# Tagged Questions

**1**

vote

**0**answers

40 views

### Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by:
$$
...

**0**

votes

**0**answers

43 views

### solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...

**4**

votes

**2**answers

148 views

### Equivalent Norms on Sobolev Spaces

When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right) :$
$$
\left\Vert \left( -I+\Delta\right) ...

**-1**

votes

**0**answers

63 views

### Clear estimate is not so clear [closed]

In a paper I found the estimate (there it is said that the estimate is clear) for $U \subset \mathbb{R}^n$ and $u \in W_0^{2,2}(U)$ saying that for all $\varepsilon >0 $ we have
$$\int_{U} ...

**0**

votes

**0**answers

65 views

### Differentiating and integrating an infinite series arising from a PDE

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given $u \in H^{\frac ...

**2**

votes

**2**answers

140 views

### Extremal functions for Gagliardo-Nirenberg inequality

Recently I read about the Gagliardo-Nirenberg inequality. And I would like to ask about the attainability and the maximizers of the GN inequality:
$(∫|u|^{r}dx)^{\frac{1}{r}} \leq ...

**1**

vote

**1**answer

87 views

### Elliptic pde with bilaplacian; boundary conditions.

I am interested in the solvability of
$$ \Delta^2 u + u = f(x) \mbox{ in } \Omega $$ with $ \partial_\nu u = \Delta u=0$ on $ \partial \Omega$ where $ f(x)$ is some smooth bounded function on $ ...

**1**

vote

**0**answers

29 views

### Boundary conditions of PDE from SV model with stochastic interest rate

The PDE for the American put option price $P(S,\sigma ,r,t)$ is
\begin{align*}
0 =& P_t+P_SS(r-\delta)+P_\sigma a(\sigma)+P_r\alpha (r,t) \\
+& \frac{1}{2}P_{SS}S^2\sigma ^2 + ...

**0**

votes

**1**answer

81 views

### How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
...

**3**

votes

**1**answer

314 views

### Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.
$$
My question is to ...

**1**

vote

**0**answers

101 views

### Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]

Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com,
...

**2**

votes

**1**answer

58 views

### solvability of linear elliptic pde on a torus

Consider a linear non-divergent form elliptic PDE on a flat torus $\mathbf{T}^n$, $$a_{ij}\partial_{ij}u+b_i\partial_iu=f$$
where all the coefficients and $f$ are smooth. What is the condition that ...

**3**

votes

**1**answer

84 views

### Generalization of maximum principle to other norms

Consider the Laplace equation $\Delta u=0$ in $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions, i.e. $u=g$ on $\delta \Omega$. By the maximum principle we know that the solution $u$ ...

**2**

votes

**0**answers

111 views

### Complex sum of squares of vector fields (hypoelliptic operators)

Consider a compact $M$ of dimension $n$. Consider real smooth vector fields $X_0, X_1, X_2,..., X_n$ on $M$ and consider the differential operator $\mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0.$
Now, by ...

**0**

votes

**0**answers

55 views

### Regularity result for the elliptic equation with Neumann conditions

I am having troubles to justify well some inequalities related with the classical
theory of elliptic equations. Let us consider the problem
\begin{align*}
-\Delta c & =f,\text{ in }\Omega\\
...

**0**

votes

**0**answers

10 views

### Optimal relaxation parameter for the SOR method when solving Poisson equation for non-square problems

Yang and Gobbert in paper "The Optimal Relaxation Parameter for the SOR Method Applied to a Classical Model Problem" ( http://userpages.umbc.edu/~gobbert/papers/YangGobbert2007SOR.pdf ) gave proof ...

**5**

votes

**2**answers

134 views

### Separable coordinate systems for the Laplace and Helmholtz equations?

According to Mathworld, in three dimensions there are 13 coordinate systems in which Laplace's equation is separable, and 11 for the Helmholtz equation. I've read the relevant chapters of the book by ...

**2**

votes

**2**answers

417 views

### Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...

**2**

votes

**1**answer

174 views

### $BMO$-property via a John-Nirenberg type estimate?

Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also
$$
f_B:= \frac1{|B|}\int_B f \, dx.
$$
Suppose $f \in L_{\rm loc}^p(\Omega)$ for all ...

**0**

votes

**2**answers

92 views

### Separation of variables for a particular PDE

Given the partial differential equation
\begin{equation}
(1-x)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial x} \right] + (1-y)\left[- f(x,y) + \frac{\partial f(x,y)}{\partial y} \right] = 0
...

**10**

votes

**2**answers

198 views

### First eigenvalue of the Laplacian on a regular polygon

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
...

**1**

vote

**1**answer

68 views

### On the domain of functionals in measure with singular kernels

this post is concerned with functionals defined in measures. Consider the following functional
$$\mathcal{W}[\mu]=-\int_{\mathbb{R}^2}{\log\vert x-y\vert\ d\mu(x)d\mu(y)},$$
were we define ...

**1**

vote

**0**answers

52 views

### Heat equation inequality

There is an inequality that tells us that for some sufficiently smooth $f$ satisfying $(\partial_t - \Delta )f \le - \delta f^2 +K$ for $\delta,K >0$ that $f$ is bounded by some constant. ...

**5**

votes

**1**answer

92 views

### Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator
$$D:= \frac{d}{dt}+A(t)$$
and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...

**7**

votes

**1**answer

175 views

### Poincaré inequality for curl-integrable functions

Let $B=B(r)$ denote a ball of radius $r$ in $\Omega \subset \mathbb R^d$ and
$$
u_B := \frac1{|B|}\int_B u \, dx.
$$
The standard Sobolev-Poincaré inequality states that if $u \in W^{1,p}(\Omega)$, ...

**3**

votes

**0**answers

156 views

### How can I calculate the adjoint of the wave operator $\square_{g}$ in $H^{k}$?

I have a three part question, which I could only received an answer for the first part here.
The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in ...

**4**

votes

**0**answers

104 views

### $H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao.
They study the ...

**4**

votes

**1**answer

126 views

### Loss of derivative of subelliptic operator

Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. ...

**4**

votes

**1**answer

128 views

### application of factorization theorem

Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...

**6**

votes

**2**answers

162 views

### Pseudodifferential operators on spaces with boundary

Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann ...

**0**

votes

**0**answers

36 views

### Painleve test of a new PDE hierarchies

This PDE hierarchies is :
$$u_t=\sum_{i=0}^{N}c_iu^iu_x-\frac{1}{2}\sum_{i=0}^N(c_iu^i)_{xxx}$$
so far, I have proved that this equation hierarchies has Resonaces at:$$j=2N+2,4N+2$$,according to ...

**3**

votes

**1**answer

155 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**5**

votes

**1**answer

98 views

### Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...

**6**

votes

**1**answer

248 views

### Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde:
...

**4**

votes

**2**answers

126 views

### How to find an ODE with prescribed terminal values?

Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...

**2**

votes

**0**answers

68 views

### Resolvent estimate of hyperbolic Laplacian [closed]

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form
$$\Vert (-\Delta - \lambda ...

**6**

votes

**3**answers

350 views

### Derivatives of radial functions can be bounded by derivatives in terms of radial distance?

Suppose $f$ is a radial function, i.e., $f(x)=f(|x|)$,
and $f \in C^\infty(\bar{B})$, where $\bar{B}$ is the closure of the unit ball in $\mathbb{R}^n$.
Prove or disprove the following.
Given any ...

**0**

votes

**0**answers

55 views

### Global existence solutions NLS

Let's consider the following NLS in $\mathbb{R}^3$
$$i\partial_t\psi=-\Delta\psi+\vert\psi\vert^2\psi$$
How to prove that $H^2$-solutions are globally in time? Can someone suggest references to me?

**3**

votes

**0**answers

67 views

### Integrability of Continuous Tangent Subbundles

Are there any field of mathematics, except dynamical systems, where one needs to integrate continuous sub-bundles of the tangent space?
More specifically given a smooth manifold of $M$ and a ...

**5**

votes

**2**answers

150 views

### Schwartz space of functions with values in a Frechet space

While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in ...

**4**

votes

**1**answer

109 views

### Gradient estimate for elliptic equation

Given:
1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$
2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$
3)$g=(g_i,\ldots,g_n)\in ...

**0**

votes

**0**answers

37 views

### comparison principle for viscosity solution to linear nonlocal equation with drift

I met a problem about comparison principle for nonlocal equation when I study SDEs driven by Levy noise. Since, I have no background about PDEs, it may be a stupid question:
$$Iu=\lambda ...

**1**

vote

**0**answers

70 views

### Gilbarg-Trudinger's book Theorem 4.13

I am reading Gilbarg-Trudinger's book "Elliptic Partial Differential Equations of Second Order". I do not understand the proof of Theorem 4.13.
Theorem 4.13 is a special case of Kellogg's theorem in ...

**5**

votes

**2**answers

262 views

### Weak solutions for a PDE of fourth order

I deal with two-dimensional Kirchhoff equation with $L^\infty$ coefficient and distributional right hand side:
$$
\Delta\Delta w+u(x,y)\left(\alpha^2\frac{\partial w}{\partial ...

**2**

votes

**1**answer

117 views

### elliptic boundary regularity, tangential regularity

A have a question related to the boundary regularity of a solution of a Poisson equation on a bounded domain. But to make the question easier to pose I will state it on $ R_+^2:=\{ x \in ...

**3**

votes

**0**answers

86 views

### Regularity in PDE theory

I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...

**3**

votes

**1**answer

144 views

### Mixed (anisotropic) Sobolev spaces

Consider real variables $x, y$ and a function $f(x, y) \in H^s(\mathbb{R}^2)$, say for some $s \in (0, 1)$. I am trying to get an understanding of mixed Sobolev spaces of the form $H^s_x(H^s_y)$, ...

**1**

vote

**0**answers

90 views

### Asymptotics of “heat” semigroup

Consider a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary. Consider a second order elliptic operator $L$ on $L^2(\Omega)$, defined by either the Dirichlet or Neumann boundary ...

**5**

votes

**0**answers

115 views

### Local version of the Hardy-Littlewood-Sobolev theorem for Riesz potentials: $\|I_\alpha(f)\|_{L^q} \le C \|f\|_{L^p}$?

Recently, I have been studying the properties of the Riesz potential
$$
I_\alpha(f)(x) = c_{d,\alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}} \, dy.
$$
The classical Hardy-Littlewood-Sobolev ...

**2**

votes

**0**answers

36 views

### Appropriate BCs of First Order Hyperbolic Semi-Linear Equation [closed]

The following pde is (approximately) the leading order homogenized form of the local mass transport equation with a non-linear metabolism (in symmetrical spherical co-ordinates):
\begin{equation*}
...