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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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1answer
33 views

Easy to understand real world example for pde with only weak solutions

After taking a course of ODEs, I began reading about the theory of weak solutions. Without any examples the author claimed that i.e. the function being differentiable twice in the interior of the ...
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28 views

compute $H^{3/2}(\partial\Omega)$-norm for smooth $u$ and $\Omega$

I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple ...
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46 views

Map a random variable to a Gaussian

If I have a random variable $X \in \mathbb{R}^n$, under which conditions is there a $C^1$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\varphi(X) \sim \mathcal{N}(0, I_n)$ (...
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11 views

Boundary behavior of Green’s functions of bounded planar domains

Given a smooth, bounded planar domain $D$, we can define a Green’s function $G_D:D\times D\to\mathbb R$ such that for any $\omega\in L^\infty(D)$, the integral $$ G*\omega(x)=\int_D\! G(x,y)\omega(y)\...
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35 views

$\langle curl \ u, curl \ v \rangle + \langle div \ u, div \ v\rangle=\langle f,v\rangle$ and $\operatorname{div} f=0 \Rightarrow div \ u=0$

We consider a solution $u \in H_0(curl)\cap H(div)$ of $\langle curl \ u, curl \ v \rangle + \langle div \ u, div \ v\rangle=\langle f,v\rangle$ (1). Here $f \in L^2(\Omega), div f=0, i.e. \langle ...
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2answers
54 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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32 views

$\Delta u = f , \operatorname{div} f=0 \Rightarrow \operatorname{div}u=0$ on non convex domain.

I am specifically referring to this paper and why equation (7.7) is the weak formulation of (7.6). My question is why $ \operatorname{div} u=0$ is implied by formulation (7.7) on a general domain. If ...
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1answer
2k views

Density of smooth compactly supported functions in Sobolev space over unbounded domain.

Prove that $C^{\infty}_ {c}(\mathbb{R}^n)$ is dense in $W^{k,p}(U)$ for any open $U\subset \mathbb{R}^n$ with $\partial U\in C^1.$ In which $p\in [1,\infty)$ Note: In Lawrence Evans's PDE text, the ...
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3answers
278 views

Weak solutions to $\Delta u=f$ are in $W^{2,2}$

I believe the following statement is true. Let $\Omega$ be a smoothly, bounded domain in $\mathbb{R}^{n}$. The statement: Let $u\in H^{1}(\Omega)$ so that there exists $f\in L^{2}(\Omega) \;s....
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2answers
239 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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15 views

Regularity of Schrodinger free equation solution

I have proved the following for $f\in {L^2(\mathbb{R})}$ $$ \sup_{x_o\in\mathbb{R},R>0}\left( \frac{1}{R} \int_{-\infty}^\infty \int_{x_0-R}^{x_0+R} \vert D_x^{1/2}e^{it\Delta} f\vert ^2 \ dxdt \...
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1answer
59 views

Solution of the non-linear Heat Equation

How to find $v$ such that $$u(x,t)=t^{-\alpha}v(xt^{-\beta})$$ is the solution of the non-linear Heat equation : $$u_t-\Delta(u^{\gamma})=0$$where $\frac{n-2}{n}<\gamma<1$ , $x$ $\in R^n$ ...
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1answer
19 views

Pointwise bound of the gradient of solutions of heat equations in the half-space.

I want to investigate the decay of $L(x)$: $$L(x) := \int_{\mathbb{R}^3_+} \nabla_x \Phi(x-y,1/2)(\eta(y)g(y))dy,$$ where $g:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is infinitely smooth away from ...
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1answer
25 views

Elliptic regularity for $-\nabla \cdot(a(x)\nabla u)$

Define $Du= -\nabla \cdot (a(x)\nabla u)$ where $a$ is a smooth function which is away from $0$ and bounded. Suppose I have $u \in H^1(\Omega)$ on a smooth bounded domain and I also know that $Du \in ...
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1answer
29 views

weak convergence and compactness

please how can I prove that if a sequence $u_n \to u$ in $L^{\infty}(\mathbb{R}^+; H^1(\Omega)) $ weak * and $\partial_t u_n \to \partial_t u$ in $L^2(0,T, L^2(\Omega)) $ weak for all $T>0$ ...
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27 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}^+,...
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0answers
23 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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2answers
138 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 ...
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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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1answer
421 views

Weak solution for equation $-\Delta u = f$.

Suppose $f \in L^{2}$. I know that $$\left\{\begin{array}{c} −\Delta u = f(x) & \text{on }\Omega \\ u(x)=0 & \text{on } \partial\Omega \end{array}\right.,$$ where $\Omega \subset \mathbb{R}...
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1answer
30 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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0answers
25 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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1answer
22 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
18 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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27 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
136 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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1answer
14 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 ...
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40 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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27 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
28 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 ...
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1answer
86 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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43 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 } ...
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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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1answer
44 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
20 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} \...
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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}...
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1answer
52 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
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
57 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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0answers
16 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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19 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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22 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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1answer
34 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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0answers
32 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
75 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
42 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
29 views

Globally hypoelliptic operator

Are the eigenfunctions of a globally hypoelliptic operator in the Schwarz space $S$.
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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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0answers
37 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}\...