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10 votes
0 answers
422 views

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 \...
Akira's user avatar
  • 835
8 votes
0 answers
115 views

optimal regularity for the Neumann heat equation on Lipschitz domains

$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
leo monsaingeon's user avatar
8 votes
0 answers
177 views

Understanding spaces of negative regularity

I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here. Let $C^k(\mathbb{R}^n$) be the ...
CBBAM's user avatar
  • 721
6 votes
1 answer
696 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
leo monsaingeon's user avatar
6 votes
1 answer
245 views

The sharpest regularity result of elliptic PDEs: conditions on the variable coefficients

Let $\Omega \subset \mathbb{R}^n$ be open and bounded with a sufficiently smooth boundary. Let $L$ be a second order differential operator with variable coefficients, given by $$Lu = \partial_i(a^{ij}...
MathsGoose's user avatar
5 votes
0 answers
213 views

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 ...
G. Blaickner's user avatar
  • 1,429
4 votes
1 answer
442 views

Need a regularity result for parabolic PDE, want $u' \in L^\infty((0,T)\times \Omega)$

Let us assume $\Omega \subset \mathbb{R}^n$ is as nice as required. Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b < \...
student's user avatar
  • 183
4 votes
0 answers
194 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 $...
soup's user avatar
  • 307
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 ...
leo monsaingeon's user avatar
3 votes
1 answer
540 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 $\epsilon>0$, ...
user143090's user avatar
3 votes
1 answer
845 views

Moser estimates?

Consider $u$, an $L^2$ solution to the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on a ball $B_1$ of radius 1 centered at $(t_0, x_0)$, say, where $t$ can be treated as a "time" variable. I ...
student's user avatar
  • 31
3 votes
0 answers
125 views

Partial regularity for transmission problem in corner domains

Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
PeteAgor's user avatar
  • 143
2 votes
1 answer
75 views

How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$

Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$. Assume ...
Arghya kundu's user avatar
2 votes
1 answer
102 views

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-...
alia's user avatar
  • 23
2 votes
0 answers
189 views

Smoothing property of the heat kernel on the one-dimensional torus

Let $G=G(x,t)$ be the heat kernel on the one-dimensional torus $\mathbb{T}^1,$ with $x \in \mathbb{T}^1$ and $t \in (0,T].$ $G$ is given by \begin{equation} G(x,t) = (4 \pi t)^{-1/2} \sum_{k \in \...
kumquat's user avatar
  • 185
2 votes
0 answers
83 views

3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$

Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$. Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
Himanshu Garg's user avatar
2 votes
0 answers
111 views

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 ...
Akira's user avatar
  • 835
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 ...
cork_twist's user avatar
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 $$...
Fozz's user avatar
  • 287
1 vote
1 answer
250 views

Moser/Schauder estimates for coercive boundary conditions

Consider the uniformly elliptic equation $(\partial_t^2 + L)u = 0$ on $(0, \infty) \times \Omega$, where $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary, and $L$ is a ...
user95844's user avatar
1 vote
0 answers
91 views

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}\...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
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 }\...
Curious student's user avatar
1 vote
0 answers
251 views

Regularity of a Fokker-Planck PDE with unbounded coefficient

Let $A$ be a positive definite symmetric matrix, let $b\in C^1(\mathbb R^d\!\times\!(0,\infty))\cap C(\mathbb R^d\!\times\![0,\infty))$ taking values in $\mathbb R^d$. Consider the parabolic PDE $$ \...
tituf's user avatar
  • 311
1 vote
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 +...
Marc Vaisband's user avatar
0 votes
1 answer
417 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 ...
Curious student's user avatar