**14**

votes

**2**answers

593 views

### Vanishing eigenvalues of Jacobian

Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$?
Any ...

**4**

votes

**2**answers

202 views

### A question on certain elliptic PDE

Consider the elliptic PDE "CR"
$$\begin{cases} U_{xx}=V_{yy}\\U_{yy}=-V_{xx} \end{cases}$$
And its consequence "LAP" $$U_{xxxx}+U_{yyyy}=0$$.
Somehow, these equations are similar to the Cauchi ...

**2**

votes

**1**answer

130 views

### Strong maximum principle for weak solutions

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that ...

**0**

votes

**0**answers

43 views

### Properties of rearrangement maps

I have the following question to ask, concerning some properties of Schwarz radially decreasing rearrangements. It is well known that the map $u\rightarrow u^{\ast}$, being $u^{\ast}$ the Schwarz ...

**0**

votes

**0**answers

51 views

### Function Related to Jordan Curves

I am looking for a solution to the following problem:
given
a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$
an integrable function $g(s): c(s)\mapsto ...

**6**

votes

**0**answers

121 views

### Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that
$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...

**9**

votes

**0**answers

126 views

### Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
...

**4**

votes

**0**answers

171 views

### Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions
Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...

**1**

vote

**0**answers

82 views

### Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< ...

**4**

votes

**1**answer

92 views

### Besov Characterization of Strichartz Estimate.

On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to
$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), ...

**0**

votes

**1**answer

134 views

### Weak solution of a heat equation is zero?

I work on a bounded domain in $\mathbb{R}^n$. Let $u \in H^1(0,T;H^{-1})\cap L^2(0,T;H^1)$ be a solution of the heat equation:
$$\langle u', v \rangle + \int \nabla u \nabla v = 0$$
for each test ...

**0**

votes

**0**answers

70 views

### inequality involving gradient of two harmonic functions

My question is about the last inequality in the case i) of the proof of lemma 2.3 of this paper:
Existence of classical solutions to a free boundary problem for the p-Laplace operator, (I) by A. ...

**0**

votes

**0**answers

89 views

### Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$

My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...

**2**

votes

**0**answers

111 views

### Newtonian potential for continuous $f$

Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$.
Consider the Newtonian potential of $f$ (at least I think this is what it is called)
$$ v(x)=\int_{R^N} ...

**0**

votes

**0**answers

25 views

### convection/transport with different velocities

What is the prototypical model for convective transport of a quantity whose constituents move with constant but varying velocities?
In order to illustrate what a mean:
Suppose that a large number of ...

**1**

vote

**0**answers

108 views

### Stokes operator without dirichlet boundary condition

Let $\Omega$ be a domain, then the following stokes operator is quite well known :
$\mathcal{H} \rightarrow \mathcal{V}_{\sigma} $
$f \rightarrow u$ such that $ - \Delta u = f $
where $ ...

**1**

vote

**0**answers

99 views

### Compact embedding

Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact.
Is it true that the embedding $H^1_0(\Omega) \rightarrow ...

**1**

vote

**1**answer

132 views

### Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...

**4**

votes

**1**answer

233 views

### Spectral multipliers vis-a-vis Differential geometry

Let us mention two papers for examples: this one by Seeger and Sogge and this by Cheeger, Gromov and Taylor. One can also mention papers by Stein, for example, this one. There are also many others of ...

**4**

votes

**2**answers

133 views

### Heat kernel asymptotics for the sublaplacian on a contact Riemannian manifold

Let $\Delta$ denote a Laplace-type differential operator on a compact Riemannian manifold $(M,g)$. The asymptotics of the heat kernel and the heat operator trace of $\Delta$ are well-known (cf. ...

**0**

votes

**0**answers

81 views

### Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$.
What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...

**1**

vote

**1**answer

148 views

### Interior gradient estimate for uniformly elliptic equations

I am struggling with a problem like this: In dimension $n\geq 3$,
For the following uniformly elliptic equation, do we have interior gradient estimates?
$$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where ...

**2**

votes

**0**answers

51 views

### Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic).
The standard ...

**2**

votes

**1**answer

153 views

### BMO spaces on the torus

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as
$$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$
where $u_Q$ is ...

**0**

votes

**0**answers

43 views

### $L^\infty$ estimate for a fourth order (hyperbolic) equation

Consider the following fourth order equation
$$u_{tt}+u_t= d\Delta u-\Delta^2u+f,$$
with Dirichlet or Navier boundary conditions, that is on $\partial\Omega$, we assume that
...

**9**

votes

**1**answer

196 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

**1**

vote

**1**answer

105 views

### Decay of Solutions to the Heat equation

Consider the heat equation
$$ (\partial_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.
Consider the semigroup ...

**-3**

votes

**1**answer

149 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 ...

**0**

votes

**1**answer

142 views

### $\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$? [closed]

(This may be very simple question for MO; I had post it to math stack exchange few days back but I could not get any answer(or comment) to it)
It is well-known that, for $f,g \in L^{1}(\mathbb R).$ ...

**0**

votes

**0**answers

37 views

### reading request on linear elliptic systems of pdes, strong solutions

Does anyone have some references where I could find results on strong solutions to linear elliptic systems of pdes ?
Regards

**5**

votes

**0**answers

81 views

### $L^p$ regularity for wave equations with coercive boundary conditions

Suppose we have the wave type equation
$$\partial^2_tu - L u = 0$$ on a compact manifold with boundary, where $L$ is a second order strongly elliptic operator with coercive boundary conditions (not ...

**0**

votes

**0**answers

130 views

### Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$
I want to show for $r \geq 2$ that
...

**0**

votes

**1**answer

158 views

### When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...

**3**

votes

**0**answers

79 views

### Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...

**4**

votes

**0**answers

136 views

### Is the universal constant in Caccioppoli's inequality one?

If you go through the classical proof of the fact that if $\Delta u= 0$ on $B_R$, then for every $R^\prime<R$ there holds
$$
\int_{B_{R^\prime}}|\nabla u|^2 \leq ...

**0**

votes

**1**answer

170 views

### A question about PDE argument involving monotone convergence theorem and Sobolev space

I'm reading this paper. In it there is the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...

**1**

vote

**1**answer

206 views

### Quadratic PDE Systems

(First time asking question on this forum so please kindly let me know if this is out of scope/inappropriate etc.)
I have a problem that leads me to the following quadratic system of PDEs:-
$
c_1 ...

**0**

votes

**0**answers

63 views

### Compatibility of initial and boundary conditions

Suppose we consider the heat equation $$\partial_t u = \Delta u, x \in \text{int}D^2, t > 0$$
where $D^2$ is the closed unit disc in $\mathbb{R}^2$, subject to Neumann type boundary conditions ...

**2**

votes

**0**answers

80 views

### Variant form of the gronwall inequality

I know the following statement for gronwall inequality:
Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have,
$f' \leq \phi f$ and $f(0)=0$ then $f=0$
Now is ...

**1**

vote

**0**answers

71 views

### $L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...

**0**

votes

**0**answers

87 views

### elliptic regularity when right hand side in weak $L^p$

I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose $u,f$ are smooth functions defined on $B_1$ and $ \Delta u = f$ in $B_1$ with $u=0$ on $ ...

**0**

votes

**1**answer

146 views

### Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary [closed]

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|_{L^p} \leq ...

**0**

votes

**0**answers

131 views

### $b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$

This question stems from the proof of Theorem A.1 on page 425 of this paper.
Let $Q=(0,T)\times \Omega$. Suppose $b_n \rightharpoonup b$ in $L^q(Q)$ for any $q < \infty$ and $b_n \to b$ in ...

**0**

votes

**0**answers

77 views

### Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and
$\lambda\leq \sigma_1, \sigma_2 \leq ...

**1**

vote

**1**answer

144 views

### Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result

Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)
Let $\Omega ...

**6**

votes

**2**answers

151 views

### Relativistic Control Theory

I am looking for literature that combines General relativity and control theory.
So far I found a video lecture on "Integrability meets Control Theory: Harmonic maps in GR", other than that not so ...

**2**

votes

**0**answers

57 views

### Holder continuity of Poisson equation with divergence free drift

I am interested in the following PDE.
Suppose $u_m$ is a smooth solution of a elliptic equation of the form
$$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on ...

**2**

votes

**0**answers

106 views

### Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...

**4**

votes

**3**answers

163 views

### What to read for many-body problems in 3D Schrodinger equation

I am a graduate student just started learning dispersive PDE in MSRI's summer program. I roughly finished reading the paper by Klainerman and Machedon "ON THE UNIQUENESS OF SOLUTIONS TO THE
...

**4**

votes

**2**answers

175 views

### Solution to Schrödinger equation

I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.
My goal is to solve this PDE for $f:[-1,1] ...