# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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questions

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### Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator

I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why ...

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**1**answer

37 views

### Computing the fractional Laplacian of power function

Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?

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**1**answer

122 views

### Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753.
Consider the following PDE: ...

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**1**answer

80 views

### Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...

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184 views

### Uniqueness of solution of the wave equation

Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$
with initial conditions
$$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$
Does ...

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**1**answer

129 views

### Is there any nontrivial characterization of weakly differentiable functions?

When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...

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37 views

### A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...

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69 views

### fractional Laplacian estimates

Suppose $(-\Delta)^s u=f \geq 0$ in a ball $B_2$ and $u=0$ in $ R^N \setminus B_2.$ Also suppose $u$ is $C^{s}$ non-negative and $(-\Delta)^s u=0$ in $B_2 \setminus B_1$ and $u\leq a$ on $\partial B_1$...

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62 views

### Inhomogeneous wave equation - a reference

Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...

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117 views

### Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...

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64 views

### Poincaré-type Inequality

In Lieb's paper "On the lowest eigenvalue of the Laplacian
for the intersection of two domains" one finds the following remark:
Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...

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126 views

### Decay of Fourier coefficients of real analytic functions

I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, ...

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89 views

### What is the example of non-regular boundary point?

I am studying in PDE and I have next definition :
Definition. Let $\Omega\subset\mathbb{R}^n$ open, connected. Then $\xi\in\partial\Omega$ is regular if there exists a superharmonic function $p$ in ...

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77 views

### A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...

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121 views

### Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?

Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...

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61 views

### Initial value problems on manifolds around submanifolds (reference)

I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...

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388 views

### Solutions of PDE under changing topology

Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we ...

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**1**answer

67 views

### On $s$-harmonic functions

Is this statement true?
A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...

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97 views

### Existence theory with an integral equation

I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system ...

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272 views

### A question in elementary differential geometry

Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation
$$ \partial_t u + |\nabla u|^2 = 0. $$
Can we give a precise estimate of a ...

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105 views

### inequality involving the fractional Sobolev space

Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality
$$|u(x)...

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143 views

### How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...

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160 views

### Uniqueness condition for Hamilton-Jacobi equation?

Let $f= f(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function such that
$$
\partial_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R}...

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83 views

### Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...

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46 views

### Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...

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39 views

### A nonlinear PDE on a matrix domain involving a maximum eigenvalue operator

In a research problem in optimal control
$^{\ast}$ I came across the following nonlinear first-order PDE:
$$\frac{\partial V}{\partial t}=\max\text{eig}\left[\sum_{i=1}^m P_i\frac{\partial V}{\partial ...

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1k views

### Can learning Riemann surfaces be more beneficial than numerical analysis for an analyst?

I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, ...

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96 views

### Spectrum of Laplacian-like operator

Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...

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77 views

### Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...

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297 views

### Is there a diffeomorphism of the disk with constant sum of singular values?

This question is a relaxed version of this question.
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$.
Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...

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136 views

### Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity

In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...

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45 views

### improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...

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36 views

### When does the solution to the Fokker-Planck equation admit a density wrt Lebesgue measure?

Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation
$$\frac{\partial u(t,x)}{\partial t}= \...

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69 views

### Strichartz estimate for the Schrödinger equation

Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t ...

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77 views

### Wave equation for smooth Schwartz kernels

Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...

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69 views

### Approximating functions in $H^1_0(U) \cap H^2(U)$ via $H^1$ norm and $L^2$ projection

Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \...

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60 views

### Bound on number of linearly independent eigenvectors of adjoint of composition operator

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via
$$
\begin{aligned}
C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...

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156 views

### Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...

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61 views

### Solution existence for two-dimensional parabolic PDEs

I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...

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33 views

### Are there maximum principles related to the third boundary condition?

I am working on this problem
\begin{equation}
\begin{cases}
u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\
u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0,
\end{cases}
\end{...

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177 views

### Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...

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**1**answer

81 views

### Weyl symbol of product

Are there explicit formulas for the Weyl symbol of $-f(x)D_x^2 $ where $D_x:=-i\partial_x $ and $\partial_x$ is the derivative and $f$ some sufficiently smooth function?
In the standard quantization ...

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139 views

### Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:
\begin{equation}
u_{t} = \...

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35 views

### First eigenfunction of the p-Laplacian in an interval $(a,b) \subset \mathbb R$

What is the explicit expression of the first eigenfunction $u$ of the $p$-Laplacian ($p>1$) in a bounded interval $(a,b) \subset \mathbb R$ (up to multiplicative constant)?
\begin{equation}
\begin{...

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100 views

### Stability and symmetries

In the theory of stability of solitary waves I have seen many times that people mention some of the symmetries of the equation in order to introduce the "right" notion of stability. For instance, if ...

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43 views

### Invertibility of the characteristic flow in Hamilton-Jacobi equations

We are in the context of Hamilton Jacobi equations, in particular I was reading the characteristic method. We want to solve the problem of the special form (Hamiltonian only depending on the "$p$" ...

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78 views

### Evolution equation generated by Fourier multiplier

I am on the hunt for techniques regarding a field which I am not familiar with.
More precisely, I am considering equation of the form
$$ i \partial_{t} u(t,x) + p(D)u(t,x) = 0, \ \ u_{|_{t=0}}=u_0(x)...

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215 views

### A boundary Schauder estimate

According to Theorem 1.1' in this paper we have the following estimate on classical solutions $u \in C^2(\overline{B_1^+})$ of $-\Delta u = f \text{ in } B_1^+ = B_1 \cap \{x _n \ge 0 \}$ and $u = 0 \...

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51 views

### Elliptic foliations of the plane

A $1$ dimensional foliation of the plane $\mathbb{R}^2$is called elliptic if it admits a non vanishing smooth tangent vector field $X$ with the following properties:
The differential operator ...

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44 views

### Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

Suppose I have an equation in some reflexive separable Banach space $X$:
$$Au=f$$
for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...