Questions tagged [regularity]
regularity of solutions of PDEs.
159
questions
1
vote
0answers
57 views
While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?
For example, consider the third boundary value problem:
\begin{align}
&\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n ...
3
votes
1answer
154 views
“Reversed” Bernstein Inequality
I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below:
...
2
votes
1answer
57 views
How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?
I have encountered the following problem.
Let $\chi:=\chi_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$.
$\nabla\cdot ((1+\chi_{B(0,1/2)})\nabla u )=0 $ in $...
1
vote
1answer
73 views
Poisson equation in a periodic strip
Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem
$$-\Delta u=f$$
with boundary conditions
$$...
1
vote
0answers
27 views
Condition on Functions on Overlapping patches of a Domain
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $\{p_i\}_{i=1}^m$ for some finite $m\in\mathbb{N}$ be the overlapping open sets of $\Omega$ such that $\Omega =\bigcup_{i=1}^mp_i$.
Now, ...
0
votes
1answer
88 views
Existence of solutions of a system of first order PDEs
Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset.
Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions.
That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
2
votes
0answers
40 views
Hypoellipticity or parabolic regularity for vector bundles
Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients ...
3
votes
2answers
198 views
Gradient estimates for a boundary value problem
$\newcommand{\avint}{⨍}$
Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE:
$$
\begin{cases}
-\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\
w=0 \...
1
vote
0answers
39 views
Regularity and existence linear parabolic fractional equation
\begin{equation}
\begin{cases}
a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x), & \text{in } \mathbb{R}^n
\end{cases}
\end{...
3
votes
1answer
164 views
Alternative proof of Liouville theorem for harmonic functions
From Prove Liouville theorem without using mean value property the following question arises:
To prove the Liouville theorem
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
3
votes
1answer
118 views
Duality argument for elliptic regularity
M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
2
votes
0answers
46 views
Infimum of the Dirichlet energy and deformation of metrics
Consider the Dirichlet energy $E(\phi; g,h)=\frac{1}{2}\int_M|d\phi|^2$ of a map $\phi:(M,g)\to(N,h)$ between Riemannian manifolds, where we assume $M$ and $N$ are compact. Given a path $g(t)$ of ...
2
votes
0answers
35 views
A question for regularity of solutions to wave equation
let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation
\begin{equation}\label{pf0}
\begin{aligned}
\...
0
votes
0answers
13 views
Compatibility condition for the global well-posedness of coupled Stokes equations
Consider the following coupled Stokes-Darcy systems in a simplified domain $\Omega=[-1,1]\times[-1,1]$, WLOG, we may assume the horizontal variable to be periodic. The Stokes flow occuplied the upper ...
0
votes
1answer
90 views
Regularity in Navier Stokes from $L^2$ bound on vorticity
How would one show that if $\omega$ is the vorticity associated to $\partial_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and
$$\omega\in L^\infty([0,...
4
votes
0answers
48 views
Interior regularity for parabolic systems in divergence form
Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth, bounded domain. Suppose $N \in \mathbb N$, $D \subset \mathbb R^N$ and that $a_{ij} : D \to \mathbb R$ are smooth for $i, j \in \{1, \...
2
votes
0answers
156 views
regularity; elliptic pde
Consider $N \ge 3$ a sequence of solutions $u^m$ to
$$-\Delta u^m(x) - \frac{\alpha u^m_{x_N}(x)}{x_N+\epsilon_m} = f^m(x) \quad \mbox{ in } B_2^+$$ with $ u^m=0$ on $ \partial B_2^+$ (here $B_R^+:=\{...
3
votes
0answers
35 views
Looking for a proof of a geometric regularity criteria: generalization of the exterior cone condition / Zaremba's criterion
The topic is Perron's method for the Dirichlet problem.
I am looking for a proof of the following statement:
Let $\Omega$ be an open bounded set in $\mathbb{R}^n$ with $n \geq 3$ and $0 \in \partial \...
3
votes
0answers
62 views
Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
2
votes
0answers
224 views
Singularity of L^1-solutions to elliptic PDEs on the puntured ball
Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^n$. Then it is true that if $u\in L^1(\mathbb{B})$ such that $\Delta(u)=0$ on $\mathbb{B}\backslash\{0\}$, then $\Delta(u)$, as a distribution on $\...
5
votes
1answer
138 views
Different ways to prove $L^p$-estimates for the heat equation
Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
\partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-...
3
votes
1answer
302 views
Reference to a Classical Regularity Theorem
(Edited)
I need a reference to the following result:
If $u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$ satisfies
\begin{cases}
{\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ ...
4
votes
2answers
149 views
Regularity of a conformal map
Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (...
2
votes
0answers
81 views
Reference request : Global boundedness of weak solution for Neumann problem
I have some question on global boundedness of weak solution to Neumann problems.
Let $u\in W^{1,2}(\Omega)$ is a weak solution for Neumannn problem
$$ \mathrm{div} (A \nabla u )= \mathrm{div}\, F\quad ...
2
votes
0answers
50 views
$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation
Consider a Fokker-Planck equation:
$$
\partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0,
$$
with initial condition ...
1
vote
0answers
73 views
Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
2
votes
0answers
51 views
wave equation with non-smooth coefficients
Let us consider the equation
$$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$
subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
2
votes
1answer
231 views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
2
votes
1answer
104 views
Regularity and normal trace of “Hdiv” measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
3
votes
1answer
223 views
Inequality for initial data
I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
2
votes
0answers
95 views
$W^{2,p}$-estimates for Neumann boundary condition to Poisson equation
Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$:
$-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$.
Here $F\in L^p(\Omega)$.
...
1
vote
0answers
44 views
Axis regularity in cylindrical coordinates: conditions on the non linear terms?
I've been working on this for months and can't find a good answer.
I'm looking at the incompressible Euler equations in cylindrical coordinates ($r$, $\theta$, $z$), and I am looking at the non ...
2
votes
1answer
103 views
regularity of p-harmonic functions
We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.
Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
1
vote
0answers
27 views
free boundary of a p-harmonic function
let $u$ be a p-harmonic function in $\Omega \subset \mathbb R^N$.
We already know that the set $\{u=0\}$ is locally a $C^{1,\alpha}$ hypersurface at the points where $\nabla u\neq 0$.
What can be ...
2
votes
0answers
302 views
Regularity of locally finite Borel measure
Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...
0
votes
1answer
149 views
Estimate for Laplace equation with Neumann boundary on manifold with corner
Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.
...
1
vote
0answers
41 views
Integrability condition on function determining PDE domain
I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf
On page 2, there appears a PDE of the form
$\partial_t u +...
2
votes
0answers
39 views
Time derivative in parabolic Hölder spaces
Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$.
Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
2
votes
1answer
263 views
Embedding of weighted sobolev space with exponential weights
In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. ...
2
votes
0answers
80 views
A question about how to use the convexity condition?
At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say that since $p$ is convex, we can deduce that
$$ \...
7
votes
0answers
288 views
Regularity result for the boundary value problem for the heat equation
Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...
2
votes
1answer
165 views
Reference request: Schauder estimates for parabolic equations
Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)?
Any reference would be highly appreciated.
2
votes
0answers
74 views
Wave equation regularity
I have an equation of the type
$$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$
for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...
4
votes
1answer
192 views
Oscillation and Holder continuity
Where can I find a proof of the follwing fact?
If
$$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$
for some function $u(x)$ satisfies
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \...
1
vote
0answers
79 views
Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?
Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
5
votes
1answer
307 views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...
0
votes
1answer
84 views
$2n$-regular graphs with maximal chromatic number
Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ ...
0
votes
0answers
97 views
Biharmonic equation
Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation
\begin{equation}\label{\star}
\Delta^2u+V(x)u=g(x, u)+K(x)u,
\end{equation}
where $|g(x,s)|\leq \varepsilon|...
4
votes
1answer
208 views
Regularity of harmonic forms on manifolds-with-boundaries
Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
2
votes
1answer
126 views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)...
