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PDE where the square of gradient of the unknown equals a given positive function

Let $V(x)$ be a non-negative smooth function defined in a open domain $U\subset\mathbb{R}^n$. Suppose that $V(x)=0$ only at a given point $x_0\in U$. Consider the PDE $$|\nabla u|^2=V$$ with ...
Homurism's user avatar
3 votes
1 answer
79 views

Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$

Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
P. P. Tuong's user avatar
5 votes
0 answers
608 views

What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?

1. On the $L^\infty$ calculus of variations: The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
Nate River's user avatar
  • 6,215
1 vote
1 answer
122 views

distance in the matrix algebra w.r.t. the nuclear norm

Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
Krzysztof's user avatar
  • 375
2 votes
1 answer
204 views

A continuous analogue of the notion of Hilbert basis

Let $X$ be a locally compact space, let $H$ be a Hilbert space and let $\beta:X\to H$ be a continuous function such that the linear subspace of $H$ spanned by $\beta(X)$ is dense in $H$. I would like ...
P. P. Tuong's user avatar
8 votes
0 answers
115 views

optimal regularity for the Neumann heat equation on Lipschitz domains

$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
leo monsaingeon's user avatar
1 vote
0 answers
72 views

How to understand "sparse graph limits"

For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph. For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
tom jerry's user avatar
  • 349
0 votes
0 answers
96 views

Sufficient condition for weak convergence in Banach spaces

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
an_ordinary_mathematician's user avatar
1 vote
0 answers
55 views

Characterizing one-sided M-projections on real C*-algebras

Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map $$ v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
Neal B's user avatar
  • 11
4 votes
1 answer
308 views

A certain solution for Sine-Gordon Equation

I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
RWien's user avatar
  • 245
2 votes
1 answer
66 views

How many non-trivial solutions can a semilinear elliptic equation have on a smooth star-shaped bounded domain with 0-Dirichlet boundary conditions?

I am not an expert in elliptic partial differential equations, but while studying the attractor structure of evolutionary PDEs, I frequently encounter problems related to elliptic equations. ...
monotone operator's user avatar
2 votes
1 answer
75 views

How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$

Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
Arghya kundu's user avatar
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
5 votes
2 answers
267 views

Positivity for the mild solution of a heat equation on the torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 185
8 votes
0 answers
103 views

Sobolev embedding theorems in vector bundles on non-compact manifolds

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
G. Blaickner's user avatar
  • 1,429
2 votes
1 answer
121 views

Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$

For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite, $$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$ Using the triangle inequality $|\eta-\xi|\le |\eta|...
Dispersion's user avatar
2 votes
0 answers
40 views

Characterization of critical point of an integral operator

I have an integral operator and I wonder how I can characterize the critical point. I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question. ...
user8469759's user avatar
0 votes
0 answers
42 views

Geometric alignment of adaptive models on evolving manifolds

Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
CollisionGeometry's user avatar
4 votes
0 answers
113 views

SPDE Renormalization

some SPDE (in higher dimensions) can only be interpreted in a "renormalised" sense. For example considering $\Phi_2^4$ on $\mathbb{R}_+\times \mathbb{T}^d$ the solution is defined as the ...
mathex's user avatar
  • 573
4 votes
0 answers
148 views

Some questions on Hardy's spaces

In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
A beginner mathmatician's user avatar
0 votes
0 answers
54 views

Functional equations with coupled arguments and additive sructure

Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation $$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$ for all $x, y \...
Chandler Halderson's user avatar
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
2 votes
1 answer
225 views

Some questions on a paper of Rellich

I was trying to read the paper "Über das asymptotische Verhalten der Lösungen von $\Delta u+\lambda u =0$ in unendlichen Gebieten" by Franz Rellich (MR17816, Zbl 0028.16401). Since it is in ...
Emmie's user avatar
  • 41
12 votes
3 answers
2k views

Why is resonance such a widespread phenomenon?

It is easy to mathematically describe the motion of a mass which is attached to a spring and also pushed around by a sinusoidal force. We get a differential equation of the form: $$\frac{\mathrm{d}^2x}...
semisimpleton's user avatar
4 votes
1 answer
132 views

Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
Martin Argerami's user avatar
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
1 vote
0 answers
90 views

Euler-Lagrange equation of fractional Laplacian

The following result is in "An extension problem related to the fractional Laplacian" Section 3.2 by Caffarelli-Silvestre. I’m confused how to show it and wish to have some help. Suppose $u:\...
Holden Lyu's user avatar
3 votes
1 answer
219 views

Moment problem, ergodicity and spectral gap on the space of tempered distributions

Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$. Just below formula [5] in p.122 of the Fröhlich paper, ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
141 views

(Sub)Optimality of random transport

Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
yfful's user avatar
  • 25
2 votes
0 answers
65 views

Generalized Fourier transforms associated to Schroedinger operators

Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
Ali's user avatar
  • 4,145
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
0 votes
0 answers
20 views

Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?

See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
S-F's user avatar
  • 63
3 votes
1 answer
287 views

Expectation comparison inequality for concave function of symmetric random variables

Suppose that $X_i$, $i\in[n]$ are independent symmetric random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
Aryeh Kontorovich's user avatar
3 votes
0 answers
117 views

Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?

Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
Stefan Schrott's user avatar
0 votes
0 answers
112 views

Vector field connecting two points

I'm now working on somehow an inverse problem of an ODE: Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t). Now there is a ...
Sqr's user avatar
  • 1
2 votes
0 answers
80 views

Surjectivity of kernel operator with kernel in $L^1(\nu \times \mu)$

Let $ \mu $ and $ \nu $ be two finite and non discrete measures. Let's begin with a well-known fact. Let $ k \in L^2(\nu \otimes \mu) $, then we can define an operator $ \tilde{T} $ as follows: $$ \...
thibault jeannin's user avatar
2 votes
1 answer
79 views

There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?

Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
Ilovemath's user avatar
  • 677
4 votes
1 answer
152 views

Hölder Gradient Estimates for Linear Elliptic Equations in higher dimensions

I was looking at Theorem 12.4 of Gilbarg and Trudinger's Elliptic partial differential equations of second order (MR1814364, Zbl 1042.35002): Theorem 12.4. Let $u$ be a bounded $C^2(\Omega)$ solution ...
Student's user avatar
  • 537
-1 votes
1 answer
86 views

how take weak derivative of norms in hilbert spaces?

Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$. Let $u∈L^2 ([0,T];V); ...
Alucard-o Ming's user avatar
2 votes
1 answer
128 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
Bastien's user avatar
  • 23
2 votes
1 answer
142 views

Bounded differentiation operator on compact intervals with $L^2$ norm

It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
graham's user avatar
  • 153
0 votes
0 answers
42 views

Fiber-wise mappings composed with projection map $\pi$

Let $M^2=(0,1)^2$. Recall that a chart is a diffeomorphism $\varphi:M^2 \to M^2$. Given a chart $\varphi:(M^2,g_0)\to (M^2,g_0)$ for $g_0$ the Euclidean metric, consider the curves $\varphi^{-1}(u,t)=\...
John McManus's user avatar
2 votes
0 answers
90 views

Positivity for a kinetic PDE

Let us consider the following kinetic equation: $$ \partial_t f + v \cdot \partial_x f = \rho[f] \, M[T] - f $$ for a the phase space density $f=f(x,v,t)$ on $\mathbb{T}^1 \times \mathbb{R} \times (0, ...
kumquat's user avatar
  • 185
3 votes
1 answer
375 views

Dimensionality reduction for total variation

Let $P_i,Q_i$, $i\in[n]$, be distributions on a finite set $\Omega$. We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions. For each $i\in[n]$, define the dimensionally-...
Aryeh Kontorovich's user avatar
4 votes
1 answer
54 views

Krein-Rutman for integral transforms: proof of convergence to leading eigenvector

Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory. Consider an integral ...
Plemath's user avatar
  • 312
5 votes
1 answer
261 views

Counter example for Hadamard Differentiability

I am having a hard time while trying to fully understand Hadamard differentiability. I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
Matthis's user avatar
  • 53
2 votes
0 answers
55 views

Distance between a Hölder function and a Sobolev ball

Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively. My question ...
Drew Brady's user avatar
7 votes
1 answer
959 views

a claim for a proof of the invariant subspace problem [closed]

Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states Does every bounded operator on a separable Hilbert space have a non-trivial ...
euleroid's user avatar
1 vote
0 answers
105 views

KdV/KP-II equation with upper semicontinuous initial data and viscosity solutions

In the article "KP governs random growth off a 1-dimensional substrate", they study the KP-II equation: the function $\phi(t,x,r)=\partial_{r}^{2}\log(F)$ satisfies $$\partial_{t}\phi+\frac{...
Thomas Kojar's user avatar
  • 5,474
1 vote
0 answers
45 views

Adding a data-dependent term to the porous medium equation while retaining an explicit solution

I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by: $ \frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1 $ For this equation, ...
VargM's user avatar
  • 11

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