**0**

votes

**0**answers

59 views

### Boundary conditions for Klein-Gordon equation [on hold]

Let us consider the Klein-Gordon equation
$$(\Box +m^2)u=0,$$
where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$.
...

**4**

votes

**0**answers

48 views

### I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...

**0**

votes

**0**answers

55 views

### why $\varphi''\in L^{2}(R)$ [on hold]

I have the following question: Let $T_{c}$ be an unbounded operator with domain $D(T_{c})=\{u\in L^{2}(R), T_{c}(u)\in L^{2}(R)\}$.
If $\forall \varphi \in \mathcal{C}^{\infty}_{0}(R): \|\varphi''\...

**0**

votes

**0**answers

45 views

### estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...

**2**

votes

**0**answers

38 views

### Conservation laws for modified Degasperis-Procesi equation

It is known that the Korteweg-de Vries equation
$$u_{t}+uu_{x}+u_{xxx} = 0,$$
with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely,
$$E(u)=\frac{1}{2}\int_{0}^{...

**5**

votes

**1**answer

67 views

### Decay estimates for wave and Klein-Gordon equation in “generic” curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...

**2**

votes

**1**answer

91 views

### $H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...

**1**

vote

**0**answers

45 views

### How much can the integrability at zero tell about the decay rate around zero? [migrated]

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...

**0**

votes

**0**answers

39 views

### Half strip neighbourhoods for regular surfaces [on hold]

Let $S$ be a regular compact surface in $\mathbb{R}^3$. It is well known that such surfaces admit a global tubular neighborhood, of thickness $\epsilon>0$ (for a suitable $\epsilon>0$).
In ...

**2**

votes

**1**answer

55 views

### IBVP with transformed boundary conditions

I am trying to use the following result
Theorem:
A pde of the form
$$\frac{\partial w}{\partial t} = F\{x, \frac{\partial w}{\partial x},\frac{\partial^2 w}{\partial^2 x}\}$$
has an ...

**1**

vote

**0**answers

44 views

### Mean Value Property for harmonic functions [migrated]

I would appreciate any insights on this matter;
Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...

**5**

votes

**1**answer

183 views

### finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...

**2**

votes

**1**answer

79 views

### Step 2 of The Strichartz's Estimates in Cazenave's Book

My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...

**0**

votes

**0**answers

64 views

### Solving a system of Laplace equations

Let $u_0$ and $u_1$ to be smooth functions defined on $\Omega\subset\mathbb{R}^n$, consider the following system of equations
$$\triangle u_1 = C_1(\partial_{ij}u_0),$$
$$\triangle u_0 = C_0u_1,$$
...

**4**

votes

**1**answer

69 views

### weak convergence in $H_0^1$ and strong convergence in $L^2$

I'm reading a hand-waving argument in a proof of Chapter 7 of the Navier-Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly.
Let $\Omega\subset{\mathbb{R}^n}$ ...

**1**

vote

**0**answers

60 views

### Method of characteristic for a system of first order PDEs

I am working with this system of first order PDEs:
\begin{equation}
\left\{
\begin{aligned}
%Suscettibili
&\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...

**4**

votes

**0**answers

139 views

### Contact manifolds and pseudodifferential operators

By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...

**4**

votes

**0**answers

66 views

### Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer:
$$
...

**0**

votes

**1**answer

121 views

### a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(R^{2})$?
Thank you in advance.

**1**

vote

**1**answer

141 views

### Lipschitz functions and $W^{1,\infty}$

I am not sure my question is research type, but I am sure I can find here an answer.
So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295:
Theorem 4 (...

**11**

votes

**1**answer

163 views

### Harmonic analysis, compute that this integral tends to $0$

We have the following setting.
$U$ is a bounded Lipschitz domain in the complex plane.
Consider the following classical Dirichlet problem for the Laplace operator:
$$\begin{align}
\Delta{}u&=0 \...

**4**

votes

**1**answer

166 views

### Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in
$$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$
...

**0**

votes

**0**answers

63 views

### existence of an initial-boundary value problem with nonhomogeneous boundary conditions

Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP:
$u_t=\Delta u$, for $x\in \Omega$, $t>0$;
$u(x, 0)=f(x), x\in\...

**1**

vote

**0**answers

42 views

### Dirichlet conditions - Proof of theorem $4$ [migrated]

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...

**7**

votes

**1**answer

272 views

### Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...

**0**

votes

**0**answers

64 views

### compact injection

Put:
$D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$
Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...

**0**

votes

**0**answers

43 views

### stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g.
$ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$
$ \nabla \cdot u = 0 \; \text{in} \; \Omega$
$ u = 0 \; \text{on} ...

**0**

votes

**0**answers

32 views

### Mixed PDE/finite difference equation

I have the following mixed pde/finite-difference equation for $f(t,x,y)$:
$a x^2 f_{xx} + bxf_x + f_t - bxy + c\sinh(d\delta) = 0$
subject to $f(T,x,y)=0$, $x>0,\ t\geq 0,\ y\in\mathbb Z$,
...

**3**

votes

**1**answer

185 views

### Critical case of Sobolev Embedding

I got stuck in the following lemma:
Lemma: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$.
As we know this is ...

**2**

votes

**1**answer

97 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...

**0**

votes

**0**answers

21 views

### how can we extend this result [duplicate]

Let $T_{a},a\in C$ be a closed operator defined on $D$ subspace of $L^{2}(R)$ onto $L^{2}(R)$ $(T_{a}: D\rightarrow L^{2}(R) )$ with
$D$ contains a Schawrz space $S$
$\Big<\psi,T_{a}\varphi\...

**1**

vote

**0**answers

83 views

### Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$.
The most ...

**3**

votes

**0**answers

62 views

### Strengthening of the local smoothing estimates for the free Laplacian

The classical local-smoothing estimates for the free Laplacian asserts that:
$$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$
where $B\subset\mathbb{...

**2**

votes

**2**answers

112 views

### Symplectic formulation of compressible Euler equation

It has been widely known that the compressible Euler equation can be cast into the Hamiltonian form. For example, in the book "Dubrovin B A, Fomenko A T, Novikov S P. Modern geometry—methods and ...

**0**

votes

**0**answers

60 views

### Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on $\...

**0**

votes

**0**answers

201 views

### Isothermal coordinates

Is there an application or interest in studying the isothermal surfaces where the metric is $ds^2=E∗(du^2+dv^2)$ and where $E>0$ is an harmonic function?
I know that this metric is a special kind ...

**4**

votes

**0**answers

148 views

### Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...

**3**

votes

**0**answers

59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$:
\...

**11**

votes

**2**answers

470 views

### Unexpected regularity of the distance from a $C^2$ submanifold

Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...

**0**

votes

**0**answers

50 views

### Is $\Delta u+f\in (H^1(\Omega))^*$ with $u\in H^1_0(\Omega)$ and $f\in L^2(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...

**3**

votes

**0**answers

75 views

### Donnelly-Fefferman growth of eigenfunctions

Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda ...

**9**

votes

**0**answers

239 views

### Are harmonic mappings of Riemannian manifolds always non-singular outside a set of measure zero?

Let $(M,g)$ be an $n$-dimensional, connected, compact, oriented, smooth Riemannian manifold with boundary. Assume we are given an immersion $f \colon M \to \mathbb{R}^n$ (note that $n=\dim M$).
Let $...

**3**

votes

**1**answer

134 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...

**1**

vote

**2**answers

66 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

**1**

vote

**0**answers

40 views

### Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...

**1**

vote

**0**answers

114 views

### The Yamabe problem and $\phi^4$ scalar field theory?

The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...

**1**

vote

**1**answer

76 views

### Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...

**2**

votes

**1**answer

241 views

### upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.
You can skip examples below and read from "General setting" at ...

**0**

votes

**0**answers

50 views

### High dimensional partial differential equation

I encountered the following partial equation. Let $f(z,x_1,\cdots,x_n)$ be a function with $n+1$ entries.Let $a_i,b,c$ be constants.
$$
\sum_{i=1}^n \frac{a_i}{(x_i-z)^2}+\frac{b}{z(z+1)}-\frac{\...