Questions tagged [regularity]
regularity of solutions of PDEs.
115 questions
5
votes
1answer
215 views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
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-1
votes
1answer
59 views
$2n$-regular graphs with maximal chromatic number
Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ ...
0
votes
0answers
67 views
Biharmonic equation
Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation
\begin{equation}\label{\star}
\Delta^2u+V(x)u=g(x, u)+K(x)u,
\end{equation}
where $|g(x,s)|\leq \varepsilon|...
2
votes
1answer
81 views
regularity of harmonic forms on manifolds-with-boundaries
Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
2
votes
1answer
64 views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)...
2
votes
1answer
122 views
Bounded solution for parabolic equation
Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
4
votes
3answers
276 views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
0
votes
0answers
50 views
About well-posedness and regularity for a wave equation with nonhomogeneous Dirichlet boundary condition?
I want to get any result about well-posedness and regularity for the following wave equation with nonhomogeneous Dirichlet boundary condition described by
Let $\Omega \subset
%TCIMACRO{\U{211d} }%
%...
3
votes
1answer
136 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
vote
1answer
124 views
Elliptic regularity of harmonic forms in $L^1$
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This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...
3
votes
2answers
127 views
Density on Hölder spaces whose elements vanish on the boundary
I would like to ask the following problem.
Let $\Omega$ be a $C^{r+1,\alpha}$ domain, $r\in \mathbb{N}, 0<\alpha<1.$ We denote $$C^{r,\alpha}_{0}(\overline{\Omega})=\{f\in C^{r,\alpha}(\...
1
vote
0answers
82 views
2nd oder evolution equations and regularity results of their solution
I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...
0
votes
0answers
33 views
What is the minimum setting in which regularity results are available for the solutions of Poisson's equation?
Let the generator $L$ of a diffusion process be given in Hörmander form, i.e.
$$L=\frac{1}{2}\sum_{i=1}^k X_i^2+X_0,$$
where $k\leq n$ and $X_i$, $i=0,1,...,k$, are vector fields on $\mathbb{R}^n$. ...
7
votes
1answer
161 views
+50
Is a Sobolev map with smooth minors smooth on the whole domain?
Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
...
0
votes
0answers
34 views
Higher regularity of weak solution of Laplace equation with Robin condition
From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...
0
votes
1answer
87 views
Regularity of Laplace equation with Dirichlet data on a part of the boundary
From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system
\begin{align}
-\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\
u &= g &\text{on}\ \...
0
votes
1answer
75 views
$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...
2
votes
1answer
106 views
Higher regularity of solutions for Laplace equation with mixed boundary condition
Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\...
6
votes
0answers
227 views
Is a Sobolev map with invertible smooth minors smooth?
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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 ...
2
votes
2answers
334 views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
2
votes
1answer
190 views
Is this approach for establishing regularity of harmonic maps between manifolds valid?
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While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
1
vote
0answers
23 views
Hidden regularity for the coupled wave equation with dynamaic boundary condition
We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\...
0
votes
0answers
45 views
Looser condition for regularity for Neumann problems
If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then
$$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$
Let's say we are studying rotationally symmetric solutions to ...
1
vote
0answers
53 views
Dependence of the Hölder exponent in De Giorgi-Nash-Moser
I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity.
More precisely: suppose $u$ satisfies weakly
$$
D_i(a^{ij}D_ju)=f
$$
on the $d$-...
1
vote
0answers
34 views
Harnack type Estimates for a p-Poisson equation with constant source term
Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE
$$
-\Delta_p u=1\,\,\mbox{in $B$}
$$
Let another function $f$ be such that
$$
\begin{cases}
-\Delta_p f =1 \;\;\mbox{in $B$}\\
f=0 \...
3
votes
0answers
88 views
Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?
Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$.
Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).
Suppose we are given a $W^{2,2}$ isometric immersion $...
1
vote
1answer
89 views
Is there any “extra regularity” to the solution to Poisson's equation posed on a 3-dimensional polyhedron?
I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form
$$
\|u\|_{H^{1+\epsilon}(\Omega)} \le C \|f\|_{L^2(\Omega)}
$$
for some $\epsilon >0 $ ...
8
votes
2answers
291 views
Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?
It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$.
Is there a $W^{2,2}$ isometric embedding? (i.e an isometric ...
1
vote
1answer
70 views
Advection equation regularity (2D and time independent)
I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$. One such example is:
$$
\frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) ...
0
votes
0answers
71 views
Consequence of John and Nirenberg's lemma?
The lemma I'm referring to in the title is the following:
John and Nirenberg's lemma: Let $C_0 \subset \mathbf{R}^n$ a finite cube. Let $u \in L^1(C_0)$ and assume there exists a constant $k$ such ...
3
votes
0answers
157 views
Elliptic regularity for Robin boundary conditions
Suppose I have a (non-smooth) domain $\Omega$ on which I have a $H^1$ solution $u$ of a constant coefficient elliptic PDE $L$. Suppose also that $\Gamma$ is a smooth portion of the boundary $\partial\...
9
votes
1answer
330 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 ...
3
votes
0answers
74 views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
5
votes
3answers
269 views
Request for reference for some proofs about Gowers' norm
For any map $f : \mathbb{F}_2^n \rightarrow \mathbb{C}$ we define its $d^{th}-$Gowers' Norm (for $1 \leq d \leq n$) as, $\|f\|_{U^d(\mathbb{F}_2^n)}^{2^d} = \mathbb{E}_{L : \mathbb{F}_2^d \rightarrow \...
1
vote
0answers
43 views
Entire solutions of first order linear homogeneous evolution PDEs
I've posted this question on MSE, and haven't got any feedback yet, so I will try again here:
I'll start off with the following example for $u(x,t):\mathbb{C}^2 \to \mathbb{C}$:
$$\begin{align}
u_t&...
13
votes
1answer
467 views
Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?
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$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\...
14
votes
2answers
399 views
Do curvature differences obstruct a.e orientation-preserving isometries?
Is there an example of a pair $M,N$ of connected, oriented equidimensional Riemannian manifolds with the following properties:
$M$ is everywhere non-flat, $N$ is flat.
There exist a map $f:M \to N$ ...
2
votes
2answers
513 views
Simultaneous extensions of strictly convex functions
21/03/2017: I have decided to accept Denis Serre's answer, even though it does not exactly answer my question, however I like its simplicity and I'd say it is close enough to the desired claim. Of ...
3
votes
1answer
156 views
Uniqueness conditions for linear transport equation with nonconstant velocity
Considering the following equation,
$$
u_t + \operatorname{div} \, (u \, \mathbf{b}(\mathbf{x},t)) = 0
$$
in a cylinder $K = \{(\mathbf{x},t) \in \Omega \times (0,T) \}$ where $\Omega \subset \mathbb{...
20
votes
1answer
585 views
A differentiable isometry is smooth?
I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...
0
votes
0answers
109 views
Boundary regularity of the solution of a Poisson equation in a polyhedron
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
0
votes
0answers
187 views
Implicit Function Theorem
Let $f$ be a $C^2$ function defined on a neighborhood of $0$ in $\mathbb R^n$ such that $f(0)=0, df(0)\not=0$. By the Implicit Function Theorem, it is easy to get (after a rotation) that near 0
$$
f(x)...
1
vote
0answers
104 views
Does $u\in H^{3/2}(\Omega)$ imply continuity of $\nabla u\cdot\overrightarrow{n}$ across an interior interface?
When investigaing the regularity of certain functions, I encountered this problem:
if $u\in H^{3/2}([0,1]\times [0,1])$,
what can we say about the continuity of $\nabla u\cdot\overrightarrow{n}$
...
0
votes
2answers
300 views
Local L^p regularity theory for elliptic operators
I've been studying the elliptic regularity theory using $H^s$ spaces as done in Folland's "Introduction to Partial Differential Equations".
At the end of section C, chapter 6, Folland affirms that we ...
1
vote
1answer
142 views
Moser/Schauder estimates for coercive boundary conditions
Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a ...
1
vote
0answers
92 views
$L^\infty(\Omega)$-regularity for strongly damped wave equation
I am interested in the following IBVP for the strongly damped wave equation:
\begin{equation}
u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\
u=0 \quad \text{on} \ ...
0
votes
0answers
279 views
Regularity for a div-curl system
Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
12
votes
2answers
596 views
Unexpected regularity of the distance from a $C^2$ submanifold
Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...
3
votes
0answers
143 views
Interior regularity for elliptic operators with non smooth coefficients
I need a pretty standard interior regularity result for a second order elliptic operator of the form
$$
-\nabla^b \cdot (A(x) \nabla^b v)+c v=f,
\qquad
\nabla^b=\nabla+ib(x)
$$
where $A(x)$ is a ...
1
vote
0answers
154 views
An H2 estimate for Helmholtz equation
How to show the following statement?
Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation,
$$
-\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...
