# Questions tagged [regularity]

regularity of solutions of PDEs.

115 questions

**5**

votes

**1**answer

215 views

### Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?

$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...

**-1**

votes

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

124 views

### Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$
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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

227 views

### Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$
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

**2**answers

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

**1**answer

190 views

### Is this approach for establishing regularity of harmonic maps between manifolds valid?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...

**1**

vote

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**0**answers

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

**1**answer

467 views

### Is this expression for the Laplacian of conformal maps between Riemannian manifolds known?

$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
$\newcommand{\Hom}{\operatorname{Hom}}$
$\newcommand{\tr}{\operatorname{tr}}$
$\newcommand{\TM}{\operatorname{T\M}}$
$\newcommand{\TN}{\...

**14**

votes

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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 \...