13
votes
Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?
It is a result of Pakzad's that $W^{2,2}$ isometric immersions $f$ of bounded regular convex domains with Lipschitz boundary $\Omega\subset\mathbb{R}^2$ must be developable; more precisely, for every ...
- 13.6k
11
votes
Accepted
Elliptic regularity on compact manifold without boundary
This follows from the following regularity estimate for the flat Laplacian case (which is, I believe, proved in Warner's book using Fourier series on a torus but also in most standard texts on ...
- 27.5k
9
votes
Accepted
Dependence of the Hölder exponent in De Giorgi-Nash-Moser
(By scaling we can take $\lambda=1$ for simplicity. I will also take $f=0$, and discuss just the interior estimates.)
The precise exponent is known in two dimensions (see Piccinini & Spagnolo 1972)...
- 2,155
8
votes
Do curvature differences obstruct a.e orientation-preserving isometries?
There is a discontinuous map $f\colon\mathbb{S}^2\to\mathbb{R}^2$ such that $d_xf$ is defined and isometric for almost all $x$. (If you want a continuous one then I am sure the answer is "no")
To ...
- 45k
8
votes
Elliptic regularity on compact manifold without boundary
This result is true. This is Theorem 6.30 in:
F.W Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-...
- 28k
7
votes
Accepted
Existence and uniqueness of geodesics in low regularity
There is a classical example by Hartman that shows failure of uniqueness for $C^{1,\alpha}$ metrics. (P. Hartman, On the local uniqueness of geodesics, Amer. J. Math. 1950).
You could lower the ...
- 659
7
votes
Regularity of harmonic forms on manifolds-with-boundaries
The answer depends on your definition of harmonic; if you mean $\Delta \alpha=0$ then you can not conclude that $\alpha$ is smooth. It is easy to find counter examples on the unit disk in $\mathbb{R}^...
- 676
7
votes
Accepted
$2n$-regular graphs with maximal chromatic number
Example 1. Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.
Example 2. Let $n=2$, $m=15$. Then $c=5$,...
- 13.4k
7
votes
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf which I state below. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 ...
- 28k
7
votes
Accepted
Nonsmooth version of Hopf boundary point lemma
I think this is just the comment following Lemma 3.4 of Gilbarg and Trudinger (specifically equation 3.11).
I should add that lowering the regularity of the boundary seems like a harder problem (and ...
- 2,478
7
votes
Weak Hessian of the distance function
The following result is due to Asplund [A, p.235].
Theorem. If $\varnothing\neq E\subset\mathbb{R}^n$ is closed, and $d(x)=\operatorname{dist}(x,E)$, then $f:\mathbb{R}^n\to\mathbb{R}$ defined by
$f(...
- 28k
6
votes
Accepted
A differentiable isometry is smooth?
By the nontrivial fact that $f$ is a local homeomorphism, we can assume without loss of generality that $f$ is a bijective homeomorphism. The usual textbook proof of the formula for the differential ...
- 3,146
6
votes
Accepted
Regularity of a conformal map
EDIT. I just realized that my previous answer was incorrect.
The slits in the Kobe theorem that you consider are bounded,
since the image under the mapping function with your normalization
has $\infty$...
- 91.8k
6
votes
Accepted
Main utility of the monotonicity formula for generalized surfaces
A basic answer is that "the monotonicity formula places constraints on the shape of a minimal surface" e.g., you cannot have a lot of area concentrated in a ball if then later there is a (...
- 7,197
5
votes
Accepted
Reference for De Giorgi-Nash-Moser theory
The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities.
The elliptic case ...
- 28k
5
votes
Accepted
Oscillation and Hölder continuity
Just prove it yourself:
Take $r=1$. Then
$$w(x_0,2^{-n})\leq \lambda^n w(x_0,1)=:C\lambda^n.$$
To estimate $|u(x_0)-u(y)|$, where $y$ is close to $x_0$, choose $n$ so that
$|x_0-y|\in[2^{-n-1},2^{-n}]...
- 91.8k
5
votes
Accepted
Gluing of two solutions to the same parabolic equation
Absolutely not! Taking the difference $v=u_1-u_2$, you see that $v(x,t)$ solves
$$
\begin{cases}
(L-\partial_t) v=0 & \mbox{for }(x,t)\in(0,1/2)\times(0,T];
\\
v(0,t)=f(t) & \mbox{for }x=0,\,...
- 5,380
5
votes
Accepted
Smoothness of critical elliptic problem
The result is true. As you suggest, the starting regularity assumption $u \in H^1_0$ is critical in the sense that, if you additionally knew $u \in L^p$ with $p>2^*$, then you could bootstrap to ...
- 381
5
votes
Accepted
Can solution of heat equation become constant in finite time
Not only is this possible, but more is true: For any initial datum, there exist boundary data which make it so. Google "boundary null controllability" for the heat equation.
- 13k
5
votes
Reference Request for global Hölder continuity of solutions to elliptic PDEs
Such a global Hoelder regularity result requires some minimal assumptions on the geometry of $\Omega$ but no smoothness assumptions on the coefficients.
The classical assumption is a measure density ...
- 2,670
4
votes
Accepted
Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$
The following result is well known. It is Theorem 5.1 in [1]. It is proved by the method of duality solutions sue to Stampacchia.
Theorem. Let $\mu$ be a signed Borel measure with the finite total ...
- 28k
4
votes
$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
Without further assumptions, no. Consider the case $a_1=a_2=0$ and $\Omega$ smooth. If $u\in H^2(\Omega)$, then $\partial u/\partial n$ is in $H^{1/2}(\Gamma)$ by the trace theorem. However, $g_1\in H^...
- 13k
4
votes
Different ways to prove $L^p$-estimates for the heat equation
I would say that methods 1) and 2) are very close each other but in both cases the proof is a bit harder than the elliptic counterpart since one has to use the Marcinkiewicz multiplier theorem instead ...
- 6,025
4
votes
Accepted
Regularity of transport map
This is an open question. The difficulty is that Monge's cost function is very degenerate, so doesn't satisfy the assumptions of the standard regularity theory of optimal transport.
The first ...
- 6,001
4
votes
Optimal assumption on H^2 regularity
One cannot weaken the regularity hypothesis on $A$ to $C^{\alpha}$. Consider for example the 1D case: when $f = 0$ the solution $u$ satisfies
$$u'(x) = \frac{const.}{A(x)},$$
which is not in $H^1$ for ...
- 4,889
4
votes
Accepted
Reference Request for global Hölder continuity of solutions to elliptic PDEs
Although not the most accessible ones, Theorem 8.27 and 8.29 of the book of Gilbarg and Trudinger should give what you want.
3
votes
Higher regularity of solutions for Laplace equation with mixed boundary condition
This is more of an extended comment, but maybe it is helpful.
At least for the case of mixed boundary conditions involving Dirichlet and Neumann conditions where the corresponding boundary parts ...
- 2,670
3
votes
Accepted
Is this approach for establishing regularity of harmonic maps between manifolds valid?
Regularity of harmonic map is true in dimension with $f\in W^{1,2}$ but it is subtle see Helein, Harmonic maps, conservation laws and moving frames.
It is false in higher dimension, even if f\in W^{...
- 914
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