The regularity tag has no wiki summary.
3
votes
1answer
103 views
Distance function from a topological submanifold
Let $(M,g)$ be a Riemannian manifold, and let $N\subset M$ be an embedded sphere that is everywhere smooth except for a single point at which the embedding will only be $C^0$.
How much regularity can ...
1
vote
1answer
54 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 ...
1
vote
1answer
163 views
On the Hölder regularity of an integral function
Let $n\geq 3$. Let $\Omega$ be an open and bounded subset of $\mathbb{R}^n$. Let define $X_0$ as the space of functions $f:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $f(x,\cdot)$ is ...
4
votes
1answer
82 views
Approximation of sets by sets with regular border
What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...
1
vote
1answer
81 views
Topological description of the regular values of a differentiable function
Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms?
In particular, is the set of regular values ...
1
vote
0answers
22 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 ...
2
votes
1answer
182 views
Does the implicit function theorem hold for discontinuously differentiable functions?
(This was posted on math.SE over 5 days ago and has not been answered,
although a comment mentioned a similar question on this site.)
Wikipedia's statement of the implicit function theorem requires ...
1
vote
1answer
87 views
W^{2,∞} regularity of solutions of Poisson's equation if the right hand side is in L^{∞}
Let $u$ be solution of $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial n} = 0$ on $\partial \Omega$.
Is it true that if $f \in L^{\infty}(\Omega)$ then $u \in W^{2,\infty}(\Omega)$?
...
0
votes
0answers
54 views
Reference for a regularity result for linear parabolic equations under mixed Neumann-Dirichlent boundary condition
Could someone please help me to find a precise reference for $L^{p}%
-$regularity results for linear parabolic equations under mixed
Neumann-Dirichlet conditions? More precisely, I would like to have ...
2
votes
0answers
147 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 ...
8
votes
5answers
1k views
Are all almost regular graphs obvious?
Let the maximum and minimum degress of a graph be denoted (as usual) by $\Delta$ and $\delta$ respectively.
A graph is almost regular if $\Delta-\delta=1$.
Now, here is a simple way to generate ...
0
votes
0answers
70 views
mixed Dirichlet Neumann regularity for an elliptic equation
Here is a problem which may be easy for some of you but not for me.
Statement of the problem:
Denote $\Omega := \{ (x,y) \in (0, \infty) \times (-\infty,\infty) \}$.
Let $f \in L^2(\Omega)$ then by a ...
0
votes
1answer
76 views
Optimal Regularity for Invariance of Curvature under Isometries
It is well known that sectional curvature is an invariant under isometries. I wonder what the optimal regularity for this result to hold is (in terms of Hölder-spaces)?. It is classical that ...
3
votes
2answers
257 views
Physical and real life interpretation of the concept of regularity used in differential equations?
I guess the title kind of speaks for my questions: I'm curious to know what could be the physical interpretation or real life application of the concept of regularity that arises in PDE: take for ...
0
votes
1answer
139 views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
3
votes
0answers
75 views
Integrability of $D^2u$ for $\infty$-harmonic function $u$?
Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with
$$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
2
votes
2answers
390 views
Elliptic theory on compact manifolds
Maybe this is silly.
On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation
$$ \Delta u=f \quad\text{ in $\Omega$}$$
$$ u=0\quad\text{ on $\partial\Omega$}.$$
One has the following ...
3
votes
0answers
163 views
Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$
Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < ...
2
votes
0answers
83 views
Constant in Maximal sobolev regularity
We know the following evolution equation
\begin{equation}
\left\{
\begin{array}{llc}
v_t=A v+f,\\
v(0)=0.
\end{array}
\right.
\end{equation}
$A$ generates a bounded analytic semigroup on a Banach ...
4
votes
1answer
398 views
What's wrong with the Courant nodal domain theorem
The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. ...
-1
votes
1answer
89 views
Regularity of solutions for a non linear elliptic equation
Let $v_k$ be a radial sequence of function that satisfies in $\Omega\subset\mathbb{R}^4$
$(-\Delta)^2 v_k=e^{v_k}$
$v_k(x)\leq v_k(0)=0$
$\left\Vert (-\Delta)v_k\right\Vert_{L^1(B_R(0))}=O(1)\qquad ...
2
votes
0answers
96 views
Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain
Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
2
votes
1answer
180 views
Proof of regularity for bounded elliptic problem
We consider the boundary value problem for potential in the form:
$$-\Delta u(\boldsymbol{x})=0,\quad \boldsymbol{x}\in \mathbb R^3\smallsetminus S,$$
with boundary conditions
$$\nabla ...
3
votes
1answer
131 views
sub and super-levelset regularity for Sobolev functions
I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:
Assume ...
2
votes
0answers
146 views
Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules
Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on ...
8
votes
1answer
193 views
Failure of Fredholm property of elliptic PDE systems
Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...
2
votes
2answers
118 views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
11
votes
4answers
619 views
Einstein field equations in perspectives from PDE and functional analysis
The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
3
votes
2answers
284 views
Non symmetric coefficient matrix for elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form
$$ D_i(a_{i,j}D_ju)=0 \tag{1}$$
where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
5
votes
1answer
500 views
Possible mistake in De Giorgi's paper on Holder's regularity
$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one.
$I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...
4
votes
1answer
243 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 ...
0
votes
2answers
600 views
Interior regularity for elliptic equations
The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for ...
5
votes
3answers
393 views
Divergence form Elliptic PDE Removable Singularity/Regularity Question
Idea
Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) ...
9
votes
1answer
623 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 ...
6
votes
1answer
372 views
If a compact Kahler manifold $(M,g)$ has constant scalar curvature, is the metric $g$ real analytic?
Hi to all!
Perhaps it is a silly question, if so i'll delete this post.
Suppose we have a compact Kahler manifold $(M,g)$ of complex dimension $m$ with constant scalar curvature with respect to its ...
2
votes
0answers
138 views
regularity for viscosity solutions of second order parabolic equations
I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre .
Here F is ...
2
votes
2answers
482 views
Does regularity of the boundary imply interior sphere condition
In the article of Massari presented here there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition:
There exists $\rho>0$ such that for every ...
0
votes
1answer
276 views
A property of sets of finite perimeter
I have a question regarding sets of finite perimeter. I feel that it should be true, but I didn't manage to prove or find a reference about it. Suppose $D$ is an open, bounded subset of $\Bbb{R}^n$, ...
6
votes
0answers
280 views
Compactness of solutions to parabolic equations (parabolic regularity)
I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.
For each $s>0$, I have a ...
7
votes
2answers
466 views
Boundary regularity for the Dirichlet problem
Let us consider the unit ball $B^n$ of $\mathbf{R}^n$ and $K = B^{n-1}(0,1/2) \times \{0\}$ the ball of dimension $n-1$ and radius $1/2$ lying on the equator.
We wish to solve the Dirichlet problem ...
2
votes
0answers
158 views
Manifolds with a lower degree of regularity
Hello guys,
I've been reading a paper about regularity theory for a P.D.E in a non-smooth domain(see the reference below).
There, the authors consider domains of $R^n$ with regularity of class $W^2 ...
7
votes
1answer
454 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 ...
3
votes
0answers
221 views
Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?
Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...
2
votes
1answer
314 views
What is the regularity of the argument of a complex function?
Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for ...
1
vote
2answers
818 views
Regular vs. Irregular Vertices in a Mesh
Hi everybody,
Reading about Geometry Processing, I have realized that people in this area are very interested in regular vertices(degree=6) rather than irregular ones.
Can anybody give me reasons ...
6
votes
2answers
2k views
Moser iteration for elliptic systems
I heard that De Giorgi-Nash-Moser type regularity arguments fail for elliptic systems, but do not know where to start looking for more substantial information. Why does the regularity fail? Is there ...
