**6**

votes

**1**answer

111 views

### Time decay for Hartree equation with Coulomb potential

Are there any time-decay results for the solution of the Hartree equation
\begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in ...

**5**

votes

**1**answer

135 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 ...

**0**

votes

**1**answer

86 views

### separable BV space for PDE's, Whats stopping us? [on hold]

Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...

**4**

votes

**1**answer

360 views

### traces of sobolev spaces under additional assumptions

Let $p\in [1,\infty]$, $\Omega$ an open bounded domain with (smooth, if necessary) boundary $\partial \Omega$.
Is there a subspace $X\subset L^p(\Omega)$ - a simply describable space, ideally a ...

**2**

votes

**0**answers

39 views

### Positivity of semiclassical pseudodifferential operators

Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)
Let $m\in\mathbb{R}$, and ...

**0**

votes

**0**answers

33 views

### Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...

**1**

vote

**1**answer

86 views

### Uniqueness of $\partial_t u -u\Delta u=0$ with $u(0,\cdot)=1$

Is there anything known about uniqueness of classical solutions to
$$
\partial_t u -u\Delta u=0\quad u(0,\cdot)=1
$$
on smooth domains $[0,T]\times D$ without boundary conditions? I know that ...

**3**

votes

**1**answer

137 views

### Progress on isospectral plane domains

Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit ...

**3**

votes

**0**answers

309 views

### Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention.
Note 2: This is my research problem, but the original problem ...

**9**

votes

**2**answers

208 views

### Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...

**1**

vote

**1**answer

77 views

### How to prove the Hölder continuity of a function $u$ by evaluating $\int_{B_{\rho}(x_0)}\frac{|Du(x)|^{2}}{|x-x_0|^{n-2}} dx$?

I'm looking at a video on thin obstacle problem given by Arshak Petrosyan.
In his lecture, he uses the following results:
Let $0<\alpha<1$, and $B_1$ be the unit ball centered at origin in ...

**1**

vote

**0**answers

44 views

### Regularity on Neumann problem on polygonal domain

I asked a similar question before but didn't get any responses. So I will attempt again (the prior question was regarding Holder continuity).
Let $ \Omega$ denote a cube in $ R^n$ and consider ...

**4**

votes

**1**answer

135 views

### Questions about the regularity of the “norm” associated to a convex set

Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g_K$ has all ...

**2**

votes

**2**answers

583 views

### Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...

**0**

votes

**0**answers

61 views

### A PDE problem on the Heat-Like differential equations

I came across the following questions in part of my work:
Consider the Heat-Like equation of the form $\frac{\partial u}{\partial t}=\hat{H}u + f(x,t)u + g(x,t)$ where $\hat{H}$ is a Sturm-Liouville ...

**0**

votes

**0**answers

38 views

### Error in the paper “Oscillating-Decaying Solutions, Runge Approximation and its Applications to Inverse Problems”?

I am going to use simpler notation then the paper. The potential problem is not very technical.
Theorem 4.2. in the paper states the following:
a) If $t < t_0$, then $I(t,\tau) \to 0$ as $\tau ...

**3**

votes

**0**answers

45 views

### Laplace Equation with Tangential Derivative Prescribed on the Boundary [closed]

I asked this question on MSE. However, I didn't get good answers there so I am seeking for it here. :)
Consider the following Laplace boundary value problem (BVP)
$$\matrix{
{{\nabla ^2}\Phi ...

**2**

votes

**0**answers

66 views

### A construction of the fundamental solution for Schroedinger equations

Does someone know some book or lecture notes useful for the reading of the paper
"A construction of the fundamental solution for the Schrödinger equation", Fujiwara, Daisuke, J. Analyse Math. 35 ...

**1**

vote

**0**answers

58 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball ...

**42**

votes

**7**answers

7k views

### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...

**2**

votes

**0**answers

322 views

### A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...

**3**

votes

**2**answers

267 views

### Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that
$$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla ...

**2**

votes

**0**answers

76 views

### A modification of Minty's trick?

I have the following result:
$$0 \leq \int_0^T (a(t)- |w(t)|)(b(t) - g^{-1}(|w(t)|))\quad\forall w \in L^2(0,T)$$
where $a$ and $b$ are both non-negative.
Does it follow that $b(t) = g^{-1}(a(t))$? ...

**2**

votes

**2**answers

109 views

### References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity.
I studied always, following Evans book "PDE", the case with ...

**2**

votes

**0**answers

77 views

### Heat equation - regularity of solutions [migrated]

Consider the heat equation on $\mathbb{R}$
$$
u_t=u_{xx}
$$
with boundary conditions $u(0,x)=g(x)$.
It is well-known that even if the function $g$ is "very bad'' (say, only bounded but not ...

**5**

votes

**1**answer

56 views

### Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$.
Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...

**2**

votes

**2**answers

111 views

### Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me.
Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...

**0**

votes

**1**answer

69 views

### Discrete maximum principle for finite difference to elliptic pde [on hold]

Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - ...

**1**

vote

**1**answer

179 views

### The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions

I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \nabla ^2 \nabla ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y ...

**-2**

votes

**0**answers

11 views

### Density results [migrated]

I have a Reflexive Banach space V that is compactly embedded into a Hilbert sapce H which is continuously embedded in its turn into a Banach space W:= V' (with W being the dual of V, that is V') and ...

**1**

vote

**0**answers

22 views

### Regularity of a flux induced by a potential

Take
$\Omega\subset R^n$ with smooth boundary (take a ball for example)
a function $f\in L^{\infty}(\Omega)$ with support strictly contained in $\Omega$ and with $\int _{\Omega} f \; dx=0$
a ...

**1**

vote

**1**answer

168 views

### Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...

**1**

vote

**0**answers

66 views

### One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...

**4**

votes

**2**answers

570 views

### Elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**0**

votes

**2**answers

205 views

### Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions.
Unfortunately, I am a ...

**3**

votes

**1**answer

215 views

### methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...

**0**

votes

**1**answer

82 views

### The monotone operator in $BV$ space

I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...

**2**

votes

**1**answer

75 views

### Getting an estimate of the form $\lVert u(t+h)-u(t) \rVert_{L^1(\Omega)} \leq \frac{Ch}{t}$ on solution of PDE

Let $u$ be a weak solution (i.e. $u \in C([0,T];L^1(\Omega))$ of some degenerate or nondegenerate parabolic equation $u' - Au = f$ on a bounded domain. (For my purpose it is enough to have this for ...

**6**

votes

**1**answer

360 views

### Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde:
...

**3**

votes

**1**answer

231 views

### Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research:
$$\Delta f - e^f \partial_s f = E(s,t)\,,$$
where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...

**9**

votes

**5**answers

446 views

### List of generic properties of Riemannian metrics

I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...

**2**

votes

**0**answers

28 views

### Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...

**0**

votes

**0**answers

42 views

### The properties of the solution pf minimizing problem with different parameters

I asked an similar problem before but received no respond. Here I modified the problem, add in more informations and assumptions, and with an extra question...
Let $\Omega\subset \mathbb R^2$ be open ...

**0**

votes

**1**answer

228 views

### Question on viscosity solution through stochastic differential equations

I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with ...

**8**

votes

**1**answer

454 views

### Asking for Advices for Choosing a Ph.D thesis problem (in PDE area)

I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn ...

**1**

vote

**0**answers

38 views

### PDE Parameter-Dependent Center Manifolds

In the book 'Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems' by Mariana Haragus parameter-dependent center manifolds are discussed. Here it is assumed ...

**3**

votes

**2**answers

213 views

### What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...

**3**

votes

**1**answer

161 views

### stability of the Monge-Ampère equation

Is there any hope to prove this conjecture (or a similar one)?
Conjecture Let $\Omega_k$ be a family of convex (smooth) domains, and let $u_k$ be convex Alexandrov solution of $$ \begin{cases}
...

**0**

votes

**1**answer

65 views

### Strong Maximum Principle for very weak supersolutions of Laplacian operator

I know classical strong maximum principles for supersolutions of Laplacian operator, which says:
Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$.
...

**2**

votes

**1**answer

79 views

### Elliptic regularity with mixed boundary conditions

I'm looking for some results about elliptic regularity with mixed boundary conditions. I know they exist with non mixed boundary conditions but where can I find some results for the mixed case?
Thanks
...