# Questions tagged [regularity]

regularity of solutions of PDEs.

202
questions

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### Smoothness of critical elliptic problem

I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of
$$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth ...

1
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0
answers

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### Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...

3
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0
answers

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### On the derivatives of the solutions of the heat equations with Neumann boundary condition

Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...

1
vote

1
answer

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### Quotient of solutions of a semilinear Dirichlet problem is $L^\infty$

I posted this on MathStackExchange, but it hasn't even got 10 views, so probably it is better to post here. I hope it is not inappropriate.
I am reading a paper of Brezis and Oswald about existence ...

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0
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### Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?

Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$.
If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...

2
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0
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### Smoothness of distance function induced by Finsler metric

Consider $\mathbb R^N$ endowed with a smooth Finsler metric $\phi:\mathbb R^N\times S^N\to (0,+\infty]$. The smoothness assumption are both on $\phi(x,\cdot)$ (being at least $C^{2,1}$) and $\phi(\...

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### Differentiability of a weak solution

Let $d$ be a positive integer with $d \ge 2$. We write $x=(x_1,\ldots,x_{d-1},x_d)=(\hat{x},x_d)$ for $x \in \mathbb{R}^d.$ The standard inner product and the Euclidean norm on $\mathbb{R}^d$ are ...

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0
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### Regularity of Fokker-Planck equation

Consider solutions $\rho_{1,2}$ of the Fokker-Planck equation
$$\begin{cases}\partial_t \rho_i = \Delta \rho_i + \nabla \cdot (\rho_i \nabla \Phi_{1,2})\\
\rho_i(0,\cdot) = \rho^0 \end{cases}$$
for ...

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0
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### Regularity conditions involving fractional spectral Laplacian

I want to check the regularity conditions
$$ (1) \quad\| (-\Delta)^{-1/2}F'(u)(-\Delta)^{1/2} w \| \leq L\cdot \| w \|,\quad u \in L^2(\Omega),\ w\in D((-\Delta)^{1/2})$$
and
$$
(2)\quad \| \Delta^{-...

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1
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### The regularity theorem, a non-regular minimizer problem

During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...

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### Elliptic regularity for a system of PDEs

I am considering a system that can be simplified to the following problem.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$, and consider the following coupled ...

3
votes

1
answer

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### $C^{1,\alpha}$ estimate for Newton potential of $L^\infty$ function

Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then
$$
u(x) = \int_\Omega \log |x-y| f(y)\ dy
$$
...

2
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0
answers

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### Smooth solutions to the Neumann problem defined on the closure of the domain

Does the Neumann-problem for the Laplace equation on a smooth bounded domain $\Omega \subset \mathbb{R}^{3}$ with smooth boundary data $g$ such that
\begin{equation}
\int_{\partial \Omega}g \, d \...

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0
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### Boundary regularity for elliptic PDE in Lipschitz domains

In section 2.6 of Fernandez-Real and Ros-Oton's book "Regularity theory for elliptic PDEs" it is stated that solutions of the Dirichlet problem with smooth data for the Laplacian are
$C^{1-...

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### A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book "Partial Differential Equations III"

Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that
$$
\partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6}
$$
Michael E. Taylor said that (proposition (1.3))
For $k=1,2, \...

3
votes

1
answer

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### Regularity of transport map

Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...

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### Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...

0
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0
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### Regularity of $\frac{d}{dt}f = Df + Bf$ in the interior of a cylinder

Suppose that $f\in L^2_1([0,1]\times \mathbb{T}^m)$ satisfies the following PDE
$$\frac{d}{dt}f = Df + Bf$$
$$f(0) = g\in L^2_{3/2}(\mathbb{T}^k)$$
where $D:L^2_1(\mathbb{T}^m)\to L_0^2(\mathbb{T}^m)...

3
votes

1
answer

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### Gluing of two solutions to the same parabolic equation

Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. Suppose I have $u_1(x,t) \in C^\infty([0,1] \times [0,T])$ solving
\begin{...

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0
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### Parabolic equation in non-cylindrical domain with cone

Let $d_1(t)$ and $d_2(t)$ be smooth functions from $[0,T]$ to $\mathbb{R}$ such that $d_1(t) <d_2(t)$ for $t \in (0,T]$ and $d_1(0)=d_2(0)$. Suppose $L$ is a uniform elliptic operator and $u(x,t) :\...

0
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1
answer

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### What is the critical exponent for irregular function in the Sobolev scale?

When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...

2
votes

1
answer

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### Regularity bound

For $\Delta f_g = g$, can we prove that $f_g \in L^\infty_{\text{per}}(\Omega)$ and
\begin{align*}
\|f_g\|_{L^\infty_{\rm per}} \le c \|g\|_{L^\infty_{\rm per}}
\end{align*}
where $c$ does not ...

2
votes

0
answers

83
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### A question of the book "Regularity Theory for elliptic PDE"

In the book "Regularity Theory for elliptic PDE", written by Fernández-Real, page 67, $ \tilde{u}_{k} $ converge to $ \tilde{u} $ only in $ C^1 $ norm, but the result is that we can get a ...

4
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0
answers

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### Minimal regularity for domains in Green's formula

The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?

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0
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### Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform

Let us consider the wave equation
$$\begin{cases}
w_{tt} -\Delta u = 0, & x \in \Omega, \ t >0, \\
w(0,x) = 0, & x \in \Omega,\\
w_t(0.x) = \phi(x), & x \in \Omega,\\
w(t,x) = 0, &...

1
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0
answers

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### $L_p$ estimate in mixed boundary problem for elliptic equation

Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...

3
votes

1
answer

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### Open problem 1.28: $W^{1,1}$ regularity for optimal transport map

While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we ...

4
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0
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### $L^\infty$ solutions for parabolic Neumann problem (heat equation)

Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...

0
votes

1
answer

102
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### Application of Green function for non linear PDE [closed]

In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$.
Is the same thing hold for ...

3
votes

1
answer

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### A regularity estimate for second-derivative

I was reading this paper (arXiv link)
On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations
Guy Barles (LMPT), Alessio Porretta, Thierry ...

1
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0
answers

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### Regularity results for non uniform elliptic equation

I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...

1
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0
answers

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### Regularity with explicit bound

Let $\Omega$ be an open, bounded with $C^2$ boundary (or smooth as we want). A result about elliptic regularity is given as follows. If $\Omega_0\subset\subset \Omega$ and $u$ is a weak solution of $...

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### Dimension of critical set of p-harmonic function

Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...

2
votes

1
answer

158
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### Main utility of the monotonicity formula for generalized surfaces

I hope not to be too simplistic.
I read about this monotonicity formula A question on the monotonicity formula for minimal submanifolds
I noticed that the monotonicity formula is often used in ...

4
votes

1
answer

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### Improving regularity of the boundary of a convex set in Riemannian manifolds

Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$...

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0
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### Regularity of a Fokker-Planck PDE with unbounded coefficient

Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\![0,\infty))$ taking values in $\mathbb R^d$. Consider the parabolic PDE
$$ \...

1
vote

1
answer

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### Characterization on smallest element in affine Sobolev subspace

Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that
$$
\phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...

0
votes

1
answer

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### Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?

Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...

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### some superharmonic function as a universal lower bound on Lipschitz domains

Question: for any bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$, does there always exist a nonnegative function $\phi\in C^2(\Omega)$ such that
$\phi$ vanishes on $\partial\Omega$
the normal ...

4
votes

0
answers

112
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### optimal regularity for elliptic pdes with $div(L^\infty)$ right-hand side (Hodge decomposition?)

Question: In a smooth, bounded domain $\Omega\subset \mathbb R^d$, is it true
that solutions $\phi_f$ of
$$ \begin{cases}
-\Delta \phi_f=\operatorname{div}f & \mbox{in }\Omega\\ \phi_f = 0 & \...

2
votes

0
answers

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### Examples of harmonic functions

I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...

3
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0
answers

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### Integral representation of solution of an elliptic PDE in divergence form

Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...

1
vote

0
answers

46
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### Regularity of Laplace equation on non-convex polyhedral domain

This might be a known problem, but I could not find a precise answer.
I have the following Laplace equation
\begin{equation}
\begin{cases}
-\Delta u = f & x \in \Omega;\\
\quad\: u = g & x \in ...

3
votes

0
answers

172
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### Variational problems living in two different Sobolev spaces

Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...

1
vote

0
answers

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### While doing Lp estimates, is the constant C monotonically increasing with respect to the parameters it depends on?

For example, consider the third boundary value problem:
\begin{align}
&\frac{\partial u}{\partial t}-\sum_{i,j=1}^n a_{ij}(x,t) \frac{\partial^2 u}{\partial x_i \partial x_j} + \sum_{i=1}^n ...

3
votes

1
answer

205
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### "Reversed" Bernstein Inequality

I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below:
...

2
votes

1
answer

74
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### How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation?

I have encountered the following problem.
Let $\chi:=\chi_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$.
$\nabla\cdot ((1+\chi_{B(0,1/2)})\nabla u )=0 $ in $...

1
vote

1
answer

150
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### Poisson equation in a periodic strip

Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem
$$-\Delta u=f$$
with boundary conditions
$$...

1
vote

0
answers

30
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### Condition on Functions on Overlapping patches of a Domain

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain. Let $\{p_i\}_{i=1}^m$ for some finite $m\in\mathbb{N}$ be the overlapping open sets of $\Omega$ such that $\Omega =\bigcup_{i=1}^mp_i$.
Now, ...

0
votes

1
answer

97
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### Existence of solutions of a system of first order PDEs

Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset.
Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions.
That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...