**-2**

votes

**1**answer

531 views

### System of first order PDE

I have a system of first-order nonlinear partial differential equations.
$$A(x,t)\frac{\partial u}{\partial t}(x,t) + B(x,t)\frac{\partial u}{\partial x}(x,t) + c(x,t) = 0$$
$$x \in \mathbb{R}, \quad ...

**4**

votes

**2**answers

437 views

### Reference on Semigroup Theory and Parabolics PDE'S

Recently started to study Semigroup Theory. My background is equivalent to the first three chapters of the Jack Hale's book, Asymptotic Behavior of Dissipative Systems.
Looking for a reference to an ...

**1**

vote

**0**answers

292 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 ...

**10**

votes

**1**answer

379 views

### Lie $2$-groups and differential equations

I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: ...

**6**

votes

**0**answers

356 views

### How to estimate isoperimetric constant?

Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties,
There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$
for ...

**19**

votes

**3**answers

1k views

### Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...

**8**

votes

**1**answer

500 views

### How does electric potential relate to mean curvature?

Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.
Now consider the electric potential generated by this uniform mass ...

**0**

votes

**1**answer

162 views

### Test function .

For a smooth test function \eta and some constant C is it possible to have an estimate like the following?
|grad \eta|^2 < C {\eta}^2 ?
Thanks.

**1**

vote

**0**answers

200 views

### A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...

**15**

votes

**8**answers

2k views

### Applications of PDE in mathematical subjects other than geometry & topology

Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau ...

**15**

votes

**3**answers

1k views

### History of fundamental solutions

I have a few questions on the history of PDE.
Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's ...

**5**

votes

**1**answer

280 views

### A nonlinear system with special structure

Suppose we have an $n \times n$ uniform grid, covering $[-1,1] \times [-1,1]$ (typically, n $\approx$ 500). We have a smooth, differentiable function $z(x,y)$ that we want to determine on the nodes of ...

**2**

votes

**1**answer

192 views

### optimality of energy estimates for non smooth metric

Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty_t H^3_x$ and $\partial_t g \in L^\infty H^2_x$, then energy estimates enable ...

**9**

votes

**1**answer

372 views

### When is a given matrix of two forms a curvature form?

Let's assume we are working over $\mathbb{R}^n$ (but feel free to change to domain to answer the question). I wish to know if the equation $F = dA + A \wedge A$ can be solved for a matrix of $1$-forms ...

**1**

vote

**0**answers

67 views

### decay for spatially discrete parabolic equations with non-constant non-self-adjoint right hand side

Consider the following uniformly parabolic lattice differential equation
$ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & ...

**11**

votes

**1**answer

619 views

### On the definition of regularity

In the litterature on D-modules, there are many definitions of regularity of holonomic D-modules.
(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...

**1**

vote

**2**answers

497 views

### Hölder estimates on solutions of non-linear elliptic PDE.

In his book "Some non-linear problems in Riemannian geometry" T.
Aubin states the following result (Theorem 3.56):
Let $A(u)=F(x,u,\nabla u,\nabla^2u)$ be a non-linear second order
differential ...

**0**

votes

**1**answer

559 views

### Units of time in the gradient flow equation?

From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow ...

**3**

votes

**1**answer

573 views

### are there soliton solutions for Euler and Navier-Stokes Equation

I'm now reading papers about the the well-posedness of Euler and Navier-Stokes Equation, so I wonder if we have soliton solutions for this two equations just like for KdV equation. I'm interested in ...

**0**

votes

**1**answer

751 views

### Functionals continuous with respect to weak convergence

It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...

**2**

votes

**1**answer

314 views

### Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient?

Hello,
I am considering the following non-linear heat equation
$$
\left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R_+\times R^d
$$
where ...

**8**

votes

**8**answers

3k views

### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**26**

votes

**1**answer

2k views

### Unconditional nonexistence for the heat equation with rapidly growing data?

Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: ...

**1**

vote

**0**answers

234 views

### Geometric description of Jacobi's theorem on complete integrals of HJ eqn.

I am not sure if this question is adapted to this site, if it is not, then I will delete it.
The Hamilton--Jacobi theory is about the connection between:
the solutions of an Hamilton--Jacobi ...

**0**

votes

**0**answers

486 views

### characteristic surface

Hello,
I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ :
(1) $G_{xy}=0$
(2) $G_{xz}=0$
(3) $G_{yz}=0$
(4) $G_{xx}-G_{yy}=0$.
It is not hard to see that the general ...

**2**

votes

**1**answer

166 views

### Convergence of elliptic operators

Let $A_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A_t$ ...

**2**

votes

**0**answers

428 views

### Comparision of cubic hermite finite element and cubic B-spline finite element (in condition nunmbers of stiffness matrix, or sth else)

Background
Consider the one dimensional second order elliptic PDE,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
...

**3**

votes

**0**answers

145 views

### Regularity properties of the derivatives of a particular function on $D \times D\to \bar{D} $ ?

This question might sound a little less rigorously formulated, but I hope the question still makes sense.
Let $h: S^1 \to S^1$ be an oriention-preserving homeomorphism and let $p(z,t) = ...

**3**

votes

**0**answers

170 views

### Numerical solution

Last time, I asked this question
but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...

**3**

votes

**1**answer

812 views

### regularity of solution of linear elliptic PDE

I am interested in the boundary regularity of solutions of $ L(u) = f(x) \ge 0$ in $ \Omega$ with zero Dirichlet boundary conditions, here $L(u) = (-\Delta)^\frac{\alpha}{2}$ where $ 0 < \alpha ...

**2**

votes

**0**answers

249 views

### A free boundary problem by finite difference method

I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...

**3**

votes

**1**answer

313 views

### Embeddings for spaces of maximal regularity

Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true
...

**9**

votes

**1**answer

1k views

### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset ...

**1**

vote

**3**answers

445 views

### Finding an $H^1$ function given its values on $\partial\Omega$

Background
I've met this problem when I was trying to convert a elliptic PDE problem
into the corresponding variational problem in order to apply finite element method.
The PDE is an elliptic PDE ...

**2**

votes

**1**answer

564 views

### Endpoint Strichartz Estimates for the Schrödinger Equation

The non-endpoint Strichartz estimates for the (linear) Schrödinger equation:
$$
\|e^{i t \Delta/2} u_0 \|_{L^q_t L^r_x(\mathbb{R}\times \mathbb{R}^d)} \lesssim \|u_0\|_{L^2_x(\mathbb{R}^d)}
$$
$$
2 ...

**7**

votes

**1**answer

393 views

### What would the best treatment of Gehring's lemma look like?

In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...

**3**

votes

**0**answers

328 views

### kernel of the conformal Laplacian

Let $M$ be a smooth, closed manifold of dimension $n>2$. Let $L_g$ be the conformal Laplacian of the metric $g$. That is, $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar ...

**2**

votes

**0**answers

254 views

### General solutions for HJB equations in a special case.

I am reading the book of Wendell Flemming in control theorem to learn the HJB equation
Here is the setting that interests me: Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ functions such that ...

**4**

votes

**1**answer

503 views

### regularity of solutions of fractional laplacian

Hello, I am looking for boundary regularity of solutions of $(-\Delta)^s u= f(x)$ in $\Omega$ with Dirichlet boundary conditions and where $f $ is nice enough say $f\in C^{1,\alpha}(\overline\Omega)$. ...

**1**

vote

**1**answer

205 views

### Is there such a priori estimates for mean curvature type equation?

I am dealing with a mean curvature type equation as following:
$\displaystyle{\sum_{i,j=1}^{2}}(\delta_{ij}-\frac{u_{i}u_{j}}{1+|Du|^{2}})u_{ij}=(1+|Du|^{2})^{\frac{1}{2}-\frac{1}{2\alpha}}$, where ...

**3**

votes

**1**answer

555 views

### Long time behavior of the heat equation on R

Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...

**8**

votes

**0**answers

231 views

### Finding a dimension-free bound for a certain multiplier on Euclidean space

The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...

**6**

votes

**3**answers

697 views

### How to motivate and interpret the geometric solutions of Hamilton-Jacobi equation?

Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.
For an H-J eqn, I mean a first order pde $H\circ ...

**5**

votes

**1**answer

354 views

### A moving boundary in rock mechanics

I'm concern a moving boundary problem in rock mechanics.
We consider a problem of unsaturated flow of an in-compressible fluid in a
porous medium(rock) like D. Moreover suppose that support of a ...

**7**

votes

**3**answers

674 views

### Applications of geometric evolution equations.

Hi everybody,
I'm looking for applications of geometric evolution equations such as the Ricci flow and the extrinsic flows by Gauss and mean curvature. Applications other than topological ...

**5**

votes

**2**answers

782 views

### Estimates on the Green function of an elliptic second order differential operator.

Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...

**6**

votes

**0**answers

231 views

### Dirichlet-to-Neumann map on $C^{k,1}$ domains

I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in ...

**8**

votes

**2**answers

489 views

### Probabilistic Solution of the Porous Medium Equation

It is well known that the transition density for standard Brownian motion $B_t$ in $\mathbb{R}^d$ yields a solution to the global Cauchy problem for the heat equation $$u_t = \Delta u$$ with initial ...

**3**

votes

**0**answers

355 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 ...

**1**

vote

**0**answers

346 views

### Maximum principle for heat eq. with boundary conditions on derivatives

The Maximum principle for parabolic eq. is based on the fact that the boundary conditions are given on u.
How can this Maximum principle be used, when having boundary conditions including ...