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 ...
- 1,313
1
vote
0
answers
27
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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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3
votes
0
answers
74
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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 ...
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1
vote
1
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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 ...
2
votes
0
answers
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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 $...
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votes
0
answers
71
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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(\...
3
votes
0
answers
74
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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
answers
26
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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
$$
(2)\quad \| \Delta^{-...
3
votes
1
answer
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The regularity theorem, a non-regular minimizer problem
During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...
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votes
0
answers
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Elliptic regularity for a system of PDEs
I am considering a system that can be simplified to the following problem.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$, and consider the following coupled ...
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3
votes
1
answer
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$C^{1,\alpha}$ estimate for Newton potential of $L^\infty$ function
Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then
$$
u(x) = \int_\Omega \log |x-y| f(y)\ dy
$$
...
2
votes
0
answers
63
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Smooth solutions to the Neumann problem defined on the closure of the domain
Does the Neumann-problem for the Laplace equation on a smooth bounded domain $\Omega \subset \mathbb{R}^{3}$ with smooth boundary data $g$ such that
\begin{equation}
\int_{\partial \Omega}g \, d \...
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votes
0
answers
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Boundary regularity for elliptic PDE in Lipschitz domains
In section 2.6 of Fernandez-Real and Ros-Oton's book "Regularity theory for elliptic PDEs" it is stated that solutions of the Dirichlet problem with smooth data for the Laplacian are
$C^{1-...
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0
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A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book "Partial Differential Equations III"
Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that
$$
\partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6}
$$
Michael E. Taylor said that (proposition (1.3))
For $k=1,2, \...
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votes
1
answer
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Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
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votes
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49
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Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
0
votes
0
answers
48
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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) :\...
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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 ...
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2
votes
1
answer
188
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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 ...
user140442
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
votes
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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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Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform
Let us consider the wave equation
$$\begin{cases}
w_{tt} -\Delta u = 0, & x \in \Omega, \ t >0, \\
w(0,x) = 0, & x \in \Omega,\\
w_t(0.x) = \phi(x), & x \in \Omega,\\
w(t,x) = 0, &...
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1
vote
0
answers
126
views
$L_p$ estimate in mixed boundary problem for elliptic equation
Let $Q$ be convex polygon, $\Gamma$ be a portion of boundary
$\partial Q$ and $H^1_\Gamma(Q)=\lbrace u\in H^1(Q):
u|_\Gamma=0\rbrace$. For $f\in (L_2(Q))^2$ consider the problem
$$
\int_Q A(x)\nabla u ...
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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 ...
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answers
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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 $...
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votes
1
answer
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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 ...
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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 ...
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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 }\...
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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 $...
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votes
0
answers
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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 ...
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votes
1
answer
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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 ...
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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
answers
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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\
- 301
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1
answer
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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 $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...
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votes
1
answer
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Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
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 ...
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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
$$ \begin{cases}
-\Delta \phi_f=\operatorname{div}f & \mbox{in }\Omega\\ \phi_f = 0 & \...
- 3,695
2
votes
0
answers
160
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Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
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votes
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answers
134
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Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
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vote
0
answers
46
views
Regularity of Laplace equation on non-convex polyhedral domain
This might be a known problem, but I could not find a precise answer.
I have the following Laplace equation
\begin{equation}
\begin{cases}
-\Delta u = f & x \in \Omega;\\
\quad\: u = g & x \in ...
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3
votes
0
answers
172
views
Variational problems living in two different Sobolev spaces
Is there a general reference concerning variational problems living in $W^{h,p}\times W^{k,p}$, with $h, k\in\mathbb{N}_0$ not coinciding? I'm thinking to problems of type:
$$\inf_{u,v}\int_{\Omega} ...
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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 ...
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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:
...
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2
votes
1
answer
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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 $...
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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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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, ...
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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_{...
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