Questions tagged [regularity]

regularity of solutions of PDEs.

Filter by
Sorted by
Tagged with
4 votes
1 answer
94 views

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 ...
user avatar
  • 1,313
1 vote
0 answers
27 views

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 ...
user avatar
3 votes
0 answers
74 views

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 ...
user avatar
  • 703
1 vote
1 answer
41 views

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 ...
user avatar
2 votes
0 answers
34 views

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 $...
user avatar
  • 163
2 votes
0 answers
71 views

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(\...
user avatar
3 votes
0 answers
74 views

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 ...
user avatar
  • 703
1 vote
0 answers
142 views

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 ...
user avatar
  • 141
0 votes
0 answers
26 views

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^{-...
user avatar
3 votes
1 answer
96 views

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 ...
user avatar
2 votes
0 answers
136 views

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 ...
user avatar
3 votes
1 answer
112 views

$C^{1,\alpha}$ 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 $$ u(x) = \int_\Omega \log |x-y| f(y)\ dy $$ ...
user avatar
2 votes
0 answers
63 views

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 \...
user avatar
  • 147
4 votes
0 answers
105 views

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-...
user avatar
  • 331
2 votes
0 answers
197 views

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, \...
user avatar
3 votes
1 answer
116 views

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 ...
user avatar
0 votes
0 answers
49 views

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)...
user avatar
0 votes
0 answers
48 views

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)...
user avatar
3 votes
1 answer
103 views

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{...
user avatar
  • 168
0 votes
0 answers
23 views

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) :\...
user avatar
  • 168
0 votes
1 answer
52 views

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 ...
user avatar
  • 8,660
2 votes
1 answer
188 views

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 ...
user avatar
2 votes
0 answers
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 ...
user avatar
4 votes
0 answers
72 views

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?
user avatar
  • 363
0 votes
0 answers
49 views

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, &...
user avatar
1 vote
0 answers
126 views

$L_p$ 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 ...
user avatar
3 votes
1 answer
131 views

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 ...
user avatar
4 votes
0 answers
113 views

$L^\infty$ 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 $...
user avatar
  • 277
0 votes
1 answer
102 views

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 ...
user avatar
3 votes
1 answer
83 views

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 ...
user avatar
  • 187
1 vote
0 answers
67 views

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 }\...
user avatar
1 vote
0 answers
66 views

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 $...
user avatar
  • 187
2 votes
0 answers
84 views

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 ...
user avatar
2 votes
1 answer
158 views

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 ...
user avatar
  • 205
4 votes
1 answer
68 views

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 $K \subset X$...
user avatar
1 vote
0 answers
186 views

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 $$ \...
user avatar
  • 301
1 vote
1 answer
99 views

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^{...
user avatar
  • 241
0 votes
1 answer
88 views

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)...
user avatar
  • 241
1 vote
0 answers
46 views

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 ...
user avatar
4 votes
0 answers
112 views

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 & \...
user avatar
2 votes
0 answers
160 views

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). ...
user avatar
  • 246
3 votes
0 answers
134 views

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 ...
user avatar
  • 241
1 vote
0 answers
46 views

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 ...
user avatar
  • 203
3 votes
0 answers
172 views

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} ...
user avatar
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 ...
user avatar
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: ...
user avatar
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 $...
user avatar
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 $$...
user avatar
  • 267
1 vote
0 answers
30 views

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, ...
user avatar
  • 111
0 votes
1 answer
97 views

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_{...
user avatar
  • 241

1
2 3 4 5