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Questions tagged [regularity-theory-of-pdes]

The concept of regularity concerns the smoothness of weak solutions to partial differential equations.

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32 views

Smoothness of solution to transport equation IVP

Consider the following PDE (the unknown is $u\in C^1(\mathbb{R}^{n+1},\mathbb{R})$): $$\partial_tu(x,t)+\sum_{k=1}^{n} a_k\partial_k u(x,t)= cu(x,t)$$ $$u(x,0)=g(x)$$ where $g \in C(\mathbb{R}...
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1answer
56 views

Continuity of mild solution

This is a question on the proof of Theorem 6.1.2, Pazy's book Semigroups of Linear Operators and Applications to Partial Differential Equations. Also the title might not be too accurate, as my ...
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1answer
46 views

About pseudo-differential operators

Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is ...
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1answer
31 views

Uniform boundedness of weak solution

Let $u_n\in W_{0}^{1,p}(\Omega)$ be a positive weak solution of the equation: $$ -\Delta_p u=\frac{f_n(x)}{(u+\frac{1}{n})^\delta}\text{ in }\Omega. $$ Let $p=N$ and $f\in L^m(\Omega)$ for some $m&...
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45 views

Physical interpretation of a singular pde

Suppose $\Omega$ is a bounded domain in $\mathbb{R}^N$ and consider the following Dirichlet boundary condition: $$ -\Delta u=\frac{f(x)}{u^\delta}\text{ in }\Omega, u>0\text{ in }\Omega; $$ where $\...
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12 views

Example of a function in the weighted Morrey space

Let $1<p<\infty$ and $w\in A_p$ which is the well-known class of Muckenhoupt weights and let $\Omega$ be an open bounded smooth domain in $\mathbb{R}^N$ ($N\geq 2$). For $t>0$ we say that $u$ ...
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13 views

eigenfunctions of an globally hypoelliptic operator

An operator $L$ is said globally hypoelliptic in the Schwartz space $\mathcal{S}(\Bbb{R}^{n})$ if $u\in \mathcal{S}'(\Bbb{R}^{n}), Lu\in \mathcal{S}(\Bbb{R}^{n})\Rightarrow u\in \mathcal{S}(\Bbb{R}^{n}...
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12 views

Solution to weighted $p$-Laplace equation

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$ where $N\geq 2$ and let $f\in L^{\infty}(\Omega)$. Then does there exist $w\in A_p$ (the class of Muckenhoupt weights) such that no solutions ...
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0answers
22 views

Existence of solution to Laplace equation

Given condition: Let $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$. Suppose there exist $w(>0)\in H_{0}^1(\Omega)$ satisfying the inequality $-\Delta w\leq C e^w$ in $\Omega$ for some ...
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1answer
67 views

A function in $W^{2,p}$ for $p>n/2$ is a.e. second differentiable

Let $B$ be the unit ball in $\mathbb{R}^{n}$ and $u\in W^{2,p}(B)$, with $p>\dfrac{n}{2}$. How can we see that $u$ is second differentiable almost everywhere in $B$? This result is claimed in Page ...
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1answer
29 views

Subsolution of Laplace equation

Let $\Omega$ be a bounded open subset of $\mathbb{R}^2$ and $w(>0)\in H_{0}^1(\Omega)$ satisfies the equation $$ -\Delta w\leq e^w\text{ in }\Omega. $$ Let $v(>0)\in H_{0}^1(\Omega)$ satisfies ...
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26 views

Globally hypoelliptic operator

Are the eigenfunctions of a globally hypoelliptic operator in the Schwarz space $S$.
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36 views

estimating the gradient of a harmonic function on a ball

Given $u$ a harmonic function on a ball of radius $r$. i.e. $$ -\Delta u=0 \qquad{\text{in $B_r(0)$}} $$ Then show that $$ |\nabla u(0)|\leq C \frac{1}{r}\def\avint{\mathop{\,\rlap{-}\!\!\int}\...
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46 views

Distributional second-order derivatives of $\frac{e^{-|x|}}{4\pi |x|}$ to show the solution of $u -\Delta u=f$ is in $H^2$

In Brezis's book "Functional Anlaysis" it is proven that the solutions of the Helmotz equation $u - \Delta u=f$ where $f \in L^2 (\mathbb{\Omega})$ belong to $H^2 (\mathbb{\Omega}) \cap H^1 _0 (\Omega)...
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1answer
40 views

completion of $C^\infty_0(D)$ w.r.t $\|\cdot\|_\nabla$

Let $D$ be an unbounded domain in $\mathbb{R}^n$. Consider the set $C^\infty_c(D)$ with two different norms: $\|\cdot\|_\nabla$ and $\|\cdot\|_\nabla + \|\cdot\|_{L^2}$. It is known that when $D$ is ...
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1answer
31 views

How can a discontinuous function belong to $C_B^1(\Omega)$, the space of continuous functions $u$ with bounded derivatives?

Let $\Omega = \{(x,y) \in \mathbb{R}^2 \ : \ 0 < |x| < 1, \ 0 < y < 1\}$ and consider the function $u$ defined on $\Omega$ by (Sobolev Spaces by Adams, page 68, Example 3.10) $$ u(x,y) = \...
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42 views

Is this PDE (Poisson/Laplace equation) well-posed considering I have a very degenerate domain (picture included)?

In case (1) in the following picture we have the standard interior Poisson equation in 2D with Neumann boundary conditions on some smooth domain $\Omega$, subject to a point source at position $y$. I ...
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1answer
36 views

Question on Sobolev extension onto boundary

Let $U \subset \mathbb R^3$ be an open, bounded and connected set with a $C^2-$regular boundary $\partial U$. I'm trying to understand the following implication: If $f\in W^{2-1/2,2}(U)$ then $f{\...
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36 views

Estimate on average with weight

Let $B_{2R}=B(x,2R)$ be the ball of radius $2R$ centered at $x$ and $v=log\,u$ for some positive function $u$ defined on $B_{2R}$. Denote by $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
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23 views

Question on optimal regularity for the elliptic Neumann problem

I 'm reading a paper at the moment and I have a really hard time understanding the following: Let $U \subset \mathbb R^3$ be open, bounded and connected with a $C^2-$ regular boundary $\partial U$...
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28 views

Regularity of the heat equation

I'd like to prove this lemma since this lemma asserts the regularity of the heat equation by using the cut-off function and mollification. Let $\Omega\subset\mathbb{R}^n$. Define $\Omega_T=\Omega\...
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1answer
23 views

Cut-off functions in Caccioppoli's inequality

Caccioppoli's inequality states that the solution $u$ of the equation $-\nabla\cdot(A\nabla u)=0$ in some bounded domain $\Omega$ satisfies $$\int_{B(0,\rho)}|\nabla u|^2dy\leq \frac{C}{(R-\rho)^2}\...
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1answer
33 views

Higher regularity for solutions of elliptic equations

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$. Let $f\in L^\infty(\Omega)$. For the problem $$-\Delta u=f\mbox{ in }\Omega\\ ~~~~~~~~~u=0\mbox{ on }\partial\Omega,$$ one could seek solutions in ...
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0answers
23 views

Inequality for the Laplace operator

Let $\Omega\subset\mathbb{R}^N$ with $N\geq 2$ be a smooth bounded domain. Define for $k>0$, $\gamma>0$, the real valued function $g_k(s):=min\{s^{-\gamma},k\}$ for $s>0$ and equals $k$ for ...
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1answer
35 views

Equivalence of norms in the space $H_\Delta(\Omega)$

Let $\Omega \subset \mathbb{R}^n$ is an open bounded domain with smooth boundary $\Gamma$. Consider the following space $$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$ with the ...
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0answers
30 views

Solution of p-Laplace equation

If $u$ is a solution of $\Delta_p u=0$ weakly, then $u^{+}$ is also satisfies $\Delta_p v=0$ weakly. To prove this result, I need to prove that $$ \int_{\Omega}|\nabla v|^{p-2}\nabla v.\nabla \phi\,...
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13 views

Boundary continuity of p-Laplace equation

For a nonnegative function $f\in L^1(\Omega)$, where $\Omega$ is abounded smooth domain in $\mathbb{R}^N$, consider for $p=N$, the p-Laplace equation $-\Delta_p u=f$ in $\Omega$ such that $u\in W_{0}^...
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26 views

Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put ...
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16 views

Extension of Euclidean distance

I want to know whether Carnot-Caretheodory distance is an extension of the Euclidean distance in $\mathbb{R}^N$ or not? Please help me with an explanation. Thanking You.
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1answer
39 views

Boundary regularity for the p-Laplace equation

Let $f\in L^m(\Omega)$ for some $m>1$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ ($N\geq 2$). Consider the equation, $$ \Delta_p u=f(x) $$for $p=N$, then $u$ is bounded in $\Omega$...
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12 views

Regularity for Poisson equation in Sobolev spaces

Let $U \subset \mathbb R^3$ an open,bounded and connected set with $C^2-$regular boundary $\partial U$. Let: $f\in \mathcal M_{+}(\partial U)$ be a non-negative measure $g$ be a continuous ...
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1answer
67 views

Minty-Browder theorem

Can you please give a reference where I can get the exact proof of the Minty-Browder theorem stated as follows: Observe that I below the operator $T$ is demicontinuous and need not be continuous. For ...
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22 views

Truncation function

Define the truncation function $$ T_k(s)= \begin{cases} s&\text{if }|s|\leq k\\ \\ k\dfrac{s}{|s|}&\text{if }|s|\geq k. \end{cases}$$ Let $w\in A_p$ the class of Muckenhoupt weights and $u_n,...
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1answer
40 views

Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
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1answer
49 views

An inequality for Sobolev functions

Let $\Omega$ be a smooth bounded domain and $f\in L^p(\Omega)$, $u\in W^{1,1}(\Omega)$ such that $p\geq 1$. Then $$ \int_{\Omega}|fu|\,dx\leq C\,\int_{\omega}\{|f(x)|dx(\int_{\Omega}\frac{|\nabla u(\...
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55 views

Uniqueness for classical solution of PDE

I have the following conservation law in my hand: $\partial_{t} u + \partial_{x}f(u) = -u$, with the associated initial data $u(x,0) = u_{0}(x)$ - where $u_{0} \in C^{1}$. I have to show ...
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2answers
160 views

Need good reference or a proof on regularity of solution to 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 ...
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21 views

For what right-hand side admits the Neumann problem a solution?

I discussed the following question with some of my collegues over lunch today and we didn't come up with a good answer. I'm sorry for the long question! Question: Let $\Omega \subset \mathbb{R}^n$ be ...
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1answer
43 views

Higher differentiability of weak solutions to 2nd order elliptic PDEs with mixed boundary conditions

I am interested in regularity results for 2nd order elliptic PDEs with mixed boundary conditions like $$\left\{\begin{array}{rl}-\text{div}(a\nabla u) =& f &\text{in }\Omega, \\ u=&\...
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45 views

Smoothness of closed geodesics

I'm trying to understand a paper about minimal surfaces and I came across the following problem: let $N$ be a smooth n-dimensional manifold embedded in an Euclidean space. Let $\gamma$ be a $C^1$ map ...
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44 views

Estimates of fundamental solution of heat equation in Sobolev space

Suppose we consider the heat equation in 2D: $u_t - \Delta u=0$ with initial value $u_0$. Then how to estimate the fundamental solution $u=e^{t\Delta}u_{0}$? More precisely, for $0<s<1$, if $u_{...
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0answers
24 views

Is this an hypoelliptic operator?

$$Lu = -\Delta u + \sum\limits_{i = 1}^na_iu(x_i)$$ Is this an elliptic operator? According to definition, it doesn't seem to fit in. Is this an hypoelliptic operator?
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76 views

Elliptic regularity of harmonic forms in $L^1$

$\newcommand{\M}{M}$ Let $\M$ be a smooth oriented Riemannian manifold. Let $\sigma$ be a differential $k$-form on $\M$ with coefficients in $L^1(\M)$. We say $\sigma$ is weakly harmonic if $$ \...
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0answers
15 views

$C^1$-regularity of the solutions of 1st order linear homogeneous PDEs with constant coefficients

Apologies if this is too elementary, I am largely ignorant on the subject of PDEs. Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$, let $n,m\geq 2$, let $U \subseteq \mathbb{K}^n$ be a ...
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39 views

Heat equation and smoothness effect

Consider the homogeneous heat system with Dirichlet B.C \begin{array}{c} u_{t}-u_{xx}=0 \\ u(0,t)=u(\pi ,t)=0 \\ u(x,0)=f(x).% \end{array} It is well known that the solution of the system above is ...
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1answer
45 views

Does a continuous function in $W^{1,p}$ for $p<n$ lie in $W^{1,n}$?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain. Suppose that $f \in W^{1,p}(\Omega)$ for some specific $1<p<n$, and that $f$ is continuous. Is it true that $f \in W^{1,n}_{loc}(\...
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37 views

negative outward normal derivative implies positiveness near the boundary.

So, this seens to be simple but I could not handle to prove it: Suppose $\Omega$ is a bounded and open domain of $\mathbb R^N$ with smooth boundary (as smooth as you want), $N\geq 2$ and that $u\in C^...
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1answer
121 views

Reference complementing Hairer's “Introduction to Stochastic PDEs”

I want to study stochastic partial differential equations, and the standard reference seems to be Hairer's notes - after all, he got the first Fields Medal for work on this topic. However, I also ...
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1answer
90 views

Is the composition of a Sobolev function and a smooth function Sobolev?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain, and let $1<p<n$. Suppose that $f \in W^{1,p}(\Omega)$ is continuous*, and $g \in C^{\infty}(\mathbb{R})$. Is it true that $g \...
2
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0answers
166 views

Gradient Blowup for a Parabolic (Heat) Equation

Let $u(x,t)$ be a solution to the following parabolic PDE: With $\alpha \in (0,1)$, \begin{align} \partial_t u(x,t) &= \alpha (1-t)^{\alpha - 1} \partial_x u(x,t) + \frac{1}{2} \partial_{xx} u(x,...