# Tagged Questions

**33**

votes

**2**answers

1k views

### Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...

**1**

vote

**0**answers

114 views

### nodal lines in the dirichlet problem

In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?
Thanks for help.

**0**

votes

**1**answer

181 views

### Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables

I am trying to read Fritz John's article, here:
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637
And for the proof of Thm 1.1 in page ...

**3**

votes

**1**answer

183 views

### Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...

**1**

vote

**1**answer

277 views

### maximum principle for a non-uniformly parabolic operator

Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator
$$
P = \partial_t - \mathrm{e}^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x} \big( ...

**3**

votes

**1**answer

281 views

### What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...

**1**

vote

**0**answers

98 views

### About definition of weak derivative in abstract PDE problems

I'm confused about weak derivative definition.
$u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff
$$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$
holds for all $\varphi \in ...

**1**

vote

**0**answers

58 views

### strong stability for the wave equation

Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...

**7**

votes

**1**answer

387 views

### Representing immersions from a surface into 3-space

Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$
Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping
...

**2**

votes

**1**answer

140 views

### Asymptotic decay for the wave equation

Consider the wave equation
$$
y_{tt} = \Delta y - \epsilon y_t
$$
on $\Omega\subset R^n$, with Dirichlet boundary conditions.
Where $\epsilon >0$.
Is it possible to find an explicit value ...

**2**

votes

**1**answer

272 views

### Trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality
$$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$
will hold.
...

**12**

votes

**2**answers

641 views

### Applications of pseudodifferential operators to PDE

I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole ...

**2**

votes

**1**answer

135 views

### The maximum in the Poisson problem on the cube with constant source

Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial ...

**1**

vote

**1**answer

154 views

### Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then
...

**2**

votes

**3**answers

312 views

### Nonharmonic solutions of Laplace's equation

Let $f \colon U \to \mathbb{R}$ be a twice differentiable function, where $U$ is an open subset of $\mathbb{R}^n$. Here twice differentiable means that all the second partial derivatives ...

**1**

vote

**0**answers

299 views

### Solving a PDE involving a mixed derivative for a partial derivative

Consider a PDE of the form
\begin{equation}
\frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right)
\end{equation}
or
\begin{equation}
\frac{\partial^2u}{\partial ...

**2**

votes

**1**answer

253 views

### Is there a lower bound for variance in terms of curvature?

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero ...

**1**

vote

**1**answer

154 views

### weak*closure of {f:||f||=1} in dual.

What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology.
So what is the weak* closure of this set. Thanks.

**0**

votes

**0**answers

122 views

### Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations.
My questions are:
(a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...

**4**

votes

**1**answer

232 views

### Sharpness of the Sobolev embedding theorem

We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, ...

**2**

votes

**2**answers

491 views

### Using Galerkin method for PDE with Neumann boundary condition?

I am wanting to show existence of solutions to
$$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$
with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ...

**1**

vote

**2**answers

524 views

### $\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

**4**

votes

**1**answer

609 views

### Density of smooth functions in Sobolev spaces on manifolds

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...

**1**

vote

**0**answers

124 views

### Trace Inequality question

There is a result in a paper I am reading :
Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that
$$\lVert n \times u\rVert_{H^{-1/2}(\partial ...

**1**

vote

**1**answer

468 views

### Showing a singular integral operator takes Holder continuous functions to Holder continuous functions (of the same order)

I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...

**0**

votes

**1**answer

180 views

### On the fractional Schrödinger equation

I wonder if there is any theory about what we can call the fractional Schrödinger equation:
$$
\mathrm{i}\frac{\partial \psi}{\partial t} = (-\Delta)^s \psi + g(|\psi|^2)\psi \quad\hbox{in ...

**3**

votes

**1**answer

364 views

### conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...

**3**

votes

**1**answer

168 views

### Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?

Hi!
Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type
$$f=1+\mathcal{O}\left( \frac{1}{|z|} ...

**1**

vote

**0**answers

180 views

### Galerkin method for existence for PDE with nonsymmetric bilinear form

Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in ...

**2**

votes

**2**answers

323 views

### Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

**2**

votes

**1**answer

171 views

### Manifold structure for the set of solutions to a first order elliptic system?

Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...

**1**

vote

**0**answers

180 views

### Stationary Phase and Propagation Speed (Reference)

I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if ...

**23**

votes

**3**answers

1k views

### “Wild” solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions. In ...

**18**

votes

**10**answers

4k views

### Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math ...

**1**

vote

**1**answer

220 views

### In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation
$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...

**0**

votes

**1**answer

77 views

### Integral of a harmonic function on a manifold with two non-parabolic ends

Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that ...

**2**

votes

**0**answers

87 views

### P-laplacian equation

Hi guys,
in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ?
Thank you!

**10**

votes

**1**answer

537 views

### Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...

**2**

votes

**0**answers

187 views

### Extension divergence-free, curl-converging vector field

Hi.
Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...

**22**

votes

**1**answer

1k views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...

**1**

vote

**0**answers

82 views

### Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.
Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...

**2**

votes

**2**answers

243 views

### Regularity of parabolic equation; divergence free drift

I am curious about weak solutions of the parabolic problem
$ u_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is ...

**1**

vote

**2**answers

179 views

### Stokes problem; naive question on the regularity of pressure term

I am attempting to recall some basic knowledge related to Stokes' problem.
In particular I am following along in Evans PDE book in section 8.4.
So lets assume that
$-\Delta u + \nabla P = f $ in $ ...

**1**

vote

**0**answers

164 views

### multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ ,
is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $
for $\left | \sigma \right |\leqslant ...

**0**

votes

**1**answer

133 views

### Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following
Theorem: ...

**8**

votes

**1**answer

302 views

### Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...

**0**

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**0**answers

144 views

### Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...

**3**

votes

**1**answer

156 views

### Coarea formula in a subelliptic context

Consider smooth vector fields $X_1,..,X_k$ in ${\mathbb R}^n$, satisfying the H\"ormander condition, i.e. for all $x$, the Lie algebra generated by $X_1(x),...,X_k(x)$ is ${\mathbb R}^n$. Do you know ...

**2**

votes

**1**answer

151 views

### How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

**0**

votes

**1**answer

318 views

### Existence of a function

[also asked here http://math.stackexchange.com/questions/307197]
All arguments are in $\mathbb{R}^3$.
Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. ...