**1**

vote

**1**answer

71 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 ...

**1**

vote

**1**answer

71 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

38 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

59 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

34 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 [on hold]

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 ...

**1**

vote

**0**answers

59 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

57 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 ...

**-8**

votes

**0**answers

50 views

### connectedness in Euclidean Space [closed]

f:[0,1] to R is a map defined by f(x)=sin(1/x),for x is not zero and
f(0)=0.X={(a,f(a))|a is in [0,1]}.prove that,X is a connected subset of
R^2 where the metric is usual euclidean metric on R^2.

**2**

votes

**0**answers

74 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

**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 ...

**3**

votes

**1**answer

134 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

122 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

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 ...

**3**

votes

**0**answers

293 views

+50

### 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 ...

**2**

votes

**2**answers

573 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 ...

**-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

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

**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

74 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

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 ...

**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 ...

**9**

votes

**5**answers

445 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

63 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

159 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

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
...

**3**

votes

**0**answers

87 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

125 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

56 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

61 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

31 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

81 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

84 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

72 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

150 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

234 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

70 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 ...

**3**

votes

**0**answers

40 views

### Foliation by Umbilic surfaces

Suppose $(M,g)$ denotes a Riemannian manifold with boundary that is a foliation by Umbilic surfaces. (As an example consider a manifold where the exists a unit parallel vector field) .
Is it ...

**0**

votes

**1**answer

67 views

### Discrete maximum principle for finite difference to elliptic pde

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} - ...

**0**

votes

**0**answers

66 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 ...

**8**

votes

**1**answer

453 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

63 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

31 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

74 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 ...

**3**

votes

**0**answers

44 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 ...

**8**

votes

**1**answer

183 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

77 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

178 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 ...