**5**

votes

**0**answers

94 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**4**

votes

**1**answer

169 views

### $C^0$ estimate for solutions of elliptic PDE with Neumann BC

I am interested in a reference for (or counterexample to) a particular
$C^0$ estimate for solutions of the Laplace equation with Neumann
boundary conditions. More precisely, let $(M,g)$ be a ...

**2**

votes

**0**answers

84 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...

**2**

votes

**1**answer

115 views

### Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...

**2**

votes

**0**answers

118 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...

**4**

votes

**1**answer

105 views

### Liouville type theorems; linear PDE with decaying potential

Dear Mathoverflowers,
I am interested in the following pde:
$$ -\Delta u(x) + C(x) u(x) = 0 $$ in $ R^N$. Lets assume that $ C(x)$ is bounded and (smooth if you like) and satisfies the ...

**4**

votes

**0**answers

135 views

### Partial differential Equation over characteristic p

I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...

**3**

votes

**1**answer

120 views

### Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...

**5**

votes

**2**answers

135 views

### Inverse of partial differential operator as a smooth tame map

Tameness for maps is one of the main ingredients for the Nash-Moser inverse function theorem. A linear map $f: X \to Y$ between Fŕechet spaces with fixed seminorms is called tame if we have an ...

**4**

votes

**0**answers

156 views

### Reference for Hodge decomposition

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...

**6**

votes

**0**answers

113 views

### Local solvability of nonlinear elliptic boundary value problems

Malgrange proves the following statement regarding local solvability of (determined or underdetermined) nonlinear elliptic systems:
Let $F_i(x,D^\alpha u)=0$ be a nonlinear elliptic system of order ...

**1**

vote

**0**answers

91 views

### A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...

**2**

votes

**0**answers

48 views

### Solve a PDE related to free boundary problem

I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...

**1**

vote

**0**answers

56 views

### presence of turbulent phenomena in systems of linear pde?

Are there linear systems of PDE that are known to have solutions which exhibit turbulence, or can turbulence be firmly classified as a fundamentally non-linear phenomenon, similar to solitons or shock ...

**1**

vote

**0**answers

34 views

### Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question:
Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that
$F(x,\cdot)$ is convex with respect to the second variable.
$F(\cdot,v)$ ...

**1**

vote

**0**answers

94 views

### Half Laplacian; (definitions of) and regularity

I have a question regarding the half Laplacian $ (-\Delta )^\frac{1}{2}$ on some smooth bounded domain $ \Omega$ in $R^N$. I am attempting to clarify some confusion with the various definitions. ...

**12**

votes

**3**answers

786 views

### Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for ...

**3**

votes

**0**answers

100 views

### Boundary regularity of Dirichlet Eigenfunction on bounded domains

Consider a bounded, connected and open subset $\Omega\subset \mathbb{R}^d$ and the Dirichlet Laplacian $-\Delta$ acting in $L^2(\Omega)$.
Then we know that the eigenvalues of $-\Delta$ form an ...

**3**

votes

**1**answer

154 views

### Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...

**0**

votes

**0**answers

96 views

### comparing Inverse scattering to Hirota's method

Is Hirota's method for solving non-linear partial differential equations more effective(easier to compute) than Inverse scattering? Which method is more general in terms of its ability to solve wider ...

**1**

vote

**1**answer

90 views

### 2D semilinear elliptic PDE

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. ...

**4**

votes

**1**answer

216 views

### Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?

(This may be very easy question for MO; as I am just trying to understand Besov spaces)
Let $\phi \in C^{\infty}(\mathbb R^{n})$ with
$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: ...

**14**

votes

**2**answers

598 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

205 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 ...

**3**

votes

**1**answer

138 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

129 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

131 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

181 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

86 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

96 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

137 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

72 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. ...

**2**

votes

**2**answers

109 views

### Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...

**0**

votes

**0**answers

92 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

117 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

26 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

132 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

105 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

138 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

239 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

139 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

86 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

157 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

156 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

203 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

110 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 ...