**4**

votes

**2**answers

146 views

### Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ...

**4**

votes

**1**answer

136 views

### Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into ...

**3**

votes

**2**answers

360 views

### Iwaniec's conjecture

Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...

**3**

votes

**1**answer

311 views

### Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.
$$
My question is to ...

**1**

vote

**1**answer

87 views

### Elliptic pde with bilaplacian; boundary conditions.

I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ ...

**0**

votes

**0**answers

42 views

### solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...

**-1**

votes

**0**answers

62 views

### Clear estimate is not so clear [closed]

In a paper I found the estimate (there it is said that the estimate is clear) for $U \subset \mathbb{R}^n$ and $u \in W_0^{2,2}(U)$ saying that for all $\varepsilon >0 $ we have
$$\int_{U} ...

**5**

votes

**1**answer

92 views

### Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...

**0**

votes

**0**answers

65 views

### Differentiating and integrating an infinite series arising from a PDE

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given $u \in H^{\frac ...

**2**

votes

**2**answers

139 views

### Extremal functions for Gagliardo-Nirenberg inequality

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq ...

**1**

vote

**0**answers

29 views

### Boundary conditions of PDE from SV model with stochastic interest rate

The PDE for the American put option price $P(S,\sigma ,r,t)$ is
\begin{align*}
0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\
+& \frac{1}{2}P_{SS}S^2\sigma ^2 + ...

**2**

votes

**2**answers

191 views

### compact inclusion of domains of unbounded operators

Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold.
Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset ...

**3**

votes

**1**answer

616 views

### Variational formulation for bilaplacian

I am trying to derive a variational formulation for the following problem $$\left\{ \begin{array}{ll} \Delta^2u=f, & \Omega \\ \Delta u+\rho \partial_{\nu}u=0, & \partial \Omega ...

**0**

votes

**1**answer

81 views

### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
...

**1**

vote

**1**answer

116 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R \times \mathbb R/\mathbb Z \to \mathbb R$, $f = f(s,t)$, is the unknown ...

**10**

votes

**2**answers

197 views

### First eigenvalue of the Laplacian on a regular polygon

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
...

**5**

votes

**1**answer

147 views

### Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)?
To be more detailed: if I want to show that some ...

**5**

votes

**2**answers

150 views

### Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...

**1**

vote

**0**answers

100 views

### Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com,
...

**2**

votes

**0**answers

111 views

### Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$
Now, by ...

**2**

votes

**1**answer

58 views

### solvability of linear elliptic pde on a torus

Consider a linear non-divergent form elliptic PDE on a flat torus $\mathbf{T}^n$, $$a_{ij}\partial_{ij}u+b_i\partial_iu=f$$
where all the coefficients and $f$ are smooth. What is the condition that ...

**3**

votes

**1**answer

84 views

### Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...

**6**

votes

**1**answer

1k views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**10**

votes

**2**answers

1k views

### what's the idea behind Carleman estimate

A standard Carleman-type estimate is of the form
$$
\sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty}
$$
...

**5**

votes

**1**answer

561 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**0**

votes

**0**answers

55 views

### Regularity result for the elliptic equation with Neumann conditions

I am having troubles to justify well some inequalities related with the classical
theory of elliptic equations. Let us consider the problem
\begin{align*}
-\Delta c & =f,\text{ in }\Omega\\
...

**2**

votes

**1**answer

174 views

### $BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also
$$
f_B:= \frac1{|B|}\int_B f \, dx.
$$
Suppose $f \in L_{\rm loc}^p(\Omega)$ for all ...

**0**

votes

**1**answer

122 views

### methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...

**0**

votes

**0**answers

9 views

### Optimal relaxation parameter for the SOR method when solving Poisson equation for non-square problems

Yang and Gobbert in paper "The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem" ( http://userpages.umbc.edu/~gobbert/papers/YangGobbert2007SOR.pdf ) gave proof ...

**-1**

votes

**1**answer

201 views

### Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables [closed]

I am trying to read Fritz John's article, here:
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637
And for the proof of Thm 1.1 in page ...

**5**

votes

**2**answers

133 views

### Separable coordinate systems for the Laplace and Helmholtz equations?

According to Mathworld, in three dimensions there are 13 coordinate systems in which Laplace's equation is separable, and 11 for the Helmholtz equation. I've read the relevant chapters of the book by ...

**2**

votes

**2**answers

415 views

### Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...

**4**

votes

**1**answer

362 views

### Schrodinger equation with magnetic vector potential

In many papers dealing with the Schrodinger equation with magnetic potential
$$u_t=i(\nabla+iA(t,x))^2u$$
the authors say that this equation can be studied with Kato's methods for abstract evolution ...

**7**

votes

**1**answer

175 views

### Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...

**24**

votes

**1**answer

1k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**0**

votes

**2**answers

92 views

### Separation of variables for a particular PDE

Given the partial differential equation
\begin{equation}
(1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0
...

**1**

vote

**1**answer

68 views

### On the domain of functionals in measure with singular kernels

this post is concerned with functionals defined in measures. Consider the following functional
$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$
were we define ...

**6**

votes

**3**answers

350 views

### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...

**1**

vote

**0**answers

52 views

### Heat equation inequality

There is an inequality that tells us that for some sufficiently smooth $f$ satisfying $(\partial_t - \Delta )f \le - \delta f^2 +K$ for $\delta,K >0$ that $f$ is bounded by some constant. ...

**14**

votes

**11**answers

4k views

### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**3**

votes

**0**answers

155 views

### How can I calculate the adjoint of the wave operator $\square_{g}$ in $H^{k}$?

I have a three part question, which I could only received an answer for the first part here.
The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in ...

**4**

votes

**0**answers

104 views

### $H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...

**4**

votes

**1**answer

125 views

### Loss of derivative of subelliptic operator

Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. ...

**4**

votes

**1**answer

128 views

### application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...

**6**

votes

**2**answers

162 views

### Pseudodifferential operators on spaces with boundary

Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...

**5**

votes

**1**answer

98 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**6**

votes

**1**answer

248 views

### Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde:
...

**3**

votes

**1**answer

155 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**12**

votes

**4**answers

4k views

### Can an integral equation always be rewritten as a differential equation?

Given an integral equation is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
Edit: Naively I'm hoping for ...

**0**

votes

**0**answers

36 views

### Painleve test of a new PDE hierarchies

This PDE hierarchies is :
$$u_t=\sum_{i=0}^{N}c_iu^iu_x-\frac{1}{2}\sum_{i=0}^N(c_iu^i)_{xxx}$$
so far, I have proved that this equation hierarchies has Resonaces at:$$j=2N+2,4N+2$$,according to ...