All Questions
Tagged with ap.analysis-of-pdes regularity
156 questions
1
vote
0
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103
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,
$$\nabla \cdot (\gamma(x) \nabla u)=F \text{ in } \Omega\qquad u=\phi \text{ on }\...
- 309
1
vote
0
answers
90
views
Regularity with explicit bound
Let $\Omega$ be an open, bounded with $C^2$ boundary (or smooth as we want). A result about elliptic regularity is given as follows. If $\Omega_0\subset\subset \Omega$ and $u$ is a weak solution of $...
- 375
2
votes
0
answers
147
views
Dimension of critical set of p-harmonic function
Let $\Omega\subset \mathbb{R}^n$ be a smooth domain and $u\in W^{1,p}(\Omega)$ a non-constant $p$-harmonic function, for some $1<p<n$.
Question: What is the Hausdorff dimension of the critical ...
- 91
1
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0
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251
views
Regularity of a Fokker-Planck PDE with unbounded coefficient
Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\
- 311
1
vote
1
answer
107
views
Characterization on smallest element in affine Sobolev subspace
Suppose we are given a sequence $\phi_k$ of traces (i.e. functions defined on boundary $\partial B_1$) such that
$$
\phi_k \rightarrow 0 \;\mbox{in $L^{\infty}(\partial B_1)$}
$$
(one can consider $C^{...
- 261
0
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1
answer
109
views
Does a weakly convergent sequence in $W^{1,p}(B_1)$ which also converges in $C^{0,\alpha}(B_1)$ converges strongly in $W^{1,p}(B_1)$?
Given a sequence $u_k\in W^{1,p}(B_1)\cap C^{\alpha}(B_1)$ such that $\|u_k\|_{C^{\alpha}(B_1)}\le 1$ for all $k\in \mathbb N$. Suppose we have
$$
u_k \rightharpoonup u\;\;\mbox{weakly in $W^{1,p}(B_1)...
- 261
2
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0
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297
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Examples of harmonic functions
I am looking for non-trivial examples (in the sense to be described below) of harmonic functions, which can be represented as cubes of smooth functions ($C^1$ would be also OK if this is important).
...
- 246
4
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0
answers
318
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 ...
- 261
1
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0
answers
76
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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 ...
- 51
2
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1
answer
86
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 $...
- 309
1
vote
1
answer
274
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
$$...
- 287
0
votes
1
answer
163
views
Existence of solutions of a system of first order PDEs
Let $\Omega\subset \mathbb R^N$ be an open, smooth and bounded subset.
Given a $N\times N$, bounded and elliptic matrix of Hölder continuous functions.
That is, $A(x)= \{a_{ij}(x)\}_{N\times N}$, $a_{...
- 261
2
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0
answers
68
views
Hypoellipticity or parabolic regularity for vector bundles
Let $E \to M$ be a Hermitian vector bundle (of finite rank) over a Riemannian manifold (not necessarily compact). Let $H : \Gamma(E) \to \Gamma(E)$ be a differential operator with smooth coefficients ...
- 5,407
3
votes
2
answers
364
views
Gradient estimates for a boundary value problem
$\newcommand{\avint}{⨍}$
Let $B_r$ be a call of radius $r$ and centre origin and $k<1$.$w$ satisfy the following PDE:
$$
\begin{cases}
-\Delta w = 0 \qquad \mbox{in $B_r\setminus B_{kr}$}\\
w=0 \...
- 261
1
vote
0
answers
67
views
Regularity and existence linear parabolic fractional equation
\begin{equation}
\begin{cases}
a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x), & \text{in } \mathbb{R}^n
\end{cases}
\end{...
- 11
3
votes
1
answer
376
views
Alternative proof of Liouville theorem for harmonic functions
From Prove Liouville theorem without using mean value property the following question arises:
To prove the Liouville theorem
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 ...
- 217
2
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0
answers
52
views
A question for regularity of solutions to wave equation
let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation
\begin{equation}\label{pf0}
\begin{aligned}
\...
- 4,135
6
votes
0
answers
224
views
Interior regularity for parabolic systems in divergence form
Let $\Omega \subset \mathbb R^n$, $n \in \mathbb N$, be a smooth, bounded domain. Suppose $N \in \mathbb N$, $D \subset \mathbb R^N$ and that $a_{ij} : D \to \mathbb R$ are smooth for $i, j \in \{1, \...
- 313
2
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0
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167
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regularity; elliptic pde
Consider $N \ge 3$ a sequence of solutions $u^m$ to
$$-\Delta u^m(x) - \frac{\alpha u^m_{x_N}(x)}{x_N+\epsilon_m} = f^m(x) \quad \mbox{ in } B_2^+$$ with $ u^m=0$ on $ \partial B_2^+$ (here $B_R^+:=\{...
- 1,385
3
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0
answers
129
views
Looking for a proof of a geometric regularity criteria: generalization of the exterior cone condition / Zaremba's criterion
The topic is Perron's method for the Dirichlet problem.
I am looking for a proof of the following statement:
Let $\Omega$ be an open bounded set in $\mathbb{R}^n$ with $n \geq 3$ and $0 \in \partial \...
- 31
3
votes
0
answers
117
views
Are continuous harmonic maps between Riemannian manifolds smooth up to the boundary?
Let $M,N$ be smooth, connected, compact, oriented, two-dimensional Riemannian manifolds, with $C^k$ boundaries.
Let $f:M \to N$ be a Lipschitz continuous weakly harmonic map**, and assume that $f(\...
- 6,741
2
votes
0
answers
237
views
Singularity of L^1-solutions to elliptic PDEs on the puntured ball
Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^n$. Then it is true that if $u\in L^1(\mathbb{B})$ such that $\Delta(u)=0$ on $\mathbb{B}\backslash\{0\}$, then $\Delta(u)$, as a distribution on $\...
- 577
7
votes
2
answers
987
views
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-...
- 111
4
votes
1
answer
338
views
Reference to a Classical Regularity Theorem
(Edited)
I need a reference to the following result:
If $u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$ satisfies
\begin{cases}
{\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ ...
- 175
2
votes
0
answers
108
views
Reference request : Global boundedness of weak solution for Neumann problem
I have some question on global boundedness of weak solution to Neumann problems.
Let $u\in W^{1,2}(\Omega)$ is a weak solution for Neumannn problem
$$ \mathrm{div} (A \nabla u )= \mathrm{div}\, F\quad ...
- 323
3
votes
0
answers
81
views
$C^{1,2}$-regularity of the kinetic Fokker-Planck equation/Langevin equation
Consider a Fokker-Planck equation:
$$
\partial_t m(t,x,v) + v \cdot \nabla_x m(t,x,v) + \nabla_v\cdot \big(b(t,x,v) m(t,x,v) \big) - \frac{1}{2} \Delta_v m(t,x,v) ~=~ 0,
$$
with initial condition ...
- 31
1
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0
answers
158
views
Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
- 435
2
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0
answers
75
views
wave equation with non-smooth coefficients
Let us consider the equation
$$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$
subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
- 4,135
2
votes
1
answer
408
views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
- 205
3
votes
1
answer
299
views
Regularity and normal trace of "Hdiv" measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
- 5,380
3
votes
1
answer
228
views
Inequality for initial data
I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
- 97
2
votes
0
answers
237
views
$W^{2,p}$-estimates for Neumann boundary condition to Poisson equation
Consider the following Poisson-Neumann problem in a lipschitz bounded domain $\Omega\subset \mathbb{R}^3$:
$-\Delta u=F,\quad \partial_n u\restriction_{\partial\Omega}=0$.
Here $F\in L^p(\Omega)$.
...
- 41
3
votes
1
answer
541
views
regularity of p-harmonic functions
We know that, in general, the 'best' regularity of p-harmonic functions is $C^{1,\alpha}$, $0<\alpha<1$.
Recently, I saw a method of regularized problems as follows: For each $\epsilon>0$, ...
- 41
1
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0
answers
33
views
free boundary of a p-harmonic function
let $u$ be a p-harmonic function in $\Omega \subset \mathbb R^N$.
We already know that the set $\{u=0\}$ is locally a $C^{1,\alpha}$ hypersurface at the points where $\nabla u\neq 0$.
What can be ...
- 261
0
votes
1
answer
289
views
Estimate for Laplace equation with Neumann boundary on manifold with corner
Let $(M,g)$ be a compact Riemannian manifold with boundary and corner, i.e. locally modelled in $[0,\infty)^1\times \mathbb R^{n-1}$ or $[0,\infty)^k\times \mathbb R^{n-k}$, where $n=\dim(M)$.
...
- 1,915
1
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0
answers
48
views
Integrability condition on function determining PDE domain
I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf
On page 2, there appears a PDE of the form
$\partial_t u +...
- 123
2
votes
0
answers
90
views
A question about how to use the convexity condition?
At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say that since $p$ is convex, we can deduce that
$$ \...
- 371
8
votes
0
answers
462
views
Regularity result for the boundary value problem for the heat equation
Let $\Omega$ be an open bounded subset of $\mathbb R^N$.
Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$
Consider the following boundary value problem for the heat equation:
...
user139845
1
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0
answers
128
views
Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?
Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
- 51
6
votes
1
answer
393
views
Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?
$\newcommand{\R}{\mathbb R}$
$\newcommand{\N}{\mathbb N}$
$\newcommand{\de}{\delta}$
$\newcommand{\sig}{\sigma}$
$\newcommand{\Average}[1]{\left\langle#1\right\rangle} $
$\newcommand{\IP}[2]{\Average{...
- 6,741
0
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0
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118
views
Biharmonic equation
Let us consider for $0<\alpha\leq V(x)\leq \beta$ and $0\leq K(x)<\gamma$ the equation
\begin{equation}\label{\star}
\Delta^2u+V(x)u=g(x, u)+K(x)u,
\end{equation}
where $|g(x,s)|\leq \varepsilon|...
- 69
2
votes
1
answer
201
views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)...
- 617
2
votes
1
answer
477
views
Bounded solution for parabolic equation
Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
- 571
10
votes
1
answer
1k
views
Global regularity for Neumann problem
Let $\Omega\subset \mathbb{R}^d $ be a bounded open subset ($d\in \mathbb{N}$) and denote $\partial\Omega$ its boundary which we assume to be Lipschitz. The classical inhomogeneous Neumann problem ...
- 1,101
4
votes
1
answer
940
views
Evans-Krylov theorem
Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...
- 181
1
vote
1
answer
293
views
Elliptic regularity of harmonic forms in $L^1$
$\newcommand{\M}{M}$
This is a cross-post. I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$.
Explicitly, let $\M$ be a smooth oriented Riemannian manifold.
...
- 6,741
1
vote
0
answers
96
views
2nd oder evolution equations and regularity results of their solution
I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...
- 65
7
votes
1
answer
209
views
Is a Sobolev map with smooth minors smooth on the whole domain?
Let $d\ge 3$ and $2 \le k \le d-1$ be integers, where at least one of $k,d$ is odd. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for some $p \ge 1$.
...
- 6,741
0
votes
1
answer
438
views
Regularity of Laplace equation with Dirichlet data on a part of the boundary
From the introductory part of Chapter 2 of Grisvard's book, we know that the PDE system
\begin{align}
-\Delta u &= 0 &\text{in}\ \Omega\subset \mathbb{R}^2\\
u &= g &\text{on}\ \...
- 363
0
votes
1
answer
547
views
$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
From [Grisvard, Thm. 2.4.2.7, p. 126], the BVP
\begin{align}
-\Delta u &= 0 & \text{in}\ \Omega\\
-\frac{\partial u}{\partial n} + au &= g & \text{on}\ \Gamma\\
\end{align}
where $a&...
- 363