All Questions
Tagged with geometric-measure-theory ap.analysis-of-pdes
42 questions
2
votes
2
answers
154
views
Domains of type (A) are Lipschitz?
In this article and in the book of Ladyzhenskaya et al (1968) - Linear and Quasilinear Elliptic Equations we have the following definition of what is a domain of type (A):
There is no example of a ...
- 1,759
7
votes
1
answer
152
views
Higher (BV) regularity of solutions to Poisson equation with Radon measure right-hand side?
I am trying to understand higher regularity for solutions to Poisson's equation when the right-hand side is a Radon measure. In particular:
$$\begin{cases}
\Delta u = \mu \text{ in } \Omega\\
u = 0\...
- 221
3
votes
0
answers
219
views
Strictly contracting solutions to the Eikonal equation on Riemannian manifolds
Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$.
Question: Does there exist, on every complete ...
- 6,155
2
votes
1
answer
128
views
On the existence of a complicated fractal-like set of finite perimeter
Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
- 1,245
5
votes
1
answer
241
views
Solution to the Eikonal equation with almost everywhere continuous derivative
Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?
$$|...
- 6,155
3
votes
0
answers
59
views
Intersection of $n$-dimensional minimal surfaces with two-dimensional planes
Let $M^n \subset \mathbf{R}^{n+k}$ be a smoothly embedded minimal surface. When the dimension is $n = 2$ and the codimension is $k = 1$ the intersection of $M$ with planes is well understood. If $M$ ...
- 5,038
3
votes
0
answers
123
views
A strong maximum principle for varifolds of arbitrary codimension
Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...
- 415
2
votes
0
answers
53
views
Has the nodal map been studied?
Let $D \subset \mathbf{R}^n$ be the unit disc, and $\alpha \in (0,1)$. Let $f \in C^{0,\alpha}(\partial D)$, and $u \in C^{2,\alpha}(D)$ be the harmonic function with $u = f$ on the boundary.
Define ...
- 5,038
5
votes
1
answer
170
views
Regular Lagrangian flow for "square root example": $\frac{d}{dt} X(t,x) = \sqrt{X(t,x)}$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \sqrt{X(t,x)}, &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
This is the prototype of non-uniqueness ...
- 839
2
votes
2
answers
109
views
Regular Lagrangian flow for explicit ODE with discontinuous right-hand side
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \begin{cases} - 1 & \text{ if } X(t,x) >0, \\
1 & \text{ if } X(t,x) < 0 \end{cases}, &t \in [0,T],\\
X(0,x) ...
- 839
6
votes
1
answer
425
views
Lipschitz property of the symmetric rearrangement
I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, ...
- 457
9
votes
1
answer
733
views
Calderon-Zygmund decomposition on manifolds?
The classical Calderon-Zygmund decomposition says that if $f\geq 0$ is $L^1$ on a cubes $B$, with average value $\alpha$, then there is a sequence of disjoint cubes $B_j$, such that the average of $f$ ...
- 637
0
votes
0
answers
70
views
Measure and other properties of nodal domains of Laplacian
Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$.
The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
- 161
1
vote
0
answers
72
views
Initial-boundary value problem for transport equation with $W^{1,p}$ velocity
Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation
$$ \begin{cases}
u_t + v(t,x) u_x = 0 \qquad & (...
- 61
4
votes
1
answer
266
views
Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$
Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
- 303
0
votes
1
answer
72
views
Equivalence of statements about level sets: $u|_{S \times [\tau, \infty)}$ depends only on $t$ $\iff$ $u(t,x) = \mu^{\tau}(t,u(\tau,x))$
Let $u:\mathbb R_+ \times \Omega \subset \mathbb R^N \to \mathbb R$ (sufficiently smooth). Are the following statements are equivalent?
For every $\tau >0$ and level surface $S$ of $u(\tau,\cdot)$,...
- 839
8
votes
2
answers
297
views
Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$
Recently, I asked a somewhat related question here. In the comment section, I found the formula
$$
\lim_{r\to 0}\frac{1}{r}\int_{\Omega_r} f(x)\,dx = \int_{\partial \Omega}f(\sigma)\,d\mathcal{H}^{n-1}...
- 1,245
4
votes
0
answers
151
views
Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define
$$
\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \},
$$
i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
- 1,245
0
votes
0
answers
216
views
Signed distance function
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function:
$d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
- 1,759
2
votes
0
answers
150
views
Extensions of minimal hypersurfaces
Let $B \subset \mathbf{R}^{n+1}$ be the unit ball, and $M \subset B$ be a minimal hypersurface. By this we mean that $M$ is an embedded $n$-dimensional submanifold with vanishing mean curvature. We ...
- 5,038
5
votes
0
answers
165
views
Singularities of phase interfaces in closed surfaces
Let $(\Sigma,g)$ be a compact surface without boundary. Given $\epsilon > 0$, the $\epsilon$-Allen-Cahn equation is the semilinear elliptic PDE $\epsilon \Delta_g u - \epsilon^{-1} W'(u) = 0$, with ...
- 5,038
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 1,101
3
votes
1
answer
138
views
Boundary regularity of rectifiable multiplicity 1 hypercurrents
Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
- 31
1
vote
0
answers
45
views
Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 371
2
votes
0
answers
90
views
Obstacle problems for minimal hypersurfaces
Given a compact Riemannian $n+1$-manifold $M$ with (possibly not mean convex) boundary (smooth or probably with codim $>2$ singularities). Consider the following problems,
1) fix a homology class $...
- 123
5
votes
1
answer
212
views
Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?
While analyzing a variational problem, I came to the following question:
Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ ...
- 6,741
4
votes
1
answer
172
views
Numerics for continuity equation with Sobolev vector field
Has any work been done about numerical methods for the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $...
- 839
1
vote
1
answer
169
views
Difference quotient for solutions of ODE and Liouville equation
Suppose that $\Phi$ is the solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
How does one prove that
$$\...
- 839
2
votes
1
answer
396
views
Role of the divergence of the vector field in transport equations: mass concentration?
Consider the continuity equation
$$\partial_t u(t,x) + \mathrm{div}(a(t,x)u(t,x)) = 0,$$
where $u: [0,T]\times \mathbb{R}^N \to \mathbb{R}$ is the solution and $a:[0,T]\times \mathbb{R}^N \to \...
- 839
2
votes
0
answers
187
views
Role of absolute continuity of divergence of BV function in proof of renormalization property
In the paper http://cvgmt.sns.it/paper/436/, the author proves the renormalization property for the flow generated by a vector field $a(t,\cdot) \in BV(\mathbb{R}^N; \mathbb{R}^N)$.
Heuristically, ...
- 839
1
vote
0
answers
92
views
Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case
By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user123457
4
votes
1
answer
597
views
Meaning of Alberti rank-one theorem
Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?
Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
user123457
6
votes
3
answers
541
views
About the Hausdorff dimension of removable singularities of PDE
There are some interesting phenomenons about removable singularities (or extension problems).
In the theory of functions of several complex variables, we know the classical Hartogs theorem:
Let $f$ ...
- 181
4
votes
0
answers
113
views
Are there any nontrivial examples of $C^1$ hypersurfaces with bounded (integrable) generalized mean curvature?
The definition of generalized mean curvature on $C^1$ hypersurfaces is given as follows:
Let $M$ be a closed orientable $C^1$ hypersurface in $\mathbb{R}^{n+1}$ and $\mu$ be the $n$-dimensional ...
- 1,350
7
votes
1
answer
344
views
Level sets of weakly differentiable funtions
Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose
$$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$
where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
14
votes
0
answers
632
views
Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
- 6,741
4
votes
1
answer
241
views
Is a function $u\in \mathrm{SBV}(\Omega)$ with these additional properties essentially bounded?
Some related earlier discussion can be found here.
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary, $\mathcal H^{N-1}(\partial\Omega)<\infty$ and $u\in SBV(\Omega)$. Then
$$
...
- 679
5
votes
2
answers
1k
views
What can we say about the boundary of the level set of a Sobolev function?
I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...
- 1,350
3
votes
0
answers
109
views
What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?
Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set $\...
- 1,771
1
vote
1
answer
188
views
"Schwarz symmetrization" on annulus
If $\Omega=\{x\in \mathbb R^n| 0<r_0<|x|<r_1\}$ is an annulus on $\mathbb R^n$, I am looking for a symmetrization result on $\Omega$. To be precise, for any $u \in W_0^{1,2}(\Omega)$, can we ...
- 1,441
22
votes
1
answer
4k
views
Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
- 39k
4
votes
2
answers
6k
views
Is the derivative of a Lipschitz function better than L^\infty
How smooth is the first derivative (in the distribution sense) of a Lipschitz function? Taking difference quotients and testing against an $L^1$ function, we see that $Df$ is in $L^\infty$. In ${\...
- 2,243