All Questions
1,304 questions
3
votes
2
answers
262
views
The gradient $\nabla u$ of $u\in W^{1,p}(M;N)$ is tangent to $N$ almost everywhere
Let $M,N$ be (compact) Riemannian manifolds. $N$ is viewed as an embedded submanifold of $\Bbb R^K$. The Sobolev space $W^{1,p}(M;N)$ is defined as
$$
W^{1,p}(M;N):=\{ u\in W^{1,p}(M;\Bbb R^K)\ |\ u(x)...
- 1,245
2
votes
0
answers
42
views
Analysis of coefficients for quickly vanishing analytic vector field
Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
- 749
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
1
vote
2
answers
629
views
Regularity of the mild solution of a semilinear evolution equation
Let
$T>0$
$H$ be a separable $\mathbb R$-Hilbert space
$u_0\in H$
$(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ be an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;\;\;\text{for all }n\in\...
- 167
1
vote
0
answers
110
views
Trace and second-order inverse trace on space with Gibbs measure
Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
- 73
0
votes
0
answers
273
views
Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
user123456
4
votes
1
answer
2k
views
Crandall & Rabinowitz Theorem, bifurcation curves
Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)...
1
vote
0
answers
88
views
Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
- 383
1
vote
1
answer
633
views
Existence of solution to heat equation on a compact manifold
Let $M$ be a compact Riemannian manifold (without boundary), I would like to know under which regularity conditions can we solve the heat equation
$$\begin{align}
\partial_tu-\Delta u &= f \\
u(\...
- 1,245
1
vote
0
answers
177
views
A consequence of De Giorgi oscillation lemma
The following lemma is true (see DeGiorgi oscillation lemma)
Let $u$ be a subsolution of $$\mathrm{div}(A(x)\nabla u) = 0,$$
where $A$ is bounded, measurable and uniformly elliptic ($C^{-1}\...
- 839
1
vote
1
answer
737
views
$L^2$ function in Schwartz space?
Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...
- 177
5
votes
1
answer
264
views
Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?
Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)
Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.
It is well-known that ...
- 153
12
votes
3
answers
2k
views
Reference request: Simple facts about vector-valued Sobolev space
Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\...
- 29.8k
5
votes
1
answer
232
views
Interpolation of some Lebesgue spaces
When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that
$$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
- 52.3k
5
votes
3
answers
1k
views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\...
- 307
10
votes
3
answers
1k
views
Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
5
votes
1
answer
171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 177
6
votes
2
answers
1k
views
Vector Fields in a Riemannian Manifold
Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks
- 4,115
1
vote
1
answer
728
views
Elliptic regularity of Laplace-Beltrami operator on a manifold
I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta_g$ ...
- 1,247
3
votes
0
answers
367
views
Reference on semigroup theory and fractional heat equation
Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.
Where can I find a ...
user103450
1
vote
1
answer
138
views
How to prove the following Whittaker formula
I am a theoretical physicist and
I need help in proving the alternate Whittaker formula
$W _ { k , m } ( z ) = \frac { \Gamma ( - 2 m ) } { \Gamma \left( \frac { 1 } { 2 } - m - k \right) } M _ { k , ...
- 113
3
votes
2
answers
670
views
Identifying the weak limit of a gradient (Bochner spaces)
Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f_n \to f$, $f_n(0)=f(0)=0$ and $f_n$ Lipschitz with Lipschitz constant ...
- 251
6
votes
2
answers
530
views
Schrödinger eigenfunctions are bounded
Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $.
Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
- 129
4
votes
1
answer
145
views
Power series in functions other than monomials
I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example.
Consider the interval $[-\pi,\pi]$ let's say.
...
- 536
7
votes
2
answers
219
views
Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions
I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
- 7,799
2
votes
0
answers
240
views
Discrete Sobolev embedding
It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$
Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
2
votes
1
answer
964
views
Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? [closed]
While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta_x(\varphi)=\varphi(x)$ is continuous as a functional
$$\delta_x:H^1(\...
- 95
8
votes
1
answer
712
views
Pseudo-differential operators with compactly supported symbols
If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO $...
- 728
0
votes
0
answers
324
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
0
votes
2
answers
388
views
Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
2
votes
0
answers
76
views
Weak Hessian for solutions to certain quasilinear elliptic PDEs
In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form
$$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla ...
- 3,146
3
votes
2
answers
1k
views
Orthogonality to harmonic functions
Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define
$a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$
$b_n=f\Delta^{...
1
vote
1
answer
145
views
Endpoint in commutator estimate
Let $p\in(1,\infty)$ and $J^s=(1-\Delta)^{s / 2}$ with $s>0$. Then we have the following commutator estimate by C. E. Kenig, G.Ponce and L. Vega (1991 JAMS),
\begin{equation}
\left\|J^{s}(f g)-f J^{...
1
vote
1
answer
247
views
Elliptic interface problem without conditions on the interface
Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...
user60665
0
votes
0
answers
57
views
Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?
Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
- 107
5
votes
1
answer
4k
views
Difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$
This is cross-posted on MSE: https://math.stackexchange.com/q/1596565/9464
In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$...
user14319
4
votes
1
answer
700
views
Is $L^1(\Omega)$ continuous embedded in the dual of $H^m(\Omega)$ $(m>\frac{d}{2})$?
Let $\Omega$ be a bounded domain of $R^d$ with Lipschitz boundary. If $m>\frac{d}{2}$, such that $H^m(\Omega)$ is continuously embedded in $L^\infty(\Omega)$. Is $L^1(\Omega)$ continuously embedded ...
- 43
0
votes
1
answer
381
views
Converse of Lax-Milgram theorem [closed]
Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’...
- 23
7
votes
2
answers
2k
views
Heat kernel estimates and Gaussian estimates for semigroups, good reference?
Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.
If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ ...
0
votes
0
answers
37
views
Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 943
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^...
user128095
1
vote
0
answers
80
views
One dimensional periodic travelling waves to some pde
Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am ...
- 383
1
vote
0
answers
184
views
One question about Schrodinger Semigroups-(B. Simon)
This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
- 1,915
1
vote
1
answer
211
views
Approximation of functions by tensor products
Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
- 411
0
votes
0
answers
88
views
Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$
Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
- 183
2
votes
1
answer
300
views
Optimal control theory of PDEs
This is a somewhat openly phrased question because I am not quite sure what has been done in that direction.
Imagine one has two evolution equations
$$\partial_t u = p(x,\partial_x,f)u$$
$$\...
- 536
0
votes
0
answers
326
views
Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
- 1
7
votes
1
answer
439
views
About the convergence rate for an approximation to the heat kernel
Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} \...
- 1,649
8
votes
2
answers
2k
views
when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
- 1,644
4
votes
1
answer
202
views
Removable set for Sobolev space
It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...
- 43