Questions tagged [regularity]
regularity of solutions of PDEs.
238
questions
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Continuity of the constant in maximal Sobolev regularity
Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
0
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0
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34
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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
138
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
163
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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
102
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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
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0
answers
389
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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 \...
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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:
$$\begin{cases}\...
0
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0
answers
48
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Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
4
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1
answer
171
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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
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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
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0
answers
170
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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 ...
1
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0
answers
113
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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
1
answer
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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-...
2
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0
answers
170
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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 \\
\...
11
votes
2
answers
3k
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What's wrong with the Courant nodal domain theorem?
The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. ...
1
vote
1
answer
106
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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
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0
answers
172
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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
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0
answers
112
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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
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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
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0
answers
81
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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
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0
answers
90
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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
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0
answers
114
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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
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0
answers
60
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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
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0
answers
101
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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\},
$$ where
$|K(x)|\leq A$ in $B_1\setminus \{0\}$, and
$u\geq 0$ in $B_1\setminus \{0\}$.
Can we ...
4
votes
0
answers
87
views
Are Sobolev isometries in Minkowski space smooth
Let $\Omega\subset\mathbb{R}^d$ be an open regular domain and let $f\in W^{1,\infty}(\Omega;\mathbb{R}^d)$ satisfy that $df\in\operatorname{SO}(d)$ almost-everywhere. It was proved by Reshetnyak (in a ...
2
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0
answers
146
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Regularity of a weak solution to an elliptic PDE with mixed boundary condition
I have a question on the regularity of a weak solution to an elliptic PDE with mixed boundary condition.
Let $\alpha \in (0,1]$ and let $D$ be a bounded $C^{1,\alpha}$-domain. Let $x \in \partial D$ ...
3
votes
0
answers
118
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If $\frac{\partial f}{\partial t}(x,t)$ exists a.e and $\frac{\partial^2 f}{\partial t \,\partial x }$ is continuous, can we improve a.e existence?
The question is as in the title.
Let $f(t,x) : [0,1]^2 \to \mathbb{R}$ be a function which is $C^\infty$ w.r.t $x$ for each fixed $t$ and whose derivatives w.r.t $x$ are all absolutely continuous w.r....
1
vote
0
answers
78
views
Non-existence of classical solutions of Hardy PDE
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
1
vote
0
answers
158
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\begin{cases}
\frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\
u(x,0)=u_0(x) & \forall x\in\Omega \\
u(x,t)=0 & \forall x\in\partial\...
0
votes
0
answers
42
views
Boundary regularity for heat equation
Consider the heat equation $u_t - \Delta u=0$ with $u = u_0$ on $\partial B \times (0,T) \cup B \times \{t=0\}$. We consider weak solutions $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
2
votes
0
answers
121
views
Linear elliptic problems: Are gradient estimates preserved after perturbation?
(This question is a duplicate from here)
We start with the linear elliptic PDE
$$
-\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\
u=0 \quad\text{on}\ \partial\Omega
$$
We assume that $\...
5
votes
1
answer
377
views
Oscillation and Hölder continuity
Where can I find a proof of the following fact?
If
$$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$
for some function $u(x)$ satisfies
$$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
2
votes
1
answer
130
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 $...
10
votes
4
answers
2k
views
Elliptic regularity on compact manifold without boundary
Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this:
For any $u\in H^1(M)$, ...
3
votes
1
answer
251
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:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
0
votes
1
answer
88
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Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form
Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
2
votes
1
answer
150
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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\...
6
votes
2
answers
774
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Different ways to prove $L^p$-estimates for the heat equation
Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
\partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-...
5
votes
1
answer
183
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 $K \subset X$...
4
votes
1
answer
128
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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
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0
answers
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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
319
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
414
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
50
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
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 ...
3
votes
3
answers
2k
views
Does regularity of the boundary imply interior sphere condition
In the article of Massari presented here there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition:
There exists $\rho>0$ such that for every $...
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 ...