**0**

votes

**0**answers

149 views

### A-priori bound on parabolic PDE that doesn't depend on end time

I have a PDE
$$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$
where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound ...

**0**

votes

**2**answers

273 views

### Fundamental Solutions with compact support (distributions)

Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$
We also then argue that if a fundamental solution has compact support, then it is supported ...

**0**

votes

**0**answers

121 views

### Coupled system of linear parabolic PDEs

Hi,
Are there any existence results for the coupled system of linear parabolic PDEs:
$$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$
$$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$
...

**1**

vote

**1**answer

216 views

### Heat equation of spatial complex variable

Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation
$$\frac{\partial ...

**0**

votes

**1**answer

272 views

### W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation

In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI:
http://arxiv.org/abs/1111.7207
They proved the $W^{2,1}$ estimate for standard monge-ampere equation
$detD^{2}u=f$
with $f$ bounded from below ...

**1**

vote

**1**answer

303 views

### Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...

**0**

votes

**1**answer

116 views

### Reference: DaPrato and Grisvard parabolic PDEs.

Has anyone read G. DaPrato and P. Grisvard Equations d'evolution abstraites nonlineaires de type parabolique?
It's not available in my library. I am wondering if it's worth me acquiring it: is it ...

**0**

votes

**1**answer

268 views

### LINEAR Parabolic equations. Smooth dependence from initial data

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data.
More specifically I have the following problem:
CONSIDER spaces $P:=\mathbb{R}^k$ ...

**1**

vote

**1**answer

474 views

### Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...

**0**

votes

**1**answer

230 views

### Map with prescribed Jacobian

Recently I came up with the following problem.
Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions ...

**1**

vote

**1**answer

420 views

### Tensor analysis/Differential forms outside physics

There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.
Most applications are physical, ...

**2**

votes

**1**answer

175 views

### Continuation of hyperbolic Laplacian eigenfunction

The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen ...

**-1**

votes

**1**answer

176 views

### Nearly elliptic equations [closed]

If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...

**1**

vote

**0**answers

290 views

### Relation between interpolation spaces and besov spaces

Consider the following two norms:
The interpolation norm:
1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...

**4**

votes

**1**answer

220 views

### Minimizing action squared versus action

I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...

**1**

vote

**0**answers

59 views

### When can a perturbation be treated as a regular perturbation?

I am working with cauchy problem of the form
$$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$
where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...

**2**

votes

**3**answers

674 views

### Question About Harmonic Function Theory

Given a non-negative function $u $ defined on $\mathbb{R}^2 $ , and satisfies :
$ \Delta u \leq 0 $ .
How can I prove that $u$ must be constant?
Is there an easy way to do it ?
Thanks !

**24**

votes

**3**answers

1k views

### Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
...

**13**

votes

**1**answer

727 views

### Is there a Seiberg-Witten version of Donaldson-Thomas theory?

Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of ...

**3**

votes

**0**answers

489 views

### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...

**1**

vote

**0**answers

234 views

### Convolution Estimates on a Smooth Manifold

Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying
$$
\|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty.
$$
Then ...

**0**

votes

**1**answer

408 views

### Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...

**0**

votes

**2**answers

598 views

### duality argument in PDE

Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references?
Occasionally I see this term appears in ...

**2**

votes

**1**answer

281 views

### Existence of PDE system (mean curvature flow coupled with surface PDE)

Hi all,
What should I look for if I want to study existence/uniqueness of the system of PDEs:
$$u_t -\Delta u + u\nabla \cdot v = f(u) \quad\text{on $\Gamma(t)$}$$
$$X_t = \kappa N(X) + u \quad ...

**0**

votes

**0**answers

201 views

### Where to learn about parabolic Hölder spaces and when to use them

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...

**0**

votes

**1**answer

436 views

### Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces

Consider a smooth metric measure space in which the integral of a gradient is meaningful. For example in the sense of upper gradients of Heinonen, or on a riemannian manifold with the associated ...

**0**

votes

**1**answer

342 views

### Application of inverse function theorem to get short time existence

I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): ...

**0**

votes

**1**answer

208 views

### Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:
Consider an orientation-preserving ...

**5**

votes

**1**answer

881 views

### Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact?
in which book we have this fact, the number of ...

**1**

vote

**0**answers

101 views

### base change for distributions

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...

**2**

votes

**1**answer

209 views

### Wavefront set of a product

Let $H$ be the Heaviside function. If $f(x_1,x_2)=H(x_2)$ on $\mathbb{R}^2$, then $WF(f)=N^*\{x_2=0\}$. Similarly, if $g(x_1,x_2)=H(x_1^2-x_2)$, I think the wavefront set of $g$ is the conormal ...

**0**

votes

**2**answers

719 views

### Existence of solution to quasilinear parabolic PDEs

Hello.
I want to prove the existence of a weak solution to:
Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that
$$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$
...

**3**

votes

**1**answer

175 views

### Stability of Dirichlet data for Helmholtz equation

I'm dealing with the Helmholtz equation $\Delta u +k^2u=0$ in a exterior region $R^3/D$ ( $D$ opened and bounded) of a three dimensional space with Dirichlet boundary condition $u=g$ on $\partial D$ ...

**8**

votes

**1**answer

700 views

### Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...

**2**

votes

**2**answers

310 views

### Higher order partial derivatives and global regularity.

Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...

**1**

vote

**0**answers

173 views

### Pitfalls when generalizing the heat kernel of a Riemannian metric

Suppose $M$ is a Riemannian manifold with some compact quotient under isometries.
Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...

**0**

votes

**0**answers

271 views

### Pullback of harmonic forms.

If $f \colon X \to Y$ is a holomorphic map between Kaehler manifolds, then the pullback of a harmonic form on $Y$ is not necessarily harmonic on $X$, even if $f$ is an immersion. This came up during a ...

**1**

vote

**1**answer

146 views

### Uniform equicontinuity of a family of indefinite integrals

Let $f_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f_k$ is locally integrable.) Does anyone know what the phrase
uniform ...

**1**

vote

**1**answer

86 views

### Neumann problem in case f=1

Is there a solution to the following problem?
$-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$.
where $\Omega$ is bounded.

**2**

votes

**1**answer

306 views

### eigenfunction of heat operator.

Let $\Delta$ be the usual Laplacian on $\mathbb R^n$, $\Delta=-\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. Consider the heat operator $H_t=e^{-t\Delta}$. Is there an eigenfunction of $H_t$ which ...

**24**

votes

**5**answers

2k views

### Why is symplectic geometry so important in modern PDE ?

First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds ...

**1**

vote

**3**answers

857 views

### book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...

**4**

votes

**1**answer

422 views

### Can one understand the Kelvin transform conceptually?

Let $U = \mathbf{R}^n - \{ 0 \}$, $n > 2$ and consider for a function $f \in C^2(U)$ the Kelvin transform
$$f^\star(x) = r^{2-n} f\left(\frac{x}{r^2}\right),$$
where $r = \lvert x \rvert$. One ...

**13**

votes

**1**answer

443 views

### Are isospectral manifolds necessarily homeomorphic?

It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric.
Is it known if there are closed Riemannian manifolds which are isospectral but not ...

**3**

votes

**1**answer

231 views

### On a family of $C^0$-convergent Riemann metrics

I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself.
Suppose that $M$ is a smooth compact manifold of dimension $m$ and ...

**4**

votes

**2**answers

686 views

### when a pseudo-differential operators to be compact?

In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...

**1**

vote

**2**answers

201 views

### The approximation to perturbed KdV Equation

Consider the perturbed KdV Equation $$u_t-6uu_x+u_{xxx}=\epsilon u$$,I want to use perturbative expansion to construct the solution as the form $$u=u(x,t;\epsilon)=\sum_{n=0}^\infty\epsilon^n ...

**2**

votes

**2**answers

473 views

### the inverse for the trace theorem

The trace theorem says that the restriction of a $W^{1,p}(\Omega)$ function $u$, $Tu$ belongs to $W^{1-1/p,p}(\partial\Omega)$ if $\Omega$ satisfies some smooth condition, for example, $\Omega$ is ...

**2**

votes

**1**answer

425 views

### The perturbed KdV Equation

I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...

**14**

votes

**6**answers

2k views

### Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...