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Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$

The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$ Where $$u = [u_1,u_2,\ldots u_n]^T$$ Now I want to rewrite these same equations but with a new ...
user102868's user avatar
1 vote
0 answers
84 views

A Riemann Hilbert problem on the unit square

Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$. Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
Ali's user avatar
  • 4,143
2 votes
0 answers
126 views

Mixed partial derivatives of planar functions converging to delta distribution

Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
cts12's user avatar
  • 51
2 votes
3 answers
259 views

How to show continuity and monotonicity of solutions to this parametrized equation?

Let $1 \le p <2$ be a parameter. Consider the equation $$ \frac{2^{p/2} (1-\sqrt{s})^p-1}{\sqrt{s}}=-2^{p/2-1}p(1-\sqrt{s})^{p-1}. \tag{1} $$ I am rather certain that for each $1 \le p <2$, ...
Asaf Shachar's user avatar
  • 6,741
4 votes
1 answer
216 views

Finding super(sub)-harmonic functions for an elliptic operator

I am looking for a super(sub) harmonic function for an elliptic operator. Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...
sharpe's user avatar
  • 721
2 votes
0 answers
156 views

On sup over boundaries of Sobolev functions

In Gilbarg-Trudinger's section on the maximum principle for weak solutions, the sup of a boundary of a Sobolev function defined as follows: Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. For $u, ...
Yongmin Park's user avatar
2 votes
0 answers
42 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
tobias's user avatar
  • 749
5 votes
2 answers
233 views

Analytic approximations of smooth vector fields

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...
tobias's user avatar
  • 749
2 votes
1 answer
165 views

Is Sommerfeld radiation condition invariant under translations?

A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever $$ \lim_{r\...
GG1's user avatar
  • 146
5 votes
0 answers
140 views

Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$

Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where \begin{align}\label{eq:bounded-...
Guy Fsone's user avatar
  • 1,101
0 votes
1 answer
148 views

Is it a sufficient condition for linearity?

Edit: According to the comment by LSpice we come back to the initial version of this question Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a smooth function such that for every $x\in \mathbb{R}^n$ the ...
Ali Taghavi's user avatar
1 vote
0 answers
60 views

About an estimate of an oblique derivative problem of Laplace's equation

Suppose $n\geq 2$, set $B=B_r(0)\subset \mathbb{R}^n$, $B^+=\{x\in B|x_n>0\}$, $H=\{x\in B|x_n=0\}$, Let $u\in C^\infty(B^+)\cap C^1(B^+\cup H)$ be a solution of the following oblique derivative ...
lzcsl's user avatar
  • 11
1 vote
1 answer
123 views

Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where $f'=\frac{\partial f}{\partial \theta}$

Let $f,g$ are the functions of $S^{1}$. Are there $f, g$ such that $$\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$$ where $f'=\frac{\partial f}{\partial \theta}$?
liding's user avatar
  • 153
2 votes
1 answer
218 views

If an estimate is false on $L^{1}$, then it is false for the $\delta$ distribution?

Let $u=\int e^{\dot{\imath}K(x,y)} f(y) dy$. My advisor told me that we can disprove an integrability estimate $$\|u\|_{L^p}\lesssim \|f\|_{L^{1}}\label{1}\tag{1}$$ by disproving it when $f=\delta$, ...
Medo's user avatar
  • 852
5 votes
1 answer
260 views

Approximate Sobolev embedding

It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$ We then define ...
Pritam Bemis's user avatar
1 vote
0 answers
353 views

Eigenvalues of convolution matrices

Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
Sina Baghal's user avatar
2 votes
0 answers
196 views

Have you seen this PDE before?

Consider the second-order nonlinear PDE $$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$ This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ ...
fsp-b's user avatar
  • 463
2 votes
2 answers
155 views

Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...
kenneth's user avatar
  • 1,399
4 votes
1 answer
755 views

Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...
April's user avatar
  • 399
1 vote
0 answers
126 views

Continuity of Helmholtz-Hodge projection in $H^1(\Omega)$

Let $\Omega \subset \mathbb{R}^d$ (for simplicity $d = 2$ or $3$) be a bounded Lipschitz domain. For any vector-valued function $\mathbf{f} \in \mathbf{L}^2(\Omega):= \left ( L^2(\Omega) \right )^d$, ...
Simon Pun's user avatar
3 votes
1 answer
275 views

Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
user avatar
1 vote
1 answer
101 views

Poincaré-type Inequality

In Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" one finds the following remark: Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
Pádua's user avatar
  • 69
2 votes
2 answers
251 views

inequality involving the fractional Sobolev space

Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality $$|u(x)...
GabS's user avatar
  • 407
2 votes
1 answer
247 views

Weyl symbol of product

Are there explicit formulas for the Weyl symbol of $-f(x)D_x^2 $ where $D_x:=-i\partial_x $ and $\partial_x$ is the derivative and $f$ some sufficiently smooth function? In the standard quantization ...
Kung Yao's user avatar
  • 192
2 votes
1 answer
200 views

Proof of a discrete isoperimetric inequality

The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions: $$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{...
MathLearner's user avatar
1 vote
0 answers
102 views

Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$

I'm currently trying to work with the non-local operator given by $$ (-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy, $$ where $f :\mathbb ...
Kernel's user avatar
  • 446
4 votes
0 answers
220 views

A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic

I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
qwe's user avatar
  • 91
3 votes
0 answers
119 views

Second derivative estimates

I am in big trouble since I don't see how to proceed (I don't need the exact calculation) with the following estimates. In one of his papers, Lin proves the following result: Let's consider a ...
Jason's user avatar
  • 59
3 votes
0 answers
235 views

Singular integral operator

I am working on a problem involving the Biot-Savart law in fluid dynamics. I found a theorem of singular integral which is intimately related to my research. Assume that $K(x)$ is a classical Calderon-...
Kira Yamato's user avatar
4 votes
1 answer
1k views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
Giovanni Febbraro's user avatar
2 votes
0 answers
77 views

Second derivative estimates for a subsolution of linear elliptic equation

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
Giovanni Febbraro's user avatar
1 vote
0 answers
142 views

Intuition from Hopf lemma (boundary point lemma )

Consider the classical boundary point lemma: Let $L$ be an elliptic operator. Boundary Point Lemma Suppose $\Omega$ has the interior sphere property and that $u\in C^2(\Omega)\cap C^1(\bar\Omega)$ ...
Giovanni Febbraro's user avatar
1 vote
1 answer
148 views

Understanding a family of Sobolev-type inequalities

I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following: Denote the following inequality as $S_{r,s}^{\theta}$: $\...
user153765's user avatar
3 votes
1 answer
846 views

Aleksandrov maximum principle for semi-convex function

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
Giovanni Febbraro's user avatar
1 vote
0 answers
45 views

Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
Hheepp's user avatar
  • 371
2 votes
1 answer
71 views

Estimates on divergence-type operator for the matrix

Is there any result (Schauder-like estimates, $L^2$ estimates or similar) to equations of the form $$ {\rm div}(Av)=f $$ where $A$ is the "unknown" (i.e. I would like estimates on $A$ depending on $f$,...
Alessio Di Lorenzo's user avatar
2 votes
0 answers
66 views

Properties of solution to Burger's equation using Cole-Hopf transformation

I am currently looking at a $1$D-Burger's equation defined by \begin{equation} \label{ex burgers} \left\{ \begin{array}{ll} {} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...
Richard's user avatar
  • 357
1 vote
2 answers
798 views

Proof of the du Bois-Reymond lemma "by approximation" [closed]

I'm curious about the following argument in Morrey ("Multiple integrals in the calculus of variations", Lemma 2.3.1). Suppose $f\in L^1[0,1]$ and $$\int_0^1 fg\,dx=0$$ for every test function $g\in C^\...
Ryan Unger's user avatar
11 votes
0 answers
2k views

A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$ $$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...
Jacob Lu's user avatar
  • 903
1 vote
1 answer
192 views

Log-concavity of function

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ My goal is to show that $$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$ is log-concave. Let us ...
Landauer's user avatar
  • 173
0 votes
0 answers
162 views

Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper

In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if $$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...
R. N. Marley's user avatar
3 votes
1 answer
374 views

Positive part of Cauchy sequence of Sobolev functions is again Cauchy

Let $p\geq 1$ and consider the space $W^{1,p}(B)$ where $B\subset \mathbb{R}^{n}$ is the standard unit ball. Moreover, let $f_{k} \in C^{\infty}(B)$ be a Cauchy sequence in $W^{1,p}(B)$ of smooth ...
BremerH's user avatar
  • 49
5 votes
2 answers
2k views

Elementary calculus estimate or not?

Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$ $$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
user avatar
0 votes
0 answers
85 views

Could the convex hull of $\operatorname{Lip}_1(\mathbb R)$ be dense in $\operatorname{Lip}_1(\mathbb R^d)$?

$\DeclareMathOperator\Lip{Lip}$My problem is slightly different from the title, but I don't have a more straightforward title. Sorry for that. For $d\ge 1$, denote $\mathbb S^{d-1}:=\{x\in\mathbb R^...
user avatar
1 vote
0 answers
90 views

Is there any characterization of polynomials in terms of asymptotic properties of Taylor coefficients? [closed]

My formal question is Let $f(z):=\sum_{n=0}^{\infty} c_n z^n$ be a formal power series. Is there any characterization of polynomials in terms of the asymptotic properties the sequence $(c_n)$? For ...
Masik Kara's user avatar
1 vote
1 answer
200 views

Smooth approximation of a subharmonic function in the barrier sense

Let $f$ be a continuous function on $\mathbb R^n$ such that $\Delta f \ge 0$ at a point $p$ in the barrier sense. More precisely, for any $\epsilon>0$, there exists a smooth function $f_{\epsilon}$ ...
Totoro's user avatar
  • 2,535
1 vote
1 answer
2k views

Sobolev embedding in the space of continuous functions [duplicate]

Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
DanielB's user avatar
  • 11
1 vote
1 answer
167 views

Lemma from Donnelly-Fefferman's paper

I am reading the paper Nodal sets of eigenfunctions of the Laplacian on Surfaces by Donnelly and Fefferman available here. I have a problem understanding Lemma 5.10. To my understanding, what follows ...
April's user avatar
  • 399
1 vote
0 answers
84 views

Finite speed of propagation for a PDE

Let $u(s,t,x)$ solve the equation $$ i \partial_s u +\partial^2_t u - \partial^2_x u =0$$ on the set $[0,1]^3$ and suppose that $u(0,t,x)=0$ on $[0,1]^2$ and that $$ u(s,0,x)=\partial_t u(s,0,x)=0$$ ...
Ali's user avatar
  • 4,143
4 votes
0 answers
747 views

Maximum Principles in Parabolic PDE with Neumann Condition

I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
Bogdan's user avatar
  • 1,759

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