Questions tagged [regularity]

regularity of solutions of PDEs.

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57 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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1answer
154 views

“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: ...
2
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1answer
57 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 $...
1
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1answer
73 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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0answers
27 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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1answer
88 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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0answers
40 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 ...
3
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2answers
198 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 \...
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0answers
39 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{...
3
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1answer
164 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 ...
3
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1answer
118 views

Duality argument for elliptic regularity

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
2
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0answers
46 views

Infimum of the Dirichlet energy and deformation of metrics

Consider the Dirichlet energy $E(\phi; g,h)=\frac{1}{2}\int_M|d\phi|^2$ of a map $\phi:(M,g)\to(N,h)$ between Riemannian manifolds, where we assume $M$ and $N$ are compact. Given a path $g(t)$ of ...
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35 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} \...
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0answers
13 views

Compatibility condition for the global well-posedness of coupled Stokes equations

Consider the following coupled Stokes-Darcy systems in a simplified domain $\Omega=[-1,1]\times[-1,1]$, WLOG, we may assume the horizontal variable to be periodic. The Stokes flow occuplied the upper ...
0
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1answer
90 views

Regularity in Navier Stokes from $L^2$ bound on vorticity

How would one show that if $\omega$ is the vorticity associated to $\partial_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and $$\omega\in L^\infty([0,...
4
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48 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, \...
2
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0answers
156 views

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^+:=\{...
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0answers
35 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 \...
3
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0answers
62 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(\...
2
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0answers
224 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 $\...
5
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1answer
138 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-...
3
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1answer
302 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^+ ...
4
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2answers
149 views

Regularity of a conformal map

Let $D$ be a domain in $\mathbb{C}$ with $n$ boundary components. From the work of Koebe, we know that $D$ can be conformally mapped to a parallel slit domain of a specified angle of inclination (...
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0answers
81 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 ...
2
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0answers
50 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 ...
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0answers
73 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 ...
2
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0answers
51 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((...
2
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1answer
231 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}...
2
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1answer
104 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 ...
3
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1answer
223 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,...
2
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0answers
95 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 $F\in L^p(\Omega)$. ...
1
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0answers
44 views

Axis regularity in cylindrical coordinates: conditions on the non linear terms?

I've been working on this for months and can't find a good answer. I'm looking at the incompressible Euler equations in cylindrical coordinates ($r$, $\theta$, $z$), and I am looking at the non ...
2
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1answer
103 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 $\epsilon>0$, ...
1
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0answers
27 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 ...
2
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0answers
302 views

Regularity of locally finite Borel measure

Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...
0
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1answer
149 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
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0answers
41 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 +...
2
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0answers
39 views

Time derivative in parabolic Hölder spaces

Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$. Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...
2
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1answer
263 views

Embedding of weighted sobolev space with exponential weights

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. ...
2
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0answers
80 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 $$ \...
7
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0answers
288 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: ...
2
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1answer
165 views

Reference request: Schauder estimates for parabolic equations

Where can I find Schauder estimates for second order linear parabolic equations (in divergence form with potential)? Any reference would be highly appreciated.
2
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0answers
74 views

Wave equation regularity

I have an equation of the type $$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$ for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...
4
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1answer
192 views

Oscillation and Holder continuity

Where can I find a proof of the follwing 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 \...
1
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0answers
79 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,...
5
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1answer
307 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{...
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1answer
84 views

$2n$-regular graphs with maximal chromatic number

Let $n\geq 1$ be an integer. Suppose $m\geq 2n+1$ is an integer. We construct the graph $\mathbb{Z}_m = (\mathbb{Z}/m\mathbb{Z}, E_m)$ where $$E_m=\big\{\{x,y\}:x, y \in \mathbb{Z}/m\mathbb{Z} \text{ ...
0
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0answers
97 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|...
4
votes
1answer
208 views

Regularity of harmonic forms on manifolds-with-boundaries

Let $M$ be a smooth, bounded, oriented Riemannian manifold-with-boundary. Let $\alpha$ be a harmonic differential $p$-form on $M$, subject to the boundary condition $\alpha\wedge\nu^\sharp|\partial M =...
2
votes
1answer
126 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)...