Questions tagged [regularity]
regularity of solutions of PDEs.
240
questions
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
3
votes
1
answer
2k
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 ...
user5810
1
vote
1
answer
727
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 $u \in W^{2,\infty}(\Omega)$?
(...
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
11
votes
5
answers
2k
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 ...
- 6,882
1
vote
0
answers
193
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 ...
- 11
0
votes
1
answer
109
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 $C^3$-...
- 1
3
votes
2
answers
463
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 ...
- 1,417
2
votes
1
answer
735
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 ...
- 5,163
4
votes
0
answers
136
views
Integrability of $D^2u$ for $\infty$-harmonic function $u$?
Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with
$$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, \...
- 648
5
votes
3
answers
1k
views
Elliptic theory on compact manifolds
Maybe this is silly.
On a bounded set $\Omega\subset\mathbb{R}^n$ consider the equation
$$ \Delta u=f \quad\text{ in $\Omega$}$$
$$ u=0\quad\text{ on $\partial\Omega$}.$$
One has the following ...
- 327
4
votes
1
answer
429
views
Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$
Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required.
Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy
$$0 < a \leq g(x,t) \leq b < \...
- 183
3
votes
0
answers
110
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 ...
- 31
11
votes
2
answers
3k
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. ...
- 5,129
1
vote
1
answer
273
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 R&...
- 327
3
votes
0
answers
325
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 ...
- 315
1
vote
1
answer
428
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
$$\nabla u(\...
- 115
4
votes
1
answer
224
views
sub and super-levelset regularity for Sobolev functions
I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:
Assume $...
- 5,163
3
votes
0
answers
208
views
Relation between Castelnuovo-Mumford regularity for coherent sheaves and modules
Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on $X$....
- 1,573
8
votes
1
answer
285
views
Failure of Fredholm property of elliptic PDE systems
Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...
- 101
2
votes
2
answers
153
views
Is the left regularizer for elliptic BVP a left inverse for the principal part?
Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
- 101
16
votes
4
answers
3k
views
Einstein field equations in perspectives from PDE and functional analysis
The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
- 4,776
3
votes
2
answers
752
views
Non symmetric coefficient matrix for elliptic PDE
Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form
$$ D_i(a_{i,j}D_ju)=0 \tag{1}$$
where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
- 255
6
votes
1
answer
656
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 \...
- 255
6
votes
1
answer
760
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 ...
- 255
0
votes
2
answers
1k
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 ...
- 3,328
6
votes
3
answers
818
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) ...
- 1,771
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
8
votes
1
answer
749
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 ...
- 1,707
4
votes
1
answer
450
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 ...
3
votes
3
answers
2k
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 $...
- 2,102
0
votes
1
answer
519
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 $\Bbb{R}^n$, ...
- 2,102
6
votes
0
answers
360
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 ...
- 1,113
7
votes
2
answers
1k
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 (...
- 373
13
votes
1
answer
839
views
On the definition of regularity
In the literature on D-modules, there are many definitions of regularity of holonomic D-modules.
(1) Bernstein first defines regularity on a curve then says a holonomic D-module is regular if its ...
- 7,387
10
votes
1
answer
977
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 ...
- 2,031
3
votes
0
answers
238
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 $\mu$...
- 51
2
votes
1
answer
438
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 ...
- 195
1
vote
2
answers
2k
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 ...
- 111
8
votes
2
answers
3k
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 ...
- 3,292