**1**

vote

**0**answers

49 views

### method for global existance for the NLS

We consider the nonlinar Schr\"odinger equation(NLS):
$$i\frac{\partial u}{\partial t}+\Delta_{x}u +\lambda |u|^{2k}u =0, \ u(x, 0)= u_{0}(x);$$
where $\lambda \in \mathbb R, \ k \in \mathbb N,$ ...

**-1**

votes

**1**answer

140 views

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

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

**2**answers

122 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 $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\...

**0**

votes

**0**answers

70 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

105 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 $u(0,\...

**1**

vote

**1**answer

103 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

61 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 ...

**0**

votes

**0**answers

51 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

79 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 (x,y)...

**2**

votes

**0**answers

117 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

72 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 $...

**2**

votes

**0**answers

96 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 $b(t) = g^{-1}(a(t))$? ...

**4**

votes

**1**answer

180 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 ...

**4**

votes

**1**answer

202 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 ...

**5**

votes

**1**answer

76 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 ...

**4**

votes

**0**answers

355 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 ...

**3**

votes

**2**answers

665 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 ...

**1**

vote

**0**answers

27 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

**0**answers

74 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 $\mathbb{R}^n$ ...

**0**

votes

**1**answer

89 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

84 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 ...

**2**

votes

**0**answers

39 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

47 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 ...

**2**

votes

**0**answers

67 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 ...

**9**

votes

**5**answers

519 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 ...

**0**

votes

**1**answer

78 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$.
...

**3**

votes

**1**answer

210 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}
...

**2**

votes

**1**answer

113 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

**3**

votes

**0**answers

101 views

### Equations of the form $((\Delta)^{2}+\lambda\Delta+\gamma)(f)=0$

Here is the problem: I am trying to figure out if there is a non-zero solution for the system of equations:
$(\Delta^{k-1}+a)(d^{\ast}\eta)=2d^{\ast}d\xi$
$(\Delta^{k}+b)(d\xi)=2dd^{\ast}\eta$
for $...

**0**

votes

**0**answers

177 views

### Boundary conditions in the Finite Element Method

I just want to solve a Sturm-Liouville problem in 1D, i.e.,
\begin{align}
(p(x)u'(x))'+q(x)u(x) = f(x)
\end{align}
with boundary conditions
\begin{align}
u(0)=a \hspace{1cm} u'(0)=b
\end{align}
How do ...

**0**

votes

**0**answers

61 views

### How Minimal solution is obtained as limit of approximations

I have encountered a problem in the proof of a Lemma in an article. The image of Lemma and it's proof is this:
I can understand the proof, but I don't know why this solution which is obtained as a ...

**0**

votes

**1**answer

70 views

### Existence and estimates of a solution of a perturbed first order partial differential equation

My question is as follows: Let $A=\partial_x-\frac y2\partial_z$, $B=\partial_y+\frac x2\partial_z$, and $\Omega\subset \mathbb R^3$ be a smooth bounded open set. Take $g\in C^\infty(\Omega)$ (if you ...

**2**

votes

**0**answers

34 views

### Does the fast diffusion equation (or singular PME) on a manifold lose mass if the exponent is small enough?

Consider the singular porous medium equation
$$u_t - \Delta (|u|^{m-1}u) = 0$$
$$u(0)=u_0$$
given $u_0$ bounded, where $m \in (0,1)$.
When posed on $\mathbb{R}^n$, it is well known that mass is ...

**4**

votes

**1**answer

99 views

### Elliptic regularity for two dimensional domains

Suppose $ \Omega$ is a smooth bounded domain in $ R^2$. I am interested in the regularity of solutions to
$$-\Delta u(x) = f(x) \mbox{ in } \Omega$$ with $ u=0$ on $ \partial \Omega$.
If $ f \in ...

**5**

votes

**0**answers

101 views

### Methods of variational calculus in analytic number theory

What methods of calculus of variations have been used in analytic number theory?
I mean do Hamilton-Jacobi theory of PDE found usage in analytic number theory, which raises yet another question has ...

**4**

votes

**1**answer

97 views

### Regularity up to the boundary for the Poisson problem

It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...

**2**

votes

**1**answer

172 views

### comparing Laplacian and gradient of function on boundary

Consider $ E(x)$ some smooth function on $ \Omega$ (some smooth bounded domain in $ R^N$) and suppose $E=0$ on $ \partial \Omega$.
Suppose one knows that there is some $C_1,C_2 \in R$ such that
$ x ...

**5**

votes

**1**answer

238 views

### harmonic extension of a curve by different parametrization

Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...

**1**

vote

**1**answer

73 views

### degree theory for elliptic equations; special solutions

I am interested in using degree theory to examine some semilinear problems.
But instead of just looking for solutions lets assume i am looking for a certain class of solutions; for instance lets ...

**0**

votes

**0**answers

79 views

### A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it.
We consider the heat equation $$U_{t}=\Delta U\\U(x,y,0)=...

**8**

votes

**1**answer

537 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 ...

**2**

votes

**0**answers

69 views

### Regularity of $u$ in $u_t - \Delta \beta(t,u) = f$, can we get $u_t$ is a function?

I'm looking for reference discussing the regularity of the weak solution $u$ to the equation
$$u_t - \Delta \beta(t, u) = f$$
$$u(0) = u_0$$
where $\beta(t,\cdot)$ is a nonlinear function depending ...

**0**

votes

**0**answers

38 views

### Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference and hint for it.
Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and
let $u$ be a positive distributional supersolution to ...

**2**

votes

**1**answer

228 views

### Proving compatibility of two Partial differential equations

Given two PDE(s): $F(x,y,z,p,q)=0$
and $G(x,y,z,p,q)=0$
In I.A.N Sneddon's "Elements of Partial Differential Equations",If every solution of $F=0$ is a solution of $G=0$,...

**3**

votes

**0**answers

55 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(...

**9**

votes

**1**answer

230 views

### Conformal changes of metric and geodesics

Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...

**1**

vote

**0**answers

98 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \...

**1**

vote

**1**answer

272 views

### The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

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

**9**

votes

**3**answers

566 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?
...

**3**

votes

**0**answers

155 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...