All Questions
35 questions
14
votes
1
answer
2k
views
Regularity of the Maxwell equations
As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...
- 455
10
votes
1
answer
1k
views
What would the best treatment of Gehring's lemma look like?
In a course about elliptic regularity probably one sooner or later stubles into the reverse Holder inequalities, and has to introduce the Gehring lemma, which in one of its many versions improves a ...
- 2,041
10
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 ...
- 1,101
10
votes
0
answers
422
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 835
9
votes
2
answers
493
views
Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
- 133
8
votes
1
answer
1k
views
Asking for Advices for Choosing a Ph.D thesis problem (in PDE area)
I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn ...
- 223
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
8
votes
0
answers
462
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:
...
user139845
7
votes
1
answer
847
views
Caccioppoli-Leray Inequality for De Giorgi's theorem proof
I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients.
This is the translation of the original paper
De Giorgi paper
At page ...
- 265
5
votes
1
answer
363
views
Regularity up to the boundary for the Poisson problem
It seems that the following assertion is widely accepted:
For $k\in\mathbb N$, $p\geq 2$, $\Omega \subset \mathbb R^n$ bounded with $\partial\Omega\in C^{k+2}$ and $f\in W^{k,p}(\Omega)$, the weak ...
4
votes
1
answer
487
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...
- 5,380
4
votes
1
answer
338
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^+ ...
- 175
4
votes
0
answers
97
views
Techniques to estimate PDE which are elliptic in some directions and degenerate in others
I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
- 708
3
votes
1
answer
2k
views
Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions
Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
user70735
3
votes
2
answers
803
views
Non symmetric coefficient matrix for elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form
$$ D_i(a_{i,j}D_ju)=0 \tag{1}$$
where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
- 265
3
votes
1
answer
376
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 ...
- 217
3
votes
1
answer
299
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 ...
- 5,380
3
votes
1
answer
845
views
Moser estimates?
Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
- 31
3
votes
0
answers
81
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 ...
- 31
2
votes
1
answer
337
views
Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property
Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...
- 51
2
votes
0
answers
111
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
- 835
2
votes
0
answers
176
views
Visualization of an oscillation lemma
How can one visualize Theorem 4.2 on page 31 of this paper by Seregin, Silvestre, Šverák and Zlatoš?
On the other hand, I have a clear visualization of a related result about how oscillation decay ...
- 41
1
vote
1
answer
85
views
$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains
Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form
$$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$
are there any elliptic ...
- 93
1
vote
1
answer
293
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.
...
- 6,741
1
vote
1
answer
250
views
Moser/Schauder estimates for coercive boundary conditions
Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a ...
- 11
1
vote
1
answer
441
views
Elliptic regularity and inhomogeneous Neumann boundary condition
Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...
- 11
1
vote
0
answers
293
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
- 61
1
vote
0
answers
39
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
1
vote
0
answers
138
views
$L_p$ estimate in mixed boundary problem for elliptic equation
Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...
- 46
1
vote
0
answers
76
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 ...
- 51
1
vote
0
answers
158
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 ...
- 435
1
vote
0
answers
115
views
$L^\infty(\Omega)$-regularity for strongly damped wave equation
I am interested in the following IBVP for the strongly damped wave equation:
\begin{equation}
u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\
u=0 \quad \text{on} \ ...
- 11
0
votes
1
answer
289
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 $n=\dim(M)$.
...
- 1,915
0
votes
0
answers
52
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
- 697
0
votes
0
answers
376
views
Regularity for a div-curl system
Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
- 161