inhomogeneous norms?

Notation: $B^n_R$ stands for the ball of radius $R$ in $\mathbb{R}^n$. $\hat{f}$ stands for the Fourier transform of $f$. Question. The following inequality holds true for all $f\in ...
4
votes
1answer
357 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
vote
1answer
72 views

Simple Harmonic estimate

I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it. Let $u$ be a solution of $$\Delta u = f \;\;\; x \in B_4 $$ Then if we can bound $$\int_{B_4} ...
2
votes
1answer
55 views

Alternate definitions of $C^{1,\alpha}(S^1)$ and $C^{1,\alpha}(\bar{D})$ maps

My question is about the precise definitition regarding the following: Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
3
votes
1answer
239 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 ...
1
vote
2answers
75 views

Estimate on a simple-looking integral arising from harmonic analysis/harmonic extensions

Let $z\in \mathbb{D}, t\in S^1, \beta\in \mathbb{R}$. I was dealing with the following integral arising from some other calculation regarding harmonic extension on $\mathbb{D}$: ...
3
votes
1answer
203 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
3
votes
1answer
329 views

Nirenberg-Gagliardo- Sobolev inequalities

I need a small help in understanding the following that how "Nirenberg -Gagliardo-Sobolev inequalities" were used. This is a part of the paper. Denote $$ H^1=W^{1, 2}(\Omega)\\ V_1=\{ f\in H^2 ...
1
vote
0answers
59 views

Inequality for harmonic extension : Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, is the following true ? (Question 1) Let $p(z,t) = \frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ ...
3
votes
1answer
782 views

Properties of subharmonic functions

A function $f$ is called subharmonic if $f:U\rightarrow\mathbb R$ (with $U\subset\mathbb R^n$) is upper semi-continuous and $$\forall\space \mathbb B_r(x)\subset ...
1
vote
3answers
696 views

Solution of Laplace's equation in an annulus with constant Dirichlet conditions?

What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
4
votes
0answers
112 views

Curvatures of contours of solutions of 3d Poisson's equation

Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial ...
101
votes
5answers
7k 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 ...
12
votes
3answers
775 views

Do discontinuous harmonic functions exist?

A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
3
votes
3answers
203 views

What is a “domain” in the maximum-minimum principle?

The maximum-minimum principle says that A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant. Here is my question: If we restrict our attention in ...
1
vote
3answers
178 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...