Questions tagged [regularity]
regularity of solutions of PDEs.
218
questions
2
votes
0
answers
105
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 $x \in \partial D$ ...
- 619
3
votes
0
answers
112
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....
- 1,879
1
vote
0
answers
46
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
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
- 337
1
vote
0
answers
81
views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem:
$$\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\...
- 41
0
votes
0
answers
27
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 $u \in C^0(0,T;L^2(B)) \cap L^2(0,T;u_0 + W_0^{1,2}(B))$ ...
- 473
2
votes
1
answer
94
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 $...
- 263
2
votes
0
answers
99
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 $\...
- 141
0
votes
0
answers
39
views
Are $c$-edge-colored clique removal lemmas known when $c>2$?
The following is a rephrasing of the Induced Graph Removal Lemma by Alon, Fischer, Krivelevich, Szegedy:
For all $k>0$ and all $\epsilon > 0$, there is $\delta > 0$ such that the following ...
- 1,369
2
votes
1
answer
189
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:
$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\...
- 1,154
0
votes
1
answer
56
views
Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form
Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to
$$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \...
- 61
2
votes
1
answer
130
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\...
- 61
4
votes
1
answer
87
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\...
- 61
1
vote
0
answers
29
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{\...
- 11
5
votes
1
answer
256
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 $\partial\Omega$. ...
- 2,410
4
votes
1
answer
351
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 $\...
- 4,309
3
votes
0
answers
45
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
$$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
- 519
1
vote
0
answers
20
views
Regularity of solutions of a 2nd order singular integro-differential operator
I have trouble finding the regularity of the solutions to a particular equation. I define
$$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}...
- 81
0
votes
0
answers
40
views
regularity of ground state of Schrodinger operator
I have a probably naive question on the regularity of ground state of the Schrodinger operator: $\Delta u - Vu = Eu$, where $\lim_{|x|\rightarrow\infty}V(x) = +\infty$ and $V\in C^2$, and $E$ is the ...
- 31
1
vote
0
answers
120
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 ...
- 31
2
votes
0
answers
81
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 $f\in L^1(B(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup \left\{ \frac{1}{...
- 199
4
votes
1
answer
137
views
Smoothness of critical elliptic problem
I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of
$$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth ...
- 1,353
1
vote
0
answers
33
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 ...
- 111
3
votes
0
answers
93
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 ...
- 619
2
votes
1
answer
49
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
56
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 $...
- 173
2
votes
0
answers
93
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
81
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 ...
- 619
1
vote
0
answers
173
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}$$
for ...
- 213
2
votes
1
answer
115
views
The regularity theorem, a non-regular minimizer problem
During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them.
The function $f:[-1,1] \times \mathbb R ...
- 89
2
votes
0
answers
157
views
Elliptic regularity for a system of PDEs
I am considering a system that can be simplified to the following problem.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ with smooth boundary, $n \geq 3$, and consider the following coupled ...
- 938
3
votes
1
answer
179
views
$C^{1,\alpha}$ estimate for Newton potential of $L^\infty$ function
Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then
$$
u(x) = \int_\Omega \log |x-y| f(y)\ dy
$$
...
2
votes
0
answers
69
views
Smooth solutions to the Neumann problem defined on the closure of the domain
Does the Neumann-problem for the Laplace equation on a smooth bounded domain $\Omega \subset \mathbb{R}^{3}$ with smooth boundary data $g$ such that
\begin{equation}
\int_{\partial \Omega}g \, d \...
- 329
4
votes
0
answers
226
views
Boundary regularity for elliptic PDE in Lipschitz domains
In section 2.6 of Fernandez-Real and Ros-Oton's book "Regularity theory for elliptic PDEs" it is stated that solutions of the Dirichlet problem with smooth data for the Laplacian are
$C^{1-...
- 353
2
votes
0
answers
261
views
A regularity result for semilinear PDE of the form $\Delta u=f(x, u)$ in Michael E. Taylor's book "Partial Differential Equations III"
Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that
$$
\partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6}
$$
Michael E. Taylor said that (proposition (1.3))
For $k=1,2, \...
- 391
3
votes
1
answer
181
views
Regularity of transport map
Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}^n$ with first moment and suppose that both $\mu$ and $\nu$ have a densities with respect to the $n$-dimensional Lebesgue measure. Fix some ...
- 4,956
0
votes
0
answers
53
views
Regularity of solution to Cauchy problem given regular initial data
Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus)
be a solution of a Cauchy problem
$$\frac{d}{dt} f(t) = A f(t)$$
$$f(0)...
- 2,483
3
votes
1
answer
117
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{...
- 208
0
votes
1
answer
56
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 ...
- 8,748
2
votes
1
answer
191
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 ...
user140442
3
votes
0
answers
105
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 ...
4
votes
0
answers
84
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?
- 395
1
vote
0
answers
133
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
$$
\int_Q A(x)\nabla u ...
- 46
3
votes
1
answer
168
views
Open problem 1.28: $W^{1,1}$ regularity for optimal transport map
While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's User's guide: the problem to determine whether we ...
- 73
4
votes
0
answers
142
views
$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
- 277
0
votes
1
answer
145
views
Application of Green function for non linear PDE [closed]
In the case of linear PDE, say $$Lu=0$$ if we have its green function say $G(x,y)$ then using that one can give solution of non homogenous PDE i.e. $Lu_f=f$ where $u_f=G*f$.
Is the same thing hold for ...
- 309
3
votes
1
answer
97
views
A regularity estimate for second-derivative
I was reading this paper (arXiv link)
On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations
Guy Barles (LMPT), Alessio Porretta, Thierry ...
- 263
1
vote
0
answers
82
views
Regularity results for non uniform elliptic equation
I have seen some regularity result for ellptic PDE but all of them consist of uniform elliptic one. For instance,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
1
vote
0
answers
78
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 $...
- 263
2
votes
0
answers
101
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 ...
- 71
3
votes
1
answer
240
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 ...
- 215