# Tagged Questions

**-3**

votes

**1**answer

139 views

### Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature?
For example, can one make sense of
$$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 ...

**1**

vote

**1**answer

115 views

### Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...

**1**

vote

**1**answer

397 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**13**

votes

**5**answers

977 views

### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**2**

votes

**1**answer

289 views

### Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...

**7**

votes

**2**answers

246 views

### Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...

**4**

votes

**1**answer

164 views

### Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...

**2**

votes

**1**answer

203 views

### Fourier transform and spectrum of PDOs in $L^p$

Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a ...

**3**

votes

**1**answer

337 views

### Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let
$\mathrm{id}$ be the identity operator, let
$\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
$t > 0$ be a parameter.
Does ...

**1**

vote

**3**answers

844 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 ...

**1**

vote

**0**answers

309 views

### Definition of spectral gradient

Consider this differential operator
$$
\mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x}))
$$
where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...

**3**

votes

**0**answers

359 views

### PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases).
Let $\nabla^{2}$ be the vector Laplacian. Let ...

**3**

votes

**1**answer

447 views

### A name for PDE systems which are neither under- nor overdetermined?

The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither ...