All Questions
1,304 questions
5
votes
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Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 175
5
votes
1
answer
991
views
Boundary regularity of Dirichlet Eigenfunction on bounded domains
Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an ...
- 675
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\int_{B_R(0)} |u| dx \le C R^{2} \sqrt{\log(2R)} \Big( \int_{\mathbb R^2}\int_{\mathbb R^2} \frac{|u(x)-u(y)|^2}{|x-y|^{2+2s}} dxdy + \int_{B_1(...
- 109
5
votes
1
answer
166
views
Strong maximum principle for a PDE with coefficient in $L^1$
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$. Set $N = \frac{2n}{n-2}$. I am interested in the following equation:
$$
-\Delta \phi + R \phi + \phi^{N-1} = 0
$$
...
- 1,263
5
votes
1
answer
462
views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
- 931
5
votes
1
answer
232
views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
- 52.3k
5
votes
1
answer
233
views
Zeta-Determinant Theorem
Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...
- 9,853
5
votes
1
answer
573
views
Analytic perturbation of eigenfunctions
Consider a domain $\Omega_0 \subset \mathbb{R}^n$, and deformations of $\Omega_0$, called $\Omega_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is ...
- 53
5
votes
1
answer
418
views
Robin-Laplacian in unbounded domains
Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem
$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.
If $\Omega$...
5
votes
2
answers
470
views
Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
- 1,245
5
votes
1
answer
227
views
On the 'usefulness' of the 'original' definition of viscosity solution
In [CEL84, Theorem 1.1, p.489], Crandall, Evans, and Lions give three equivalent definitions of viscosity solution.
As the authors note, the first two are "more appealing in some respects and more ...
user99249
5
votes
1
answer
481
views
A continuous path between two Sobolev functions
Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in H^1(\...
- 679
5
votes
1
answer
359
views
Alternative representations of Sobolev space
Is there a way to represent a Sobolev space as the image of a fractional integral operator over an $L^p$ Lebesgue space? Yes, as it was comment, there is an answer for that in the book "Singular ...
5
votes
0
answers
213
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
5
votes
0
answers
360
views
Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
- 419
5
votes
0
answers
417
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
votes
0
answers
206
views
Equality of weak solutions for inner products inducing equivalent norms
This is a repost of a now-deleted MSE question that did not get any comments or answers.
$\textbf{Background}$: This question is mainly about two basic questions: How do we systematically obtain the ...
- 153
5
votes
0
answers
419
views
Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
- 51
5
votes
0
answers
156
views
Homotopy, contraction mapping and the inverse function theorem on Banach spaces
We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
- 179
5
votes
0
answers
135
views
Relationship between continuous vector fields and divergence measure fields in dimension $\ge 2$
Let $\Omega \subset \mathbb R^d$ with $d \geq 2$ (I am mostly interested in the case when $\Omega$ is the unit ball). A vector field in $L^p(\Omega,\mathbb R^d)$ is called a divergence measure field ...
- 437
5
votes
0
answers
168
views
Sobolev extension from a discrete set of points
Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define
$$...
- 501
5
votes
0
answers
214
views
Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
- 6,035
5
votes
0
answers
438
views
Green's formula and traces in weighted Sobolev spaces
Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let
\begin{equation}
\rho(x) = 1-|x| \quad \text{ for } x \in B_1,
\end{equation}
and let $\sigma >0$ be given. As per the comments, notice that $\...
- 309
5
votes
0
answers
242
views
Spectral gap for the Brownian motion with drift on a compact manifold
Let $M$ be a compact Riemannian manifold without boundary, $X$ a smooth vector field on $M$. Consider the Brownian motion $t\mapsto B_t$ on $M$ with drift $X$, so that its generator is $L=\Delta +X$. ...
- 3,669
5
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0
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205
views
Steklov averages in PDE: what to do when we have time-dependent elliptic operator
One may have an equation (with boundary conditions omitted below)
$$u_t - Au = f$$
$$u(0)=u_0$$
which has a weak solution $u \in L^2(0,T;V) \cap C([0,T];H)$ in the sense that
$$-\int_0^T \int_\Omega u(...
- 91
5
votes
0
answers
913
views
Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?
Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
- 103
5
votes
0
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428
views
Is there an appropriate weighted Sobolev space to include exponential map and projection map?
Observe that given a non negative function
$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted
$L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions
$f: \...
- 3,245
4
votes
2
answers
781
views
Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
- 185
4
votes
1
answer
636
views
Existence of a smooth compactly supported function
Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...
- 4,143
4
votes
2
answers
315
views
Is this an $L^p-L^{\infty}$ operator?
Let $1\leq p <\infty$ and let $p^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions:
$$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t}
\int_{|x-...
- 852
4
votes
4
answers
435
views
Must Neuman Elliptic operator has discrete spectrum ?
It is well known that the Neuman eigenvalue problem has discrete spectrum and the eigen values are
nonnegative and can be arranged in a nondecreasing order of magnitude.
Do we need any smoothness ...
- 41
4
votes
2
answers
658
views
Abstract ODE; PDE; uniqueness of solution
I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...
- 539
4
votes
2
answers
447
views
Heat equation and evolution of number of critical points
Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
4
votes
2
answers
1k
views
Trace space and Neumann boundary condition
In which sense is it possible to solve $\Delta u=0$, $\partial_\nu u=\phi$, for $\int\phi=0$ on a closed domain, say a ball $B^3\subset\mathbb R^3$?
For example would a $\phi\in L^p(\partial B^3)$, $...
- 2,041
4
votes
2
answers
283
views
Regularity of solution of $(-\Delta + w)f = 0$
I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
- 1,305
4
votes
1
answer
460
views
Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In ...
- 1,759
4
votes
1
answer
128
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 $-\log\...
- 145
4
votes
3
answers
1k
views
Using Galerkin method for PDE with Neumann boundary condition?
I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ${\partial\...
- 113
4
votes
2
answers
691
views
Uniqueness of viscosity solutions of Hamilton-Jacobi equation
Consider the following Hamilton-Jacobi (HJ) equation:
$$u_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n ...
user99249
4
votes
3
answers
1k
views
A space of distributions vanishing on the boundary
The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...
- 5,407
4
votes
2
answers
505
views
Eigenfunctions of the Laplacian on singular spaces
Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
- 43
4
votes
2
answers
382
views
Extremal functions for Gagliardo-Nirenberg inequality
Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq GN(N,p,q,r)(∫|∇u|^{p}...
- 233
4
votes
2
answers
2k
views
Inclusions of $C^{k,\alpha}$ spaces
When is $C^{k,\alpha}(\bar{\Omega})$ a
subset of
$C^{k',\alpha'}(\bar{\Omega})$?
Gilbarg and Trudinger says that "for the domains of interest in this work the inclusion will hold whenever $k + \...
- 1,771
4
votes
1
answer
379
views
A constant ratio of integrals? Part I
Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.
For $0<r\leq1$, consider the average of its Dirichlet integral
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
4
votes
1
answer
698
views
Poincare Inequality for $H^2$ function satisfying homogeneous Robin boundary conditions
Let $\Omega\subset\mathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type
$$\|u\|_{L^2}\le C \|\nabla u\|_{L^2}$$
to hold for all $u\in X\subset H^1(\Omega)$ and $C$...
- 287
4
votes
1
answer
387
views
Asymptotic formula for fractional Laplacian
For the solution of
$$
\begin{cases}
\lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\
u^\epsilon=1 & \text{on } \partial \Omega
\end{cases}
$$
Varadhan ...
- 303
4
votes
1
answer
221
views
Is a specific product function orthogonal to all harmonic functions
Suppose $\Omega=[-1,1]^3$. Let $f:[-1,1]\to \mathbb R$ and $g:[-1,1]^2\to \mathbb R$ be smooth functions and suppose that given any harmonic function on $\Omega$ (i.e. $\Delta u =0$ on $\Omega$), with ...
- 4,143
4
votes
1
answer
367
views
Dissipative operator on Banach spaces
An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is ...
- 501
4
votes
3
answers
309
views
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 1,759
4
votes
1
answer
151
views
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
- 300