# 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.

3,399
questions

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### Techniques for showing non-degeneracy results (PDE)

Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...

**3**

votes

**1**answer

61 views

### Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...

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49 views

### Decay of solutions to the wave equation $\ddot\phi(t, x)+\frac{n p}{t}\dot\phi(t,x)-t^{-2p}\Delta\phi(t,x)=0$

For the physical motivation of this question, see my question 669101 on physics StackExchange.
The question is this: Let $\hat M=\mathbb R^n$ or $\hat M =(\mathbb R/\mathbb Z)^n$ for some $n\in\...

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120 views

### Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...

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24 views

### Is the time of solution shorter as the initial data increases?

I'm reading the book superlinear parabolic problems and I came across the following situation twice: given two initial data $u_0$ and $\underline{u_0}$ with $u_0\geq \underline{u_0}$, $u_0\neq \...

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votes

**1**answer

56 views

### Elliptic equations in asymptotically hyperbolic manifolds

I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this ...

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78 views

### Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...

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votes

**1**answer

128 views

### do hyperfunction solutions always exist?

I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...

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124 views

+50

### Non-linear, hyperbolic, 2nd order system of PDEs

This is a cross-post.
In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system
\begin{...

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77 views

### Results on the eigenspace of weighted elliptic eigenvalue problems

I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...

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38 views

### Lack of conformal invariance for fractional Laplace operators on a closed Riemannian surface

Let $(\Sigma,g)$ be a compact two-dimensional Riemannian manifold with no boundary. Let us denote by $\{(\lambda_k,\phi_k)\}_{k=0}^{\infty}$ the spectral data for the Laplace--Beltrami operator on $(\...

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66 views

### Minimal regularity for domains in Green's formula

The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?

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47 views

### Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform

Let us consider the wave equation
$$\begin{cases}
w_{tt} -\Delta u = 0, & x \in \Omega, \ t >0, \\
w(0,x) = 0, & x \in \Omega,\\
w_t(0.x) = \phi(x), & x \in \Omega,\\
w(t,x) = 0, &...

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votes

**1**answer

89 views

### Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces

Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$
where $f'$ is the ...

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votes

**2**answers

95 views

### Density of traces of solutions to an elliptic equation

Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...

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**1**answer

102 views

### Proving an estimate for the Neumann problem on $\mathbb{R}^3 \setminus B_1$ in Weighted Sobolev spaces

Let $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
It is known that for every $g \in H^{\frac{1}{2}} (\partial M)$, and for an appropriate $\delta$, there exists a unique solution $u$ ...

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40 views

### How to deal with the boundary estimate for the Schauder estimates of laplacian equations?

Recently, I am learning Schauder estimates for elliptic equations and I come across a proposition as follows\
Let $ \alpha\in (0,1) $ and $ \Omega $ be a bounded $ C^2(\Omega) $ domain on $ \mathbb{R}...

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46 views

### Method of characteristics and explicit formula for an IBVP for the transport equation

Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...

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95 views

### Two identical objects circling the center of mass periodically in general relativity

In Newton's gravity we can have two identical objects circle the center of mass periodically (assuming the surroundings are vacuum).
Is something like this possible in general relativity? Is there an ...

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81 views

### Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?

$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...

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39 views

### How to find a particular solution of a non-homogeneous parabolic partial differential equation

Consider the following non-homogeneous parabolic partial differential equation (PDE)
\begin{align}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...

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**1**answer

77 views

### Fractional Laplacian on closed manifolds

Naturally given any $s\in (0,1)$, the fractional Laplacian, $(-\Delta_g)^s u$ on a closed Riemannian manifold can be defined via spectral decomposition of $-\Delta_g$. There is another formulation of ...

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**2**answers

179 views

### spaces of smooth functions for linear hyperbolic PDE

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...

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**1**answer

57 views

### Inequality involving the fractional Laplacian

Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...

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**1**answer

148 views

### Regularity of weak solutions to semi-linear elliptic PDEs

Suppose that $f:\Bbb R^2\to\Bbb R$ is a continuous non-linearity and consider the following semi-linear elliptic PDE given by:
$$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$
To ...

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votes

**1**answer

517 views

### Existence of a smooth compactly supported function

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that:
$$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$
for some $\epsilon>...

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votes

**0**answers

123 views

### Weighted Sobolev norm in terms of Spherical harmonics coefficients

Let $M = [1,\infty) \times S^2$.
Consider the weighted Sobolev space $H^k_{\delta}(M)$ with the Sobolev norm:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...

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103 views

### Is the single layer potential associated to the operator $-\mathrm{div}(A \nabla)$ a $H^{-1/2}$-elliptic operator?

Let $A$ be a $3\times 3$ symmetric definite positive real constant matrix, $\Omega\subset \mathbb{R}^3$ a bounded lipschitz domain, $\Gamma$ its boundary and $g$ the fundamental solution of the second ...

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66 views

### Is Poisson formula valid for the weak solution of Laplacian?

In the book "Regularity Theory for Elliptic PDE", here is a theorem as follows
Theorem(Harnack's inequality). Assume $ u\in H^1(B_1) $ is a non-negative, is the weak solution for the ...

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63 views

### How to prove the regular result of elliptic equation $ -\operatorname{div}(A(x)\triangledown u)=F $ in $ B(0,1) $ if $ A $ has the Holder regularity?

Consider the elliptic equation $ -\operatorname{div}(A(x)\triangledown u)=F $ in $ B(0,1) $ the ball with center $ 0 $ and radius $ 1 $ in $ \mathbb{R}^d $, where $ F\in L^p(B(0,1);\mathbb{R}) $ for ...

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116 views

### $L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...

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34 views

### Local well posedness for a stochastic wave equation

Suppose we have a stochastic wave equation, with Itô's derivative in the place of the usual Newtonian ones.
Does it make sense to talk about local well posedness?

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**1**answer

107 views

### Is the Poisson formula valid when the boundary condition is $ L^2 $?

Dirichlet problem for Laplace equation as follows
\begin{eqnarray}
\Delta{u}&=&0\text{ in }B_r(0)\\
u&=&g\text{ on }\partial B_{r}(0),
\end{eqnarray}
where $ g $ is continuous.
It is ...

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vote

**1**answer

79 views

### Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$

Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution ...

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votes

**1**answer

174 views

### A simple question on the Navier-Stokes system

The Navier-Stokes system for incompressible fluids in $\mathbb R^3$
reads as
\begin{align}
&\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=...

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**1**answer

85 views

### How to prove the reverse Hölder inequality for Laplace equations?

Let $ u\in H^1(2B) $ be a weak solution of $ \Delta u=0 $ in $ 2B $, where $ B=B(0,1) $ is a ball with center $ 0 $ and radius $ 1 $. Then there exists some $ p>2 $ such that
\begin{eqnarray}
\left(...

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votes

**1**answer

135 views

### Determine the sign (positive or negative) of an integral with the fractional Laplacian

Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R_+$ be smooth bounded functions. Assume also $\phi' \ge 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of ...

**1**

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**1**answer

86 views

### Averaging and fractional Laplacian

Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...

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**1**answer

120 views

### Harmonic functions in infinite domain in Euclidean space

EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\...

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votes

**1**answer

195 views

### Do Laplace-Beltrami eigenfunctions vary continuously with the metric?

I'm interested in Laplace Beltrami operators $$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $...

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72 views

### Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?

The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...

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45 views

### How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...

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votes

**1**answer

90 views

### About the proof of higher regularity boundary Harnack inequality

I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.
In the paper they used the Hopf lemma to show that $u_\nu>c&...

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68 views

### Green kernel vs fundamental solution

Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...

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votes

**1**answer

109 views

### Existence of a global analytic solution to a linear first order PDE

Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let
$f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following
first order, linear PDE:
$$f_1\...

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**1**answer

124 views

### Non-existence result for $p>\frac{N+2}{N-2}$

I encountered a sentence which says it is well known that problem
$$
\begin{cases}
-\Delta u =|u|^{p-1} u & in \,\, \Omega \\
u=0 & on \,\, \partial \Omega
\end{cases}
$$
have a solution for $...

**2**

votes

**1**answer

59 views

### Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE)
\begin{align}
\label{eq:42}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...

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votes

**1**answer

148 views

### Intuition for almost periodic solution and Poincaré recurrence theorem

I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...

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79 views

### Semilinear PDE - BSDE presentation via Feynman Kac formula

For a semilinear PDE, we usually have this FBSDE representation:
$\mathcal{X}_t=\mathcal{X}_0+\int^t_0 \mu (s,\mathcal{X}_s)\, ds\, +\int_0^t \sigma (s,\mathcal{X}_s)dW_s,\quad 0\leq t\leq T, \\
Y_t = ...

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93 views

### $L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...