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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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22 views

Weak convergence and compacity

Please I dont understand this. I have: $ \parallel \nabla m_n \parallel_{L^{\infty}(\mathbb{R}^+, L^2(\Omega))}\leq C$ $ \parallel \frac{\partial m_n}{\partial z} \parallel_{L^{\infty}(\mathbb{R}^+,...
5
votes
2answers
102 views

$\Delta u=3u$ then $u\equiv0$

I have the following question in which it is easy to use Fourier transform to get the answer if the function is nice enough, for example $u\in C_{0}^{\infty}(\mathbb{R}^{n})$, however here $u$ is only ...
1
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0answers
20 views

Global elliptic regularity theory on $\mathbb{R}^n$ or interior estimates for elliptic pde when $p \neq 2$

Let $p \in (1.\infty)$ and $a_{ij},b_j,c \in C^\infty(\mathbb{R}^N)$ be the bounded coefficients of the elliptic second order differential operator $$[Au](x) = - \mathrm{div}(A(x) \nabla u) + \left\...
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1answer
16 views

Support of a regularized function

Let $f$ be a function such that $supp(f)=K$. Compute $supp(f_\varepsilon)$ where $f_\varepsilon$ is the regularization of $f$. Im not sure how to do this, since we have no information of the support ...
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0answers
23 views

Existence of the solution of the 3D Micropolar equations [closed]

Please how to show the local existence for the solution of the 3D micropolar equations in a Besov space setting ? $\left\{ \begin{array}{l} \partial_tu-(\nu+k)\Delta u-2k\nabla\times w+u\nabla u+\...
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vote
1answer
29 views

Norm estimate of $(-\lambda A+ I)^{-1}$ for strictly elliptic operator

Let $\Omega$ be a smooth domain in $\mathbb{R}^n$, and $A$ be a strictly elliptic operator $$ Au=\partial_i(a^{ij}(x)\partial_j)u, $$ where $a^{ij}$ are bounded functions satisfying $$ a^{ij}(x)\...
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1answer
15 views

Harnack inequality for linear parabolic equations

I am trying to understand the proof of the Harnack inequality using the ideas of J.Nash as proved in the paper "A new proof of Moser's Parabolic Harnack Inequality using the old Ideas of Nash" by E.B....
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0answers
17 views

Justification of naive proof to weak existence of PDE

We often get a priori estimates on a given PDE and use its idea to construct and regularize weak solution. For example, Let $u$ be a smooth solution of heat equation $u_t-\Delta u=0$. $$0=\int u\...
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0answers
20 views

Infinite differentiability for a solution of the general linear parabolic pde of second order

I'm studying by myself the the chapter of second order parabolic linear equations by Evan's book, which focus in solve $$(11) \ \begin{cases} \begin{eqnarray*} u_t + Lu &=& f \ \...
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0answers
25 views

Question on parabolic smoothing of nonhomogeneous heat equation

Suppose $\Omega $ is a bounded domain in $\mathbb R^n$ and let $u$ be the weak solution of the initial value problem for the nonhomogeneous heat equation: $\begin{cases} \partial_t u-\Delta u=f,\;\...
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1answer
13 views

Regularity of linear pde with smooth coefficients

Consider $au_x+bu_y+cu_z=f$ on $\mathbb{T}^3$ where $a,b,c,f$ are in $C^\infty$ and $\forall (x,y,z)\in\mathbb{T}^3,|a|,|b|,|c|>1$. If there exists $C^1$ solution to this pde, can we say that it ...
2
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0answers
36 views

$ L^p$ regularity for weak solutions of non homogeneous heat equation

Let $U$ be a compact $C^2-$manifold and suppose $v:U\times (0,T) \to \mathbb R$ is the weak solution of: $\partial_t v= \Delta v+f$ where $f\in L^{\infty}(U\times (0,T))$ I 'm interested in the $L^...
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1answer
30 views

Function in H(curl) $\cap$ H(div), but not in H1

it is well known, that for a non-convex domain $\Omega$ the space $H^1(\Omega, \mathbb{R}²)$ is a proper subset of $H(curl) \cap H(div)$. Here, $H(curl) = \{v \in L²(\Omega)², \nabla\times v = \...
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0answers
24 views

Approximation of an element in the dual of the Sobolev Space

Let $F\in L^2(\Omega)$ be such that $-\text{div}F\in W^{-1,2}(\Omega)$ (Dual of the Sobolev Space $W_0^{1,2}(\Omega)$) be non-negative where $\Omega$ is a bounded domain in $\mathbb{R}^N$. My ...
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0answers
26 views

Splitting a PDE into two subdomains: Regularity conditions at the interface point(s)

My aim is to split a given problem into two nonoverlapping subdomains. Let us, for example, consider the initial-boundary value problem: Given $u_0 \in L_2(0,1)$ and $f \in L_2(0,1)$, find $u$ such ...
2
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0answers
42 views

$u'\in L^{\infty}(0,T;L^{2}(U))\cap L^{2}(0,T;H_{0}^{1}(U))$ regularity in the semigroup approach

Evans' PDE book 2nd, (Theorem 7.1.5) says for parabolic PDEs, if the initial condition is in $H^1_0 \cap H^2$, and the source term satisfies the regularity $\mathbf { f }, \mathbf { f } ^ { \prime } ...
6
votes
1answer
134 views

Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
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0answers
28 views

Is the divergence free condition really necessary?

In the paper "Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation" (available at https://web.math.princeton.edu/~const/cwhigh11907.pdf), the condition $\nabla\...
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0answers
18 views

What does it really mean for a wave equation to be critical?

I am trying to understand intuitively the concept of criticality in general for Wave equations. For example, consider the cauchy problem of semi-linear equation \begin{equation} \begin{cases} \...
1
vote
1answer
40 views

How to show that the mild solution to this parabolic equation is also a classical solution?

Let $U\subseteq \mathbb{R}^n$ be a smooth bounded domain and consider the following problem: $$ \begin{cases} \partial_t u = \Delta u + u &\text{in } U\times (0, \infty)\\ u = 0 &\text{on } \...
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0answers
39 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}...
3
votes
1answer
85 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 ...
2
votes
1answer
50 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 ...
1
vote
1answer
37 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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0answers
51 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 $\...
2
votes
0answers
14 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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0answers
18 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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0answers
18 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 ...
1
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0answers
28 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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vote
1answer
70 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 ...
2
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1answer
38 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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0answers
27 views

Globally hypoelliptic operator

Are the eigenfunctions of a globally hypoelliptic operator in the Schwarz space $S$.
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0answers
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}\...
2
votes
0answers
50 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
45 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
36 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) = \...
1
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0answers
49 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 ...
3
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1answer
40 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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0answers
39 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 $v_{B_{2R}}=\frac{1}{w(B_{2R})}\int_{B_{2R}}v(x)w(x)\,dx$, ...
3
votes
0answers
25 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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0answers
55 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\...
0
votes
1answer
30 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}\...
2
votes
1answer
40 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
25 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 ...
1
vote
1answer
52 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 ...
1
vote
0answers
34 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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votes
0answers
16 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}^...
1
vote
0answers
30 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 ...
0
votes
0answers
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.
0
votes
1answer
42 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 $\Omega$...