All Questions
202 questions
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
12
votes
2
answers
878
views
The ground state is signed and symmetric
Background
In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$...
- 39k
12
votes
2
answers
2k
views
Reference on Minty's trick
I am searching for a precise reference for the following result:
Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.
Assume that a sequence of nonnegative functions $(u_n)_n$ ...
- 3,425
10
votes
3
answers
1k
views
References: spectral analysis of the Laplacian operator
I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....
user89890
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'...
10
votes
0
answers
422
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 835
9
votes
1
answer
2k
views
Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
- 727
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
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
9
votes
2
answers
778
views
Rellich's theorem from compact resolvent
On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into $...
- 91
8
votes
2
answers
1k
views
Survey papers on the role played by PDE in mathematics
There are already several questions on MathOverflow that inquire about the many diverse relationships between PDE and several other 'areas' of mathematics (e.g., algebraic and differential geometry ...
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
8
votes
0
answers
260
views
Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$
I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...
- 503
7
votes
3
answers
1k
views
Reference on semigroup theory and parabolic PDEs
Recently started to study semigroup theory. My background is equivalent to the first three chapters of the Jack Hale's book "Asymptotic behavior of dissipative systems".
Looking for a reference to an ...
user11178
7
votes
1
answer
2k
views
A good reference for the wave front set
Hello,
I am wondering whether anyone know some good references for the theory of wave front set, microlocal analysis? I have some basic knowledge of distribution theory at the level of the Rudin's ...
- 1,649
7
votes
3
answers
2k
views
Collections of examples and counterexamples in (real, complex, functional) analysis, ODEs and PDEs
What books collect examples and counterexamples (or also "solved exercises", for some suitable definition of "exercise") in
real analysis,
complex analysis,
functional analysis,
ODEs,
PDEs?
The ...
user60665
7
votes
2
answers
641
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 39k
7
votes
2
answers
2k
views
Heat kernel estimates and Gaussian estimates for semigroups, good reference?
Hi, it seems like a big field and I'm having trouble getting some solid/classic references to get me started.
If $U \subset \mathbb{R}^d$ is a bounded domain with, say, $C^2$-boundary $\partial U$ ...
7
votes
1
answer
609
views
$H^s$ norm of a solution of a nonlinear Schrödinger equation
I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...
- 71
7
votes
1
answer
489
views
When the value of a function in a point is equal to its integral average over the point's neighborhood?
It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral ...
- 91
7
votes
0
answers
351
views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$
for smooth functions $h$ and $u$.
Does a similar chain rule hold (up to a reminder term) also for the ...
- 161
6
votes
2
answers
2k
views
Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
- 5,380
6
votes
1
answer
402
views
Reference request: Wasserstein metric spaces for non linear weights/mobility?
There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ...
- 5,380
6
votes
1
answer
357
views
Travelling waves for nonlinear Schrödinger equation
Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
- 61
6
votes
0
answers
110
views
Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
6
votes
0
answers
281
views
Spectral properties of Non-local Differential operators on real line
I am encountering non-local (and nonlinear) PDEs in my work. To compute stability, I am trying to numerically estimate the spectrum of linearized-but-nonlocal version of the said PDEs.
Definition: A ...
- 318
5
votes
3
answers
1k
views
Decay estimate for the heat equation: $\sup_{t>0}\int_{\mathbb{R}} t^\alpha |u_x|^2\ dx$
Let $u$ be a solution of the heat equation $$u_t - u_{xx} = 0, \quad t>0, x \in \mathbb{R}$$
with initial data $u(0,\cdot) = u_0$.
Fix $\alpha >0$. How can I estimate (without using explicitly ...
- 839
5
votes
3
answers
1k
views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\...
- 307
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
5
votes
1
answer
224
views
Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,407
5
votes
1
answer
743
views
Eigenvalues and Domain of the Laplace-Beltrami Operator
Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
- 51
5
votes
1
answer
395
views
Universal decay rate of the Fisher information along the heat flow
I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation
$$
\partial_t u=\Delta u
$$
...
- 5,380
5
votes
1
answer
453
views
Seeking for references on some PDEs
This is not a technical mathematical question. I came across some PDEs with no references nor their names.
$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$
The above ...
- 1,101
5
votes
1
answer
630
views
Uniqueness of Kantorovich potentials?
$\newcommand{\R}{\mathbb R}$Take $\Omega\subset \R^d$ bounded, convex, and smooth.
Consider the optimal transport problem with cost $c(x,y)=\lvert x-y\rvert^2$, leading to the quadratic Wassersein ...
- 5,380
5
votes
1
answer
379
views
References on the obstacle problem for the heat equation
Can you point out some references that deal with the obstacle problem for the heat equation?
$$(OP) \quad\begin{cases}
\max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \...
user108310
5
votes
1
answer
297
views
A scaled fractional ''Sobolev inequality''
Does a fractional interpolation inequality similar to $$
\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
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
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
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
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
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
$$A(r):=\frac1{\vert B_r(0)\vert}\int_{...
- 43.2k
4
votes
1
answer
150
views
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
- 300
4
votes
1
answer
418
views
Periodicity and Burger's equation
Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u_t+uu_x=u_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
...
- 43.2k
4
votes
1
answer
728
views
Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds
On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
- 43
4
votes
1
answer
129
views
Meaningful generalization of viscosity solutions to higher order equations
Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
user75426
4
votes
1
answer
1k
views
Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
I want to show:
Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align}
H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)
\end{align}
is compact.
I was able to show ...
- 437
4
votes
1
answer
364
views
$H^1$-continuity of Laplace's equation with respect to boundary data
Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
- 2,286
4
votes
1
answer
562
views
Fundamental solutions for degenerate elliptic equations
I am looking for a paper or a book that says about the existence and some estimates (like these in the non-degenerate case) of the fundamental solutions for degenerate elliptic equations $L = -divA\...
- 51
4
votes
1
answer
346
views
Reference for a Heat Process in a Wedge
I would like to ask about an explicit suggestion/reference for the following type of heat processes:
Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
- 337