# 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.

2,857 questions
Filter by
Sorted by
Tagged with
55 views

### 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 ...
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 $|x|^p$?
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: ...
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 ...
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 ...
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 $\mathfrak{D}$...
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 ...
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 $\partial B_1$...
62 views

64 views

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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
105 views

83 views

60 views

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{...
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 ...
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 ...
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} = \...
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{...
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 ...
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 "$p$" ...