Questions tagged [regularity]
regularity of solutions of PDEs.
237
questions
0
votes
0
answers
29
views
A question about regularity results in the Elliptic case which are given by Schauder theory
I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
0
votes
1
answer
129
views
About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain
I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$:
Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
2
votes
0
answers
101
views
Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$
Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it.
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
10
votes
0
answers
383
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
1
vote
0
answers
55
views
Parabolic regularity for weak solution with $L^2$ data
I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions:
$$\begin{cases}\...
0
votes
0
answers
48
views
Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
4
votes
1
answer
169
views
Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$
I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given:
\begin{align*}
\partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\
...
0
votes
0
answers
41
views
Smoothness of solutions to wave equation in a bounded domain
Consider the wave equation
\begin{equation}
\partial_t^2 u - \sum \partial^2_{x_i} u =0
\end{equation}
in a bounded domain $M$ with $C^\infty$ boundary, and the boundary conditons
\begin{equation}
u(...
0
votes
0
answers
169
views
How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
2
votes
1
answer
90
views
Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
1
vote
0
answers
111
views
Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
2
votes
0
answers
167
views
A question about the regularity of the Schrödinger equation
While reading the article [1], I noticed I don't understand part of the proof of regularity. For the Schrödinger eigenvalue problem,
\begin{cases}
-\Delta u+Vu=\lambda u, &\text{in } \Omega \\
\...
1
vote
1
answer
106
views
Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
5
votes
0
answers
172
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
1
vote
0
answers
112
views
Continuity of a minimizing measure w.r.t a parameter
Let $V_t(x)=x^2+t\phi(x)$ where $t>0$ and $\phi\in C^\infty_c(\mathbb{R})$.
My question is what can be said about the continuity of the (unique) minimizer (among probability measures) of the ...
0
votes
0
answers
24
views
Detailed estimate of the magnitude for the constant appearing in maximal regularity of the inhomogeneous heat equation
For any $T \in (0,\infty)$ and the $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ together with $1< p,q < \infty$, let us consider the following Cauchy problem:
\begin{equation}
\partial_t U - \alpha \...
1
vote
0
answers
80
views
Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space
Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...
1
vote
0
answers
90
views
Regularity of minimizing harmonic maps with no topological obstructions
So during (not really) my research I stumbled upon the following question, for which I could not find results in literature in any direction. It is not stated super precisely mathematically speaking, ...
1
vote
0
answers
110
views
Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?
I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation.
However, detailed proofs can ...
1
vote
0
answers
60
views
A question about semigroups in a Heisenberg group
I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
1
vote
0
answers
100
views
Brezis-Kato theorem
Let $n\geq 3$, and $u$ satisfies
$$
-\Delta u=K(x)u^{\frac{n+2}{n-2}}
\quad x\in B_1\setminus \{0\},
$$ where
$|K(x)|\leq A$ in $B_1\setminus \{0\}$, and
$u\geq 0$ in $B_1\setminus \{0\}$.
Can we ...
4
votes
0
answers
87
views
Are Sobolev isometries in Minkowski space smooth
Let $\Omega\subset\mathbb{R}^d$ be an open regular domain and let $f\in W^{1,\infty}(\Omega;\mathbb{R}^d)$ satisfy that $df\in\operatorname{SO}(d)$ almost-everywhere. It was proved by Reshetnyak (in a ...
2
votes
0
answers
144
views
Regularity of a weak solution to an elliptic PDE with mixed boundary condition
I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition.
Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...
3
votes
0
answers
118
views
If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?
The question is as in the title.
Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
1
vote
0
answers
75
views
Non-existence of classical solutions of Hardy PDE
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
1
vote
0
answers
155
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
0
votes
0
answers
41
views
Boundary regularity for heat equation
Consider the heat equation $u_t - \Delta u=0$ with $u = u_0$ on $\partial B \times (0,T) \cup B \times \{t=0\}$. We consider weak solutions $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
2
votes
1
answer
129
views
Global Hölder regularity
I am reading the book "Regularity theory for elliptic PDE" by Xavier Fernández-Real
and Xavier Ros-Oton, and I saw this result on page 69 about solutions of $\Delta u = f$ in $\Omega$ with $...
2
votes
0
answers
119
views
Linear elliptic problems: Are gradient estimates preserved after perturbation?
(This question is a duplicate from here)
We start with the linear elliptic PDE
$$
-\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\
u=0 \quad\text{on}\ \partial\Omega
$$
We assume that $\...
3
votes
1
answer
248
views
Are Neumann Laplacian eigenfunctions in $C(\overline{\Omega})$?
Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
0
votes
1
answer
88
views
Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form
Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
2
votes
1
answer
150
views
Optimal assumption on H^2 regularity
In many text book (Evans, Gilbarg-Trudinger for example) there is a classical result of interior regularity for weak solutions to a elliptic divergence problem $\rm{div}(A(x)u)=f$ in $\Omega\subset\...
4
votes
1
answer
124
views
Interior Sobolev regularity of parabolic solutions
In Evans book (and many others) there are a classic result about interior regularity in Sobolev spaces for solutions to uniformly elliptic problem (Theorem 1, p. 309). That is, let $\Omega\subset\...
1
vote
0
answers
31
views
Regularly of Neuman and Dirichlet problem
Let $\Omega \subset \mathbb{R}^n$, $n=2,3$ be a bounded domain with $\Gamma$ Lipschitz boundary (regularly) such that $\Gamma=\Gamma_D\cup\Gamma_N$. Assume that $h \in L^2(\Gamma)$ and $a \in C(\bar{\...
5
votes
1
answer
317
views
Weak Hessian of the distance function
If $\Omega\subset\mathbf{R}^d$ has a smooth boundary it is known that the distance function $\mathrm{d}_\Omega:x\mapsto \mathrm{d}(x,\partial\Omega)$ is smooth on a neighborhood of $\partial\Omega$. ...
4
votes
1
answer
413
views
Nonsmooth version of Hopf boundary point lemma
Let
$$
Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u
$$
be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite.
Here I'm only considering smooth coefficients, and the domain $\...
3
votes
0
answers
49
views
Regularity of subelliptic eigenfunction on characteristic domain
Background: Consider the Hörmander vector fields $X=(X_1,\cdots,X_m)$ on $\mathbb{R}^n$, and the associated Dirichlet eigenvalue problem
$$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
1
vote
0
answers
21
views
Regularity of solutions of a 2nd order singular integro-differential operator
I have trouble finding the regularity of the solutions to a particular equation. I define
$$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
2
votes
0
answers
161
views
Visualization of an oscillation lemma
How can one visualize Theorem 4.2 on page 31 of this paper by Seregin, Silvestre, Šverák and Zlatoš?
On the other hand, I have a clear visualization of a related result about how oscillation decay ...
2
votes
0
answers
83
views
Maximal function to high power
Consider the following maximal function : in dimension $n$ consider $B(0,1)\subset \mathbb{R}^n$ the unit ball, if $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
4
votes
1
answer
155
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 ...
1
vote
0
answers
37
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 ...
3
votes
0
answers
100
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 ...
2
votes
1
answer
67
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 ...
3
votes
0
answers
65
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 $...
2
votes
0
answers
125
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(\...
3
votes
0
answers
83
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 ...
1
vote
0
answers
205
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 ...
2
votes
1
answer
137
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 ...
2
votes
0
answers
180
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 ...