**1**

vote

**1**answer

85 views

### The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions

I am interested in solving the following biharmonic eigenvalue problem.
$$\begin{array}{cccc}
& \nabla ^2 \nabla ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y ...

**4**

votes

**2**answers

197 views

### Historical developement of analysis and partial differential equations (especially in the 20th century)

What are some comprehensive surveys or monographs that describe (in
enough technical detail) the historical development of the various
subareas of analysis and partial differential equations?
...

**3**

votes

**0**answers

112 views

### ricci flow on surfaces

In Hamiltons paper "Ricci flow on surfaces" there is an estimate on $|\nabla R|^2$ which shows that $|\nabla R|^2 \leq C_1 \exp{\frac{rt}{2}}$ for some constant $C_1$.
Actually for any solution of the ...

**3**

votes

**0**answers

52 views

### Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...

**0**

votes

**1**answer

118 views

### A question on the integrability of eigenfunctions of the Laplacian

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction,
$$\Delta u=-\lambda u.$$
I was wondering under what condition (for example, spaces ...

**7**

votes

**2**answers

205 views

### Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...

**0**

votes

**0**answers

109 views

### Harmonic function with Dirichlet boundary condition

Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...

**1**

vote

**1**answer

44 views

### Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me.
Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...

**2**

votes

**1**answer

187 views

### Does this PDE only have the trivial solution?

Let $(M,g)$ be a closed Einstein manifold of dimension $m>2$ and
$$
\mathrm{Ricc}(g)=\lambda g,
$$
$h$ a symmetric $2$-covariant tensor, $\Delta=\nabla^*\nabla$ the Laplacian on functions as well ...

**1**

vote

**0**answers

75 views

### A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...

**2**

votes

**1**answer

35 views

### Literature on Lateral cauchy problem

Can you please provide me with books that deal with lateral cauchy problem?
Also introductory articles are fine by me.
Is this topic covered in books like Evans', Taylor's, Hilbert-Courant's or ...

**0**

votes

**0**answers

21 views

### Relaxation methods for nonlinear PDE boundary value problem

I have the following type of PDE $$\frac{\partial u}{\partial t} = A - B u + \nabla^2 u^2 $$ For the relaxed system I need to solve therefore $$0 = A - B u + \nabla^2 u^2 $$
Could someone address me ...

**1**

vote

**1**answer

97 views

### Are solutions of the Beltrami Equations necessarily smooth?

Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The Beltrami Equations are defined as the following system of PDE’s on the domain $ ...

**2**

votes

**0**answers

24 views

### elliptic regularity of Neumann problem on Square

I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) ...

**2**

votes

**0**answers

56 views

### elliptic regularity for Neumann BVP on square

I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ ...

**3**

votes

**0**answers

52 views

### Minimisers and critical points of variational integrals

In the following we consider $\Omega\subset\mathbb{R}^n\ (n\geq2)$ to be open, bounded and with Lipschitz boundary. Consider the following regular variational integral:
\begin{equation*}
...

**14**

votes

**4**answers

880 views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**0**

votes

**0**answers

29 views

### regularity for an elasticity problem

Consider $B$ a bounded domain of $\mathbb{R}^3$ with a smooth boundary $\partial B$.
Is there any reference for the optimal regularity with respect of the boundary data of the solution of the ...

**5**

votes

**3**answers

239 views

### A space of distributions vanishing on the boundary

The revised question
After more reflection on the problem, I might have found the answer by myself. Let $U$ be an open subset of $M$, irrespective of whether it has a boundary or not. Let
$$\mathcal ...

**4**

votes

**1**answer

110 views

### Mixed norm estimate for the heat equation

Consider the inhomogeneous linear heat equation
$$\partial_tu-\Delta u=F$$
on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice ...

**4**

votes

**1**answer

171 views

### Zeta-Determinant Theorem

Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in ...

**4**

votes

**1**answer

106 views

### General solution to null-divergence equation

I am interested in whether each component of a divergenceless vector can itself be written as a divergence. My motivation for this question is the characterization of so-called trivial conservation ...

**6**

votes

**3**answers

206 views

### Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators.
So my question is something like this:
Let $i \partial_t \psi(x,t) = ...

**2**

votes

**0**answers

100 views

### Changing frames of the tangent bundle with Schwartz functions [closed]

Let's consider two global frames $\{v_{1},....v_{N}\}$ and $\{u_{1},....u_{N}\}$ of the tangent bundle $T\mathbb{R}^N$.
Now consider the matrix $\{f_{i,j}\}$that change the frame $\{v_k\}$ to ...

**2**

votes

**1**answer

124 views

### Connection between the p and q Laplacians

I'm just looking for some quick and dirty intuition(and/or reading material) about the following:
I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( ...

**0**

votes

**0**answers

51 views

### Modified Sine-Gordon on Torus

Has the modified (sign altered) Sine Gordon Equation:
$$ \alpha_{ss} + \sin \alpha = 0 $$
been found to have interesting geometrical properties?
With symmetric ODE I obtained closed loops on a ...

**2**

votes

**2**answers

189 views

### Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian ...

**0**

votes

**1**answer

93 views

### Numerical methods for solving a hyperbolic nonlinear PDE

What type of numercial methods are there to solve PDE of the sorts of:
$$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$
$$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ...

**5**

votes

**1**answer

155 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

**2**

votes

**1**answer

129 views

### Second order estimates of Monge-Ampere equations

In order to prove existence of solutions of real and complex Monge-Ampere equations in various modifications (e.g. as in the Calabi problem) one often uses the method of a priori estimates. One of the ...

**5**

votes

**0**answers

92 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**4**

votes

**0**answers

99 views

### Compensated compactness for system of conservation laws?

As far as I knew, the method of compensated compactness can be used only for one-dimensional scalar and $2\times 2$ systems of conservation laws, i.e. $u_t+f(u)_x=0$. But if I understood correctly ...

**1**

vote

**0**answers

42 views

### Existence of at least one positive solution for semilinear biharmonic equation with critical exponent

Let $\Omega \subset \mathbb{R}^N$, $N\geq 5$. Now assume the biharmonic problem with singular term as follow
\begin{cases} \Delta^2u=\lambda \dfrac{u}{|x|^4}+u^{p} & ...

**0**

votes

**0**answers

25 views

### Continuous time dynamic programming: Quadratic guess for value function

In a control problem like so:
$$J = min \int_0^{t_f} Qx^2 + Ru^2 dt $$
$$\dot{x} = Ax + Bu$$
$$x(0) = x_0$$
The regular Linear Quadratic Regulator is attained by asssuming that the optimal value ...

**2**

votes

**1**answer

156 views

### Theorems that tell if an explicit analytical solution is possible for nonlinear PDEs

Are there any theorems that tell if a particular nonlinear PDE can be solved explicitly by analytical methods?
Where analytical methods I refer to methods such as power series or any methods that use ...

**0**

votes

**1**answer

62 views

### Reference request for the focussing example

I was reading "From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE" by Terence Tao, which is a Notice of the American Mathematical Society Vol. ...

**3**

votes

**0**answers

52 views

### Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional
$$
F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1
$$
where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...

**1**

vote

**0**answers

73 views

### Construct a PDE solution from a net of approximations

Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let ...

**3**

votes

**1**answer

127 views

### Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants.
Let $Q(R) := Q(R^2,R) = B(0, R) \times ...

**4**

votes

**2**answers

270 views

### Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold.
Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator?
Thanks

**2**

votes

**1**answer

131 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...

**2**

votes

**1**answer

103 views

### Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation
$$
u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1}
$$
for a Hölder continuous coefficient $a(x,t)$ satisfying
$$
0<C_o \le ...

**3**

votes

**1**answer

47 views

### Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet.
Link:http://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded
...

**2**

votes

**0**answers

51 views

### Reference/proof for parabolic Holder spaces property

Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$.
What can be said about $u_x=\partial_x u$?
I am not ...

**4**

votes

**0**answers

215 views

### The Spectrum of certain differential operators

We fix a Hilbert space isomorphism $\phi:H^{1}\to H^{2}$. Here by $H^{s},\;s=1,2,\;$ we mean the sobolev space on $\mathbb{R}^{2}$ or $S^{2}$.
We consider the following polynomial vector field on ...

**0**

votes

**0**answers

28 views

### The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that
...

**1**

vote

**0**answers

49 views

### Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...

**1**

vote

**0**answers

44 views

### Level Set Advection with Extension Velocity

We're studying the following system of PDEs for a scalar function $F(x, t)$ with $x \in \mathbb{R}^3$ and $t \in \mathbb{R}$. The function $F(\cdot, t)$ is a level set function for a time-dependent ...

**3**

votes

**1**answer

122 views

### Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space.
I am looking for references about results for the scalar case ...

**0**

votes

**1**answer

66 views

### Does this time-dependent trace space have a name?

This question is a follow up to this question.
Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in ...