Questions tagged [regularity]

regularity of solutions of PDEs.

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34 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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35 views

When does the solution to the Fokker-Planck equation admit a density wrt Lebesgue measure?

Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation $$\frac{\partial u(t,x)}{\partial t}= \...
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0answers
112 views

continuity with respect to the diffusion coefficient of the solution of a semilinear parabolic equation

Let $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{n},n\geq 1$ be an open bounded subset has a boundary $\Gamma $ of class $% C^{2}$, $Q=\Omega \times \left( 0,T\...
1
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0answers
48 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 existence ...
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43 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
202 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
68 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
215 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
63 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 $F\in L^p(\Omega)$. ...
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28 views

Controlling oscillation of a p-harmonic function in a small ball

Given $\Omega\subset \mathbb R^N$ open. And $u:\Omega\rightarrow \mathbb R$ be a $p$-harmonic function. That is it minimizes the functional: $$ \min _{v\in W_{\varphi}^{1,p}(\Omega)}\int_{\Omega}|\...
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36 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
votes
1answer
97 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
25 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
156 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
129 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
38 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
37 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
209 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
237 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
votes
1answer
150 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
69 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
votes
1answer
143 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 \...
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0answers
68 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
297 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
83 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{ ...
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0answers
91 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|...
3
votes
1answer
188 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
120 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)...
3
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1answer
239 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)\...
8
votes
4answers
670 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 $u\in H^1(M)$, ...
5
votes
1answer
327 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 ...
3
votes
1answer
356 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 ...
1
vote
1answer
155 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. ...
3
votes
2answers
144 views

Density on Hölder spaces whose elements vanish on the boundary

I would like to ask the following problem. Let $\Omega$ be a $C^{r+1,\alpha}$ domain, $r\in \mathbb{N}, 0<\alpha<1.$ We denote $$C^{r,\alpha}_{0}(\overline{\Omega})=\{f\in C^{r,\alpha}(\...
1
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0answers
88 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)...
7
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1answer
188 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 $p \ge 1$. ...
1
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0answers
47 views

Higher regularity of weak solution of Laplace equation with Robin 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&...
1
vote
1answer
190 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}\ \...
1
vote
1answer
186 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&...
3
votes
1answer
217 views

Higher regularity of solutions for Laplace equation with mixed boundary condition

Let $\Omega \subset \mathbb{R}^2$ be an open bounded Lipschitz domain of class $C^{1,1}$ with boundary $\partial \Omega = \Gamma_i \cup \Gamma_o$, $\Gamma_i \cap \Gamma_o = \emptyset$ and dist$(\...
6
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0answers
238 views

Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
3
votes
2answers
578 views

Reference for De Giorgi-Nash-Moser theory

I am interested in Holder regularity for equations of the form $$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic. This was proved in the seminal paper of John Nash ...
2
votes
1answer
206 views

Is this approach for establishing regularity of harmonic maps between manifolds valid?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
1
vote
0answers
40 views

Hidden regularity for the coupled wave equation with dynamaic boundary condition

We have the equation \begin{equation} \left\{ \begin{array}{rrrr} u_{tt}-\Delta u=0,&\text{in} & \Omega \times ]0,T[ & \left( 1.1\right) \\ u=0, & \text{on } & \Gamma _{0}\...
0
votes
0answers
55 views

Looser condition for regularity for Neumann problems

If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then $$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$ Let's say we are studying rotationally symmetric solutions to ...
1
vote
0answers
90 views

Dependence of the Hölder exponent in De Giorgi-Nash-Moser

I am curious about the Hölder exponent obtained by the De Giorgi-Nash-Moser theory, as a function of the ellipticity. More precisely: suppose $u$ satisfies weakly $$ D_i(a^{ij}D_ju)=f $$ on the $d$-...
1
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0answers
37 views

Harnack type Estimates for a p-Poisson equation with constant source term

Let $B=B_1(0)\subset \mathbb R^N$ and let $u\geq 0$ solve the PDE $$ -\Delta_p u=1\,\,\mbox{in $B$} $$ Let another function $f$ be such that $$ \begin{cases} -\Delta_p f =1 \;\;\mbox{in $B$}\\ f=0 \...
3
votes
0answers
92 views

Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?

Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$. Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$). Suppose we are given a $W^{2,2}$ isometric immersion $...
1
vote
1answer
114 views

Is there any “extra regularity” to the solution to Poisson's equation posed on a 3-dimensional polyhedron?

I am trying to write a proof and I am out of my depth. I need an elliptic regularity result of the form $$ \|u\|_{H^{1+\epsilon}(\Omega)} \le C \|f\|_{L^2(\Omega)} $$ for some $\epsilon >0 $ ...