All Questions
1,304 questions
2
votes
0
answers
228
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
4
votes
0
answers
90
views
Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\...
- 825
0
votes
1
answer
126
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,143
1
vote
1
answer
59
views
Embedding of domain of fractional power of Laplacian into Sobolev space for cylindrical domains
On a bounded domain $\Omega \subset \mathbb R^d, d\geq 2$ with smooth boundary, it is well known that for the Dirichlet Laplacian $\Delta_D$, $D((-\Delta_D)^\frac12) = H^1_0(\Omega)$.
I'm interested ...
- 153
1
vote
0
answers
84
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
2
votes
0
answers
43
views
Distributions and time-kernels
Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
- 1,429
2
votes
0
answers
191
views
Smoothing property of the heat kernel on the one-dimensional torus
Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation}
G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
- 185
3
votes
1
answer
182
views
Tensor product of a slowly increasing smooth function and a tempered distribution converging to a co-located product
Let $T$ be a tempered distribution on $\mathbb{R}$ and $g$ be a smooth function on $\mathbb{R}$ whose derivatives of all orders are all polynomially bounded (a.k.a. slowly increasing).
For any pair of ...
- 3,477
3
votes
0
answers
196
views
Parabolic smoothing for semilinear PDE
Consider the semilinear energy-critical parabolic PDE in $\mathbb{R}^3$
\begin{align}
\partial_t u &= \Delta u + |u|^{4/(n-2)}u = \Delta u + u^5\\
u(0,x) &= u_0\in \smash{\dot{H}}^1(\mathbb{R}^...
- 537
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
0
votes
0
answers
55
views
Compactness and Leray-Schauder degree
What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
- 471
9
votes
2
answers
418
views
Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
4
votes
3
answers
310
views
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Let $w_1,w_2\in W^{1,p}(\Omega)$ be two functions with $w_1,w_2>0$ and $\dfrac{w_2}{w_1},\dfrac{w_1}{w_2}\in L^{\infty}(\Omega)$, where $\Omega\subset\mathbb{R}^N$ is a bounded domain (i.e. open ...
- 1,759
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
6
votes
1
answer
248
views
The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients
Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by
$$Lu = \partial_i(a^{ij}...
- 91
3
votes
1
answer
377
views
A more general product rule for weak derivatives?
Consider that $u_1,u_2:\Omega\to (0,\infty)$ where $\Omega\subset\mathbb{R}^N$ is an open set. We know that $u_1,u_2\in W^{1,p}(\Omega)$ for some $p>1$ and $\dfrac{u_1}{u_2},\ \dfrac{u_2}{u_1}\in L^...
- 1,759
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}(...
- 1,171
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 ...
- 677
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 ...
- 101
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
3
votes
1
answer
176
views
Question about Lebesgue Bochner spaces
Let $T>0$ and $\Omega\subset\mathbb{R}^N$ be a bounded domain. Also $p\in (1,\infty)$ is any number.
I know that $u\in L^{p}((0,T);L^p(\Omega))$ and $\nabla u\in L^{p}((0,T);L^p(\Omega))^N$. How ...
- 1,759
2
votes
0
answers
83
views
3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$.
Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
0
votes
0
answers
80
views
Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
2
votes
1
answer
117
views
Special density on $L^2$
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, and $u\in L^2(\Omega)$ with $0\leq u(x)\leq 1$ a.e. on $\Omega$. It is well known that $C^{\infty}_c(\Omega)$ is dense in $L^2(\Omega)$. Because $C^...
- 1,759
2
votes
1
answer
180
views
Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
- 452
10
votes
1
answer
518
views
Inverse function theorem for $W^{2,n}\cap W^{1,\infty}$ functions
Let $n\ge 2$, $f:B_1\subset \mathbb R^n\rightarrow \mathbb R^n$, $f\in W^{2,n}\cap W^{1,\infty}(B_1)$, $\text{det}(Df)>c>0$, where $B_1$ is the unit ball. Can we show that $f$ is a homeomorphism ...
- 435
2
votes
0
answers
103
views
A question from a proof of an inequality in Sobolev space $W^{1,1}$
I try to understand the proof the lemma given at page 54 in Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations. Here it is a screenshot:
Here is what I did:
$$-u(x)=u(y)-u(x)=\...
- 1,759
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
1
vote
0
answers
73
views
Convexity and subdifferential monotonicity
Do you know any reference where I can find some results in this sense:
Consider $W:K\to [0,\infty)$ is a functional defined on a convex cone $K\subset X$, where $X$ is a Banach space. Then the ...
- 1,759
3
votes
0
answers
90
views
Sobolev embedding on a compact manifold without boundary
I am reading M. E. Taylor, "Partial Differential Equations III", Second Edition, Springer-Verlag, New York, (1996).
In chapter 13, section 2, in Prop. 2.3 and Prop. 2.4, one finds the ...
- 311
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 8,998
3
votes
1
answer
263
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 825
2
votes
0
answers
92
views
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
- 101
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 825
2
votes
1
answer
318
views
Understanding the Schrodinger flow——Symplectic Banach manifold
This question was posted on https://math.stackexchange.com/questions/4925369/understanding-the-schrodinger-flow-symplectic-banach-manifold but recieve nothing. I really want to know the something ...
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
2
votes
2
answers
151
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
0
votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
7
votes
2
answers
2k
views
Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
- 8,998
2
votes
0
answers
84
views
Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
- 1,759
7
votes
1
answer
185
views
Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form
$$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$
with some initial condition $u(...
- 1,171
5
votes
1
answer
351
views
Does the Poincaré inequality hold on annular domains?
Does the following Poincaré inequality hold
$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$
where $B_r$ denotes a ball of radius ...
- 537
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
4
votes
0
answers
256
views
Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation
In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...