Questions tagged [regularity]
regularity of solutions of PDEs.
238
questions
6
votes
2
answers
775
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-...
4
votes
1
answer
332
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^+ ...
3
votes
2
answers
252
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
0
answers
104
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
$$ \mathrm{div} (A \nabla u )= \mathrm{div}\, F\quad ...
3
votes
0
answers
78
views
$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation
Consider a Fokker-Planck equation:
$$
\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,
$$
with initial condition ...
1
vote
0
answers
135
views
Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
2
votes
0
answers
72
views
wave equation with non-smooth coefficients
Let us consider the equation
$$ \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
1
answer
357
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}...
3
votes
1
answer
232
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
1
answer
228
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
0
answers
194
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
0
answers
63
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 ...
3
votes
1
answer
408
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
0
answers
33
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
0
answers
685
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
1
answer
261
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
0
answers
47
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
0
answers
58
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}$-...
5
votes
1
answer
662
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
0
answers
89
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
$$ \...
8
votes
0
answers
439
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:
...
3
votes
1
answer
322
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
0
answers
78
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(...
5
votes
1
answer
377
views
Oscillation and Hölder continuity
Where can I find a proof of the following 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
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
1
vote
0
answers
110
views
Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?
Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
6
votes
1
answer
377
views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...
0
votes
1
answer
91
views
$2n$-regular graphs with maximal chromatic number
Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ ...
0
votes
0
answers
118
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|...
5
votes
1
answer
318
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
1
answer
173
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)...
2
votes
1
answer
455
views
Bounded solution for parabolic equation
Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
10
votes
4
answers
2k
views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
9
votes
1
answer
1k
views
Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
4
votes
1
answer
838
views
Evans-Krylov theorem
Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
1
vote
1
answer
271
views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...
3
votes
2
answers
229
views
Density on Hölder spaces whose elements vanish on the boundary
I would like to ask the following problem.
Let $\Omega$ be a $C^{r+1,\alpha}$ domain, $r\in \mathbb{N}, 0<\alpha<1.$ We denote $$C^{r,\alpha}_{0}(\overline{\Omega})=\{f\in C^{r,\alpha}(\...
1
vote
0
answers
95
views
2nd oder evolution equations and regularity results of their solution
I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...
7
votes
1
answer
206
views
Is a Sobolev map with smooth minors smooth on the whole domain?
Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
...
0
votes
1
answer
389
views
Regularity of Laplace equation with Dirichlet data on a part of the boundary
From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system
\begin{align}
-\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\
u &= g &\text{on}\ \...
0
votes
1
answer
489
views
$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...
3
votes
1
answer
760
views
Higher regularity of solutions for Laplace equation with mixed boundary condition
Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\...
6
votes
0
answers
254
views
Is a Sobolev map with invertible smooth minors smooth?
$\newcommand{\Cof}{\text{cof}}$
Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
3
votes
2
answers
1k
views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
2
votes
1
answer
276
views
Is this approach for establishing regularity of harmonic maps between manifolds valid?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
1
vote
0
answers
43
views
Hidden regularity for the coupled wave equation with dynamaic boundary condition
We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\...
0
votes
0
answers
74
views
Looser condition for regularity for Neumann problems
If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then
$$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$
Let's say we are studying rotationally symmetric solutions to ...
5
votes
1
answer
427
views
Dependence of the Hölder exponent in De Giorgi-Nash-Moser
I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity.
More precisely: suppose $u$ satisfies weakly
$$
D_i(a^{ij}D_ju)=f
$$
on the $d$-...
1
vote
0
answers
46
views
Harnack type Estimates for a p-Poisson equation with constant source term
Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\Delta_p u=1\,\,\mbox{in $B$}
$$
Let another function $f$ be such that
$$
\begin{cases}
-\Delta_p f =1 \;\;\mbox{in $B$}\\
f=0 \...
3
votes
0
answers
99
views
Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?
Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$.
Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).
Suppose we are given a $W^{2,2}$ isometric immersion $...
1
vote
1
answer
185
views
Is there any "extra regularity" to the solution to Poisson's equation posed on a 3-dimensional polyhedron?
I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form
$$
\|u\|_{H^{1+\epsilon}(\Omega)} \le C \|f\|_{L^2(\Omega)}
$$
for some $\epsilon >0 $ ...