Questions tagged [regularity]
regularity of solutions of PDEs.
17
questions
6
votes
1
answer
633
views
Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$
The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to
$$
-\Delta u=f\hspace{3cm}(1)?
$$
I'm of ...
- 5,163
21
votes
1
answer
1k
views
A differentiable isometry is smooth?
I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...
- 6,621
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
14
votes
1
answer
818
views
Existence and uniqueness of geodesics in low regularity
Consider a Riemannian manifold $(M,g)$.
How much regularity is required of $g$ so that for any $x\in M$ and $v\in T_xM$ with $|v|=1$ there exists a unique geodesic $\gamma\colon(-\epsilon,\epsilon)\to ...
- 8,046
13
votes
1
answer
710
views
Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\...
- 6,621
12
votes
2
answers
839
views
Unexpected regularity of the distance from a $C^2$ submanifold
Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...
- 3,163
10
votes
0
answers
395
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 \...
- 851
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 ...
- 1,033
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 ...
- 6,621
5
votes
3
answers
1k
views
Elliptic theory on compact manifolds
Maybe this is silly.
On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation
$$ \Delta u=f \quad\text{ in $\Omega$}$$
$$ u=0\quad\text{ on $\partial\Omega$}.$$
One has the following ...
- 327
3
votes
1
answer
351
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
2
answers
752
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 ...
- 255
3
votes
1
answer
772
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
1
answer
326
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.
- 193
2
votes
1
answer
553
views
Regularity of solution to a hyperbolic pde
I have a question concerning 2nd order evolution equation of the form $u''(t)+A(t)u(t) = f(t)$ in $L^2(0,T;V^*)$, where $f\in\ L^2(0,T;H)$ holds. Under what assumptions is it possible, to guarantee a ...
- 65
2
votes
0
answers
62
views
Capacity approximations by sets with regular boundary
Suppose I have a continuous, compactly supported function $f : \mathbb{R}^2 \to \mathbb{R}_{+}$ and I define the set $S := f^{-1}([a,\infty)) \subset \mathbb{R}^2$ for some $a > 0$. It is a ...
- 321
0
votes
1
answer
68
views
What is the critical exponent for irregular function in the Sobolev scale?
When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...
- 9,150