All Questions
16 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
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
296
views
Failure of Fredholm property of elliptic PDE systems
Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...
- 101
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 ...
3
votes
2
answers
2k
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 ...
- 455
3
votes
1
answer
344
views
elliptic regularity of Neumann problem on Square
I asked a similar question the other day, but I will be more precise now.
Consider $ \Omega:=(0,1 ) \times (0,1)$ and consider
$$ - u_{xx}(x,y) - u_{yy}(x,y) + a(x) u_x + b(y) u_y + u = f(x,y) \mbox{...
- 1,385
3
votes
0
answers
125
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
- 143
2
votes
2
answers
153
views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
- 101
2
votes
1
answer
408
views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
- 205
2
votes
0
answers
142
views
elliptic regularity for Neumann BVP on square
I am interested in the regularity of ellitpic equations like
$$ -\Delta u(x) +a(x) \cdot \nabla u(x) + C(x) u(x) =f(x) \quad \Omega$$ with $ \partial_\nu u =0$ on $ \partial \Omega$ where $ \Omega=(...
- 1,385
2
votes
0
answers
114
views
biharmonic equation with L^1 data and Navier Condition
I am reading an article that, a section of it is mentioned below . I have some question about this section. I will ask my question after the section below. I am thanksed if some one could help me , ...
- 371
1
vote
1
answer
149
views
Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
- 3,477
1
vote
0
answers
131
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
- 11
1
vote
0
answers
39
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
0
votes
1
answer
217
views
About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
- 111
0
votes
0
answers
76
views
Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
- 2,533