# Questions tagged [regularity]

regularity of solutions of PDEs.

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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 vote
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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 ...
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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
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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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### 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 ...
1 vote
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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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### 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 ...
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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 ... 83 views

### 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 ...
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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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### 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 ...
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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). ...
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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
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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 ...
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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 64 views ### 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 views ### "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 views ### 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 views ### 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$$with boundary conditions$$...
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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, ...
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_{...