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
3
votes
0answers
29 views
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
1answer
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
\...
3
votes
0answers
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\...
0
votes
0answers
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 ...
0
votes
0answers
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 \...
4
votes
1answer
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 ...
2
votes
0answers
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 :
$$\begin{cases}
\textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
2
votes
1answer
128 views
do hyperfunction solutions always exist?
I have two questions---the second question (which is what I'm really interested in) is a generalization of the first, but I think the first may be more likely to get an answer. I'll be happy with an ...
3
votes
0answers
124 views
+50
Non-linear, hyperbolic, 2nd order system of PDEs
This is a cross-post.
In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system
\begin{...
1
vote
0answers
77 views
Results on the eigenspace of weighted elliptic eigenvalue problems
I am considering the following eigenvalue problem in $\Omega=\mathbb{R}^n_+$
$$-\operatorname{div}(a(x)\nabla \varphi) = \lambda a(x)w(x)\varphi$$
where the weights $a>0$ and $w\in L^{\infty}$ (and ...
0
votes
0answers
38 views
Lack of conformal invariance for fractional Laplace operators on a closed Riemannian surface
Let $(\Sigma,g)$ be a compact two-dimensional Riemannian manifold with no boundary. Let us denote by $\{(\lambda_k,\phi_k)\}_{k=0}^{\infty}$ the spectral data for the Laplace--Beltrami operator on $(\...
4
votes
0answers
66 views
Minimal regularity for domains in Green's formula
The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?
0
votes
0answers
47 views
Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform
Let us consider the wave equation
$$\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, &...
0
votes
1answer
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 ...
2
votes
2answers
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 ...
1
vote
1answer
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 $u$ ...
1
vote
0answers
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}...
2
votes
0answers
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
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
2
votes
0answers
95 views
Two identical objects circling the center of mass periodically in general relativity
In Newton's gravity we can have two identical objects circle the center of mass periodically (assuming the surroundings are vacuum).
Is something like this possible in general relativity? Is there an ...
2
votes
0answers
81 views
Is there a non-degenerate solution for this PDE on $\mathbb{R}^3$?
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\R}{\mathbb{R}}$
Does there exist a smooth map $f:\mathbb{R}^3 \to \mathbb{R}^3$, which satisfies
$$\tr \big( df \otimes \delta(df \wedge df) \big)=0,...
3
votes
0answers
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 \...
2
votes
1answer
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 ...
4
votes
2answers
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:...
0
votes
1answer
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} \...
3
votes
1answer
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 ...
4
votes
1answer
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>...
2
votes
0answers
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:
$$\lVert u \rVert_{k,\delta}^2 := \sum_{n=0}^k \int_M |D^nu \,r^{n-\delta}|^2 r^{-3} dV $$
...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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?
1
vote
1answer
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 ...
1
vote
1answer
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 ...
2
votes
1answer
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=...
0
votes
1answer
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(...
2
votes
1answer
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
vote
1answer
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 ...
0
votes
1answer
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\...
4
votes
1answer
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)$$
on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $...
1
vote
0answers
72 views
Is there a generalization of the Agmon-Douglis-Nirenberg regularity theorem for elliptic equations to domains with corners?
The Agmon-Douglis-Nirenberg theorem(s) state(s) that whenever $f\in W^{m,p}(\Omega)$ where $\Omega$ is a bounded open set of class $C^{m+2}$, then there is a unique solution $u\in W^{m+2,p}(\Omega)$ ...
1
vote
0answers
45 views
How does a computer program recognize shocks given data of a solution to a conservation law?
Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
3
votes
1answer
90 views
About the proof of higher regularity boundary Harnack inequality
I’m reading a note on higher regularity boundary Harnack inequality by D. DE SILVA AND O. SAVIN and I’m kind of confused of the case k=1.
In the paper they used the Hopf lemma to show that $u_\nu>c&...
0
votes
0answers
68 views
Green kernel vs fundamental solution
Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:
The fundamental solution $\Gamma(x)$ of $L$;...
2
votes
1answer
109 views
Existence of a global analytic solution to a linear first order PDE
Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let
$f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following
first order, linear PDE:
$$f_1\...
1
vote
1answer
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
1answer
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} + ...
4
votes
1answer
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 ...
1
vote
0answers
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 = ...
4
votes
0answers
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 $...
