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Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
2 votes
1 answer
118 views

Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius

This is a follow up from this question. I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
Ryan Hendricks's user avatar
1 vote
1 answer
76 views

Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius

I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
Ryan Hendricks's user avatar
2 votes
0 answers
97 views

On the second order analog of the upper 1-Lipschitz envelope of a function

Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope $$ \hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
Castoro Moro's user avatar
2 votes
0 answers
81 views

Extension of a tangent vector field

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
MathLearner's user avatar
1 vote
1 answer
111 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
7 votes
1 answer
185 views

Question on ODE involving mollifiers from Taylor's book on PDEs

In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
B.Hueber's user avatar
  • 1,171
-1 votes
1 answer
139 views

$L^1$ convergence

Setting For $i \in \mathbb{N}$, consider two sequences $f_i,g_i \in L^1(\mathbb{R})$ such that $$ f_i \rightarrow_{L^1} f \in L^1(\mathbb{R}) $$ and also $$ g_i \rightarrow_{L^1} g \in L^1(\mathbb{R})...
Anthony's user avatar
  • 125
0 votes
0 answers
57 views

Projection measure and an integral formula for Lipschitz functions

Let $n\geq m\geq 0$ be integers and put $k=n-m$. Let $A\subset\mathbb{R}^n$ be Borel measurable, we define the projection measure of $A$ as $$\mu_k(A):=\underset{P_1, \ldots, P_r}{\sup_{A=A_1\sqcup\...
Alexander's user avatar
2 votes
0 answers
138 views

Sufficient initial conditions for "non-local" PDE

I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
DerGalaxy's user avatar
0 votes
1 answer
140 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar
6 votes
0 answers
220 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
  • 517
2 votes
1 answer
152 views

Growth rate of elementary sequences

We consider three sequences $(x_n),(y_n),(z_n)$, where $(x_n) \in \ell^1$ is positive and the other two sequences are merely assumed to be positive, i.e. $y_n,z_n \ge 0$ where $0<z_n<z_{n+1}$ is ...
António Borges Santos's user avatar
-2 votes
1 answer
217 views

If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]

This seems like it should be true but I was wondering if anyone could prove it. Thanks!
li ang Duan's user avatar
6 votes
1 answer
310 views

Surjectivity of a class of integrals in dimensions two

Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
MathLearner's user avatar
1 vote
1 answer
160 views

On an integral equation

Let $B: C^{\infty}([0,1]^3)$ satisfy $$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$ Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation: $$ \int_0^1 f(t,x)\,dx + \int_0^t\...
Ali's user avatar
  • 4,143
0 votes
1 answer
246 views

Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by $$ I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t $$ where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
Grandes Jorasses's user avatar
5 votes
2 answers
708 views

Approximation of Hölder continuous functions "from below"

We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$. I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
António Borges Santos's user avatar
1 vote
2 answers
213 views

How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$

Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that \begin{equation}\tag{1}\label{1} \int_a^b \...
Hyperbolic PDE friend's user avatar
0 votes
1 answer
161 views

Verifying the proof of a bilinear estimate in $L^2$

$\newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ds}{\displaystyle} \newcommand{\Lpn}[2]{\left\lVert#1\right\rVert_{L^{#2}}}$ $\newcommand{\Lptxy}[3]{\left\lVert#1\right\rVert_{L^{...
Mr. Proof's user avatar
  • 159
5 votes
1 answer
151 views

On existence of a concave function

Let $a$ be a strictly positive $C^\infty$ smooth function on the unit interval. Does there exist a strictly positive $C^\infty$ smooth function $f$ on $I$ such that $$ f’’(x) \leq 0\quad \text{and} \...
Ali's user avatar
  • 4,143
2 votes
1 answer
276 views

Construction of the Lipschitz function with a given Lipschitz constant, given two values and with small Lipschitz norm

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c,$ $|f(b)| = c$ and $\varepsilon > 0.$ It is easy to see that if $\|f\|_{\infty}< \frac{\varepsilon}{2} =: \...
Hpela's user avatar
  • 97
3 votes
2 answers
354 views

General version of Weyl's lemma

The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega­)$ satisfies $$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$ then $u$ is harmonic in $\Omega.$ What I want ...
W.J.'s user avatar
  • 379
1 vote
1 answer
136 views

Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $f\colon [a,b] \to\mathbb{C}$ be Lipschitz and let $|f(a)| \geq c$ and $|f(b)| = c$. Is there a Lipschitz function $g$ such that $|g| \geq c,$ $g(a)=f(a),$ $ g(b)=f(b)$ and Lipschitz ...
Hpela's user avatar
  • 97
2 votes
1 answer
455 views

A periodic integral inequality

(This problem comes in connection with a geometric problem exposed here.) Let $\gamma(x,y)$ be a (real) function on the unit disk such that $$ \frac{\partial^2\gamma}{\partial x \, \partial y} = 0\:\:\...
Daniel Castro's user avatar
1 vote
1 answer
259 views

Are separable, continuous, monotonic and scale invariant real-valued functions everywhere differentiable?

Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively ...
kakia's user avatar
  • 399
3 votes
2 answers
716 views

Do two ways to differentiate Lipschitz functions coincide?

Let $f\colon \Omega \to \mathbb{R}$ be a Lipschitz function on an open subset $\Omega\subset \mathbb{R}^n$. By the Rademacher theorem $f$ has first derivative almost everywhere. We denote it $\nabla f$...
asv's user avatar
  • 21.8k
2 votes
0 answers
201 views

Green function of a 2D exterior domain

Consider solutions of the laplace equation \begin{equation} \begin{split} -\Delta u=f, \ \ u|_{\partial D}=0, \end{split} \end{equation} where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...
W.J.'s user avatar
  • 379
0 votes
0 answers
251 views

How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...
zoran  Vicovic's user avatar
1 vote
0 answers
59 views

Identification of a limit point of a sequence of solution of ODE

Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$, \begin{align*} & v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\ &...
G. Panel's user avatar
  • 449
3 votes
1 answer
425 views

Regularity of boundary of a level set of a $C^{1,\alpha}$ function

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^{1,\alpha}$ function. Denote $S_C=\{x\in\mathbb{R}^2\mid f(x)=C \}$ the level set of $f$ with value $C$. What i want to ask is, if $S_C$ is nonempty for some $...
W.J.'s user avatar
  • 379
2 votes
0 answers
170 views

Equivalence of implicit function theorem and Peano existence theorem in ODEs?

I was recently reading a book about the implicit function theorem (IFT): The implicit function theorem: history, theory, and applications, and before that I learned that Peano's existence theorem can ...
anyon's user avatar
  • 181
2 votes
0 answers
216 views

Fourier transform of Dirac delta distribution

Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$ $$ V(...
Guido Li's user avatar
0 votes
1 answer
125 views

Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
2 votes
0 answers
117 views

Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
2 votes
1 answer
145 views

Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral $$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$ Where $\psi(2^{-k} \xi)$ is a smooth ...
Mr. Proof's user avatar
  • 159
2 votes
0 answers
66 views

Existence of saddle points under a $C^0$-perturbation of a continuous function

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and has a strict maximum point $a$ and strict minimum point $b$. Define $g(x,y)=f(x)+f(y)$ and $h_\varepsilon(x,y)$ be a family of continuous ...
W.J.'s user avatar
  • 379
4 votes
2 answers
261 views

A convexity question

Let $Q=[0,1]\times[0,1]$ and let $a$ be a positive smooth function on $Q$. Does there exist a smooth positive function $u$ On $Q$ such that there holds $$ \frac{\partial^2}{\partial x_1^2}u <0 $$ ...
Ali's user avatar
  • 4,143
0 votes
0 answers
71 views

Sufficient condition to be increasing, following a vector field

Let $f\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R})$ $(n\geq 1)$ be an observable, and let $v^1,v^2\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ be two vector fields such that for any $(x^1_t)_{t\geq 0}$ ...
G. Panel's user avatar
  • 449
3 votes
1 answer
259 views

The continuous dependence of the Green's function on a domain

Let $\Omega\in\mathbb{R}^2$ be a smooth bounded domain and $G(x,y)$ be the Green's function of $-\Delta$ in $\Omega$ with zero Dirichlet condition. Clearly $G(x,y)=-\frac{1}{2\pi}\ln|x-y|-h(x,y)$, ...
W.J.'s user avatar
  • 379
3 votes
0 answers
84 views

A weighted $W^{2,p}$ estimates

Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have $$ \|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
W.J.'s user avatar
  • 379
2 votes
2 answers
631 views

Decomposition of a positive definite matrix

Let $K(x)_{n\times n}$ be a positive definite matrix defined on $x\in D$ and $K_{i,j}(x)\in C^2(D)$ (or generally $C^k$) for any $1\le i,j\le n$. Of course for any $x$, there exists a invertable ...
W.J.'s user avatar
  • 379
1 vote
0 answers
99 views

Estimate on integral with logarithmic weight

Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have $$ \int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
Riku's user avatar
  • 839
2 votes
0 answers
55 views

An integral average condition and its relationship with BMO, VMO, and Sobolev spaces

Let $V: \mathbb R^n \to \mathbb R^n$ be a vector field which satisfies $$ \lim_{l \to \infty} \sup_{x \in \mathbb R^n} \left|\frac{1}{l^n} \int_{[0,l]^n}V(x+y) dy \right| = 0 $$ What is the ...
Riku's user avatar
  • 839
2 votes
2 answers
176 views

Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces

Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that: For $n \geq 0$, let $E_{n}$ ...
ScienceAge's user avatar
0 votes
1 answer
139 views

Build an explicit "small perturbation" of the identity satisfying some properties

How can I build (i.e. find an explicit formula) a smooth function $f_\epsilon: \mathbb R \to \mathbb R$ depending on a parameter $\epsilon >0$ which is "almost the identity" but constant ...
user avatar
7 votes
1 answer
414 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
Sascha's user avatar
  • 536
1 vote
2 answers
155 views

Construct suitable cutoff function

Let $\bar x \in \mathbb R$. Is there a cut-off function such that $\phi_\epsilon \in C^\infty(\mathbb R)$, $0 \le \phi \le 1$, and $$\phi_\epsilon(x) = \begin{cases} 1 &\text{ if } |x - \bar x| \...
Riku's user avatar
  • 839
1 vote
1 answer
134 views

If $f \circ u \in BV$ and $f$ is strictly monotone, then is $u \in BV$?

Let $f: \mathbb R \to \mathbb R$ be a Lipschitz strictly monotone (so, in particular, invertible) function. Let $u: \mathbb R \to \mathbb R$. If $f \circ u \in BV$ can we conclude that $u \in BV$?
Zac's user avatar
  • 161
0 votes
1 answer
417 views

Application of Green function for non linear PDE [closed]

In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$. Is the same thing hold for ...
Curious student's user avatar