# Tagged Questions

**4**

votes

**0**answers

50 views

### Sobolev spaces defined on non-compact Lie groups

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and ...

**5**

votes

**1**answer

90 views

### If $u \in H^1(U)$, then $Du = 0$ almost everywhere on the set $\{u = 0\}$, auxiliary result

Let $\phi$ be a smooth, bounded and nondecreasing function, such that $\phi'$ is bounded and $\phi(z) = z$ if $|z| \le 1$. Set$$u^\epsilon(x) := \epsilon \phi(u/\epsilon).$$Do we necessarily have ...

**7**

votes

**0**answers

47 views

### Continuous inclusions Sobolev theorem, question [closed]

How do I see that if $f$, $g \in H^s(\mathbb{R}^n)$ for $s > n/2$, then $fg \in H^s(\mathbb{R}^n)$ and$$\|fg\|_{H^s(\mathbb{R}^n)} \le C\|f\|_{H^s(\mathbb{R}^n)}\|g\|_{H^s(\mathbb{R}^n)},$$the ...

**3**

votes

**2**answers

127 views

### Many-Body Green's Functions for Interacting Systems of Fermions

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition:
...

**6**

votes

**2**answers

193 views

### Epsilon regularity for minimal surfaces in arbitrary Riemannian manifolds

For experts in the analysis of minimal surfaces I will state the question first; then I will follow up with details.
Question: Does the $\varepsilon$-regularity theorem of Choi and Schoen ...

**1**

vote

**0**answers

31 views

### Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

I asked this question on Mathematics network but it didn't receive any answers. So I assume it is just beyond the classic things in PDEs and I decided to ask it here too.
Consider the following ...

**1**

vote

**3**answers

142 views

### Exists $C = C(\epsilon, q)$ such that $\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)}$ for all $W^{1, 1}(0, 1)$? [closed]

Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in ...

**3**

votes

**1**answer

53 views

### Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$? [closed]

Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging ...

**0**

votes

**0**answers

65 views

### Boundedness of a function that satisfies a PDE-type inequality

Let $\Omega$ be a bounded Lipschitz domain, and let $u\colon[-T,0]\times \Omega \to \mathbb{R}$ be a function with $u(-T)=0$.
Suppose that
$$\sup_{-T \leq t \leq 0} \int_\Omega |(u(t)-k)^+|^2 + ...

**2**

votes

**1**answer

64 views

### Exactly solvable examples of diffusion equation with variable diffusivity?

There are many examples of potentials $V(x)$ for which Schrodinger's equation for a single particle in one dimension is exactly solvable, in the sense that we can give "nice" expressions for the ...

**4**

votes

**2**answers

113 views

### existence of a special conformal mapping

Sorry I don't know how to give an appropriate title.
In the complex plane, suppose there is a graph $x+if(x)$ separating the plane into two unbounded components, where $f(x)$ is smooth and bounded, ...

**50**

votes

**2**answers

3k views

### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

**1**

vote

**0**answers

51 views

### Suggestion for books in Pertubation theory with an emphasis on the theory

As the title suggest I am looking for another good coverage of the theory of Pertubation theory.
Currently I am working through Murodock's book: Pertubations: Theory and Methods.
But I am rest assure ...

**8**

votes

**2**answers

305 views

### Poincaré lemma for distributions

Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form
$$
u=\sum_{1\le j\le n} u_j dx_j,\quad ...

**2**

votes

**0**answers

111 views

### How does the $L^\infty$ norm of the solution of $-\Delta u + \lambda u =0$, $\partial_\nu u=\alpha$ depend upon $\alpha$ and $\lambda$?

Let $\lambda > 0$ be a constant and let $u$ be the weak solution on a bounded domain $\Omega$ of
$$-\Delta u + \lambda u = 0 \quad\text{in $\Omega$}$$
$$\partial_\nu u = \alpha \quad \text{on ...

**1**

vote

**0**answers

75 views

### Wave-like equation with 1st order time derivative and non-constant coefficients

We start with the following recurrence relation for complex coefficients $c_{n,m}$:
$$i\dot{c}_{n,m}(t) = \sqrt{(n+1)(n+2)(m-1)m}c_{n-2,m+2} + \sqrt{n(n-1)(m+1)(m+2)}c_{n+2,m-2}$$
where ...

**2**

votes

**1**answer

64 views

### If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?

Let $X_0 \subset X_1 \subset X_2$ be continuous embeddings, with $X_0 \subset X_1$ compact.
Suppose $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$.
Is then $u \in L^\infty(0,T;X_1)$?
To apply ...

**1**

vote

**1**answer

116 views

### $L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of
$$u_t - \Delta u = f$$
$$u(0)=u_0$$
and
$$-\Delta v = f$$
$$|\Omega|^{-1}\int_\Omega v =0$$
on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...

**2**

votes

**0**answers

151 views

### $L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of
$$au_t - 2d\,\Delta au = cv - f$$
$$bv_t - d\,\Delta bv = f$$
$$u(0)=u_0, \quad v(0)=v_0$$
where $f$ ...

**0**

votes

**1**answer

108 views

### In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made:
Let $u_m$ converges weakly to $u$ in ...

**0**

votes

**0**answers

19 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g ...

**1**

vote

**0**answers

68 views

### Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...

**2**

votes

**1**answer

118 views

### The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you!
Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...

**0**

votes

**1**answer

68 views

### Linearized stream function

I am trying to work through a paper Instability in Parallel Flows Revisited by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance ...

**2**

votes

**2**answers

199 views

### Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function ...

**2**

votes

**1**answer

151 views

### Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...

**1**

vote

**1**answer

86 views

### Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...

**0**

votes

**1**answer

106 views

### Is there any solution for this PDE system?

Let $(\mathbb{R}^2,\langle .,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with
\begin{align}
...

**3**

votes

**0**answers

81 views

### Reconstructing a vector field on the circle

Consider a ODE on the circle of the form
\begin{align*}
\frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \omega(x) \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} \begin{pmatrix} x_1 \\ ...

**6**

votes

**1**answer

188 views

### PDE characterisation of elementary symmetric functions?

For $k\leq{}n$ the elementary symmetric polynomials are defined by:
$$e_k(x_1,\ldots,x_n)=\sum_{1\leq{}i_1<...<i_k\leq{}n}x_{i_1}\cdots x_{i_k}$$
I believe I can prove (by a complex brute force ...

**2**

votes

**1**answer

85 views

### Scaling of distributions

Suppose we have a sequence of $L^1(\mathbb{R})$ functions $p_\epsilon$ with $\|p_\epsilon\|_{L^1} \leq 1$ for all $n$. Suppose we know that $p_\epsilon \to 0$ in distributions. Is it obvious that ...

**0**

votes

**0**answers

47 views

### Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...

**0**

votes

**1**answer

71 views

### A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...

**2**

votes

**1**answer

70 views

### Smoluchowski-Poisson dynamics with atomic measures

"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, ...

**3**

votes

**0**answers

101 views

### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...

**2**

votes

**0**answers

115 views

### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...

**0**

votes

**0**answers

107 views

### For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?

Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that ...

**2**

votes

**2**answers

189 views

### Backgrounds of the p-Laplacian Operator

Motivation
I encountered the following partial differential equation (PDE) in a mathematical paper
$$\begin{array}{}
u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...

**2**

votes

**0**answers

125 views

### $L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$.
Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation
$$u_t - \Delta u = 0$$
$$u(0) = u_0$$
where $u_0$ is bounded initial data. Here ...

**7**

votes

**1**answer

506 views

### $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...

**2**

votes

**0**answers

83 views

### Heat kernel on manifold with boundary

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...

**2**

votes

**2**answers

235 views

### The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...

**1**

vote

**0**answers

81 views

### Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...

**5**

votes

**0**answers

143 views

### Has anyone studied a transport equation of this form?

Let $L\colon \mathbb{R}^2 \times \mathbb{R}^+\to \mathbb R$ satisfy
$$
\frac{\partial L}{\partial t} (x,t) = \max\left\{ \frac{\partial L}{\partial x_1}, \frac{\partial L}{\partial x_2} \right\}
$$
...

**52**

votes

**2**answers

837 views

### Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...

**6**

votes

**1**answer

223 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 ...

**3**

votes

**1**answer

219 views

### difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$

This is cross-posted on MSE: http://math.stackexchange.com/q/1596565/9464
In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$ ...

**0**

votes

**0**answers

81 views

### An Estimation on the Heat Kernel

I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get
...

**1**

vote

**0**answers

63 views

### A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...

**0**

votes

**0**answers

38 views

### A $\mathcal{C}^1$ differentiable domain is $F_\sigma$?

Let us consider a domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary.
Suppose now, that for every $R>0$ the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. ...