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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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42 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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1answer
134 views

Is a Sobolev map with smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $d>2$. Let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$ where $\Omega$ is an open subset of $\mathbb{R}^d$. Let $2 \le k \le d-1$ be fixed. Suppose that $\det df>0$...
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1answer
752 views

Interior $H^2$ regularity - applying “Cauchy's inequality with $\epsilon$”

This is from PDE Evans, 2nd edition, pages 327, 328, and 330. I have a question regarding one piece of the proof. The theorem concerned is THEOREM 1 (Interior $H^2$ regularity) which is stated on ...
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1answer
176 views

Proving weak coercivity by young's and interpolation inequalities

Let be $$(P)\left\{\begin{array}{ll} &-\Delta u + V(x)u=f & \text{ in }\ \Omega\\ &u=0 & \text{ on } \ \Gamma \end{array}\right.$$ with $V \in L^r(\Omega)$, for some $1<r<\infty$....
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1answer
122 views

Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim brezis, and I am a bit confused about the proof. The theorem is stated as: Let $\Omega \subset \mathbb{R}...
2
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1answer
116 views

Scaling the NS equation: supercriticality and energy estimate

There is a ‘rescaling transformation’ that is particularly significant for the Navier–Stokes equations when they are posed on the whole space, but is also important in the local regularity theory. ...
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2answers
190 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
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1answer
197 views

Question regarding Evan's proof of Global Approximation by $C^∞(\overline{U})$ functions

The page where the proof is is on Google Books. A picture of the statement and the proof. I reproduce the statement of the result: Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. ...
7
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2answers
6k views

cutoff function vs mollifiers

Q1. What are cutoff function? What are mollifiers? I cannot distinguish the two. Could anyone give some concrete/simple examples of cutoff functions? And how they differ from mollifiers I did check ...
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1answer
172 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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2answers
744 views

Proving the range of operator is closed

I have a hard time understand (2) of the Fredholm alternative in Evan's Appendix. To prove the image of $I-K$ is closed, what result from functional analysis is used? I am lost in understand the ...
2
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1answer
390 views

Compact embedding into boundary

I was looking for a result: $W^{1,2}(Q)$ is compactly embedded in $L^{2}(\partial Q)$ ; where Q $\subset \mathbb R^{2}$ is a bounded domain & $Q \in C^{1,1}$." (which is mentioned in the book: ...
5
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1answer
290 views

Need source for elliptic regularity on unbounded domains

I need a source that provides a $W^{2,2}$-regularity result for linear elliptic systems on unbounded domains. To be more specific, I study the equation $$ -\Delta w - ic \partial_1 w + \left( \frac{c^...
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0answers
151 views

Question about equivalent norms on $W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$.

Assume $\mathbb{‎‎H}=W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ with the norm induced from inner product $$\langle u,v\rangle_{‎\mathbb{‎‎H}}=\int_\Omega \Delta u \,\Delta v\, dx$$ for any $u \in W^{2,2}(...
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1answer
174 views

Equivalence between norms in $H_0^1(\Omega)\cap H^2(\Omega)$.

Based in many questions and answers like [1, 2, 3 ] and a comment a good comment here [4]. I would like to know that the space $H=H_0^1(\Omega)\cap H^2(\Omega)$ can be equiped with this norm $$\tag{...
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1answer
1k views

uniqueness heat equation

Consider the heat equation, $(1)$ $u_t=u_{xx}+f(x,t)$, $0<x<1$, $t>0$ $(2)$ $u(x,0)=\phi(x)$ $(3)$ $u(0,t)=g(t)$, $u(1,t)=h(t)$ When one wants to Show the uniqueness of solution of ...
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0answers
150 views

Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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0answers
47 views

Faedo-Galerkin method for the 1-d von-karman system

I want to prove the existence of a solution to the following system $$\eqalign{ & {u_{tt}}-{u_{ttxx}}+{u_{xxxx}}-{({u_x}({v_x} + \frac{1}{2}u_x^2))_x}=0 \cr & {v_{tt}}- {({v_x} + \frac{...
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140 views

Conservation Laws: Difference and Reasonability Weak solutions, Integral solutions, distributional solutions

Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&...
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1answer
141 views

Effective Boundary Condition for a Heat Equation with Variable Conductivity

Consider a heat equation in one space dimension $$\frac{\partial u(t,x)}{\partial t} = \frac12\Theta(x)\frac{\partial^2u(t,x)}{\partial x^2} \tag{1}$$ where Heavyside function $$ \Theta(x) = \begin{...
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1answer
349 views

Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f''$ ...
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0answers
60 views

Nonnegative solution of a PDE with zeroth order term

Given the PDE, $x_t - \triangle x + c(m,t)x = 0$, on a bounded domain $\omega$ where $c(m,t)$ is a smooth function (bounded but can be negative). Show that if $x_{0}(m) = x(m,0)\geq 0$, then $x(m,t)\...
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1answer
242 views

Properties of the solution of the schrodinger equation

I'm considering the Free Schrodinger equation: $$ i\partial_{t}u+\Delta{u}=0,~\mathbb{R}\times\mathbb{R}$$ with $$ u(0,\cdot)=u_{0} \in C^{\infty}_{0}(\mathbb{R}).$$ The solution is given by: $$ u(t,x)...
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1answer
313 views

Does Green's (first) identity hold for Weak Derivatives?

Recall Green's First Identity: $$\int_{\Omega}v \Delta u \ (d\Omega) =\int_{\partial \Omega}v (\nabla u )\vec{n} \ d (\partial \Omega) - \int_{\Omega} \nabla u \nabla v \ (d \Omega)$$ Which ...
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1answer
412 views

Laplacian as a Fredholm operator

Let $\Omega$ be a bounded smooth domain in $\mathbb R^n$. The Laplacian $\Delta$ acts on functions on $\Omega$. From elliptic regularity (I haven't worked out all the details), we have that $$ \...
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1answer
75 views

Boundedness of solutions for the Laplacian

A solution to the equation $-\Delta u+u=f$ for $f\in L^2(\mathbb R^n)$ belongs in $H^2(\mathbb R^n)$. Is it possible to obtain a solution in $H^2\cap L^\infty(\mathbb R^n)$ if $f\in L^2\cap L^\infty(\...
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2answers
710 views

In what conditions a weak solution is a classical solution?

I'm studying elliptic equations in divergence form $$-D_{j}(a_{ij}(x)D_{i}u) + c(x)u = f(x) \text { in a domain } \Omega \subset \mathbb{R}^{n}$$ I call a function $u \in H^{1}(\Omega)$ a weak ...
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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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0answers
72 views

Regularity of the mild solution of a semilinear PDE

Let $H$ be a separable $\mathbb R$-Hilbert space $(\mathcal D(A),A)$ be a linear operator on $H$ $(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ be an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;...
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0answers
28 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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0answers
232 views

The integral of mollifier function is $1 $

\begin{eqnarray} u(x) = \begin{cases} e^{-\frac{1}{1-\|x\|^2}}& \text{ if } \|x\| < 1\\ 0& \text{ if } \|x\|\geq 1 \end{cases} \end{eqnarray} I am trying ...
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1answer
95 views

Evans PDE p.714 Change of variable and change of integration region

In the following definition of convolution involving mollification . When I make change of variable $x-y=z$, I have $$\int_{U}\eta_{\epsilon}(x-y)f(y)dy=\int_{?}\eta_{\epsilon}(z)f(x-z)dz=\int_{?}\...
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1answer
55 views

Counterexample of Sobolev traces

Does anyone know a counterexample to the following fact: let $\Omega$ be a smooth domain in $\mathbf{R}^n$. Then $T(W^{2,2}) \subset C(\partial \Omega)$. I admit that i do not know much about trace ...
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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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0answers
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
72 views

Unicity of solution for a parabolic problem?

How can I show that the parabolic problem $$ \begin{cases} \partial_xu- \Delta u=0 & \mathbb{R} \times (0,+ \infty)\\ u(x,0)=f(x) & \mathbb{R} \end{cases} $$ has a unique solution? Can I ...
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1answer
21 views

Positivity of a normal derivative of $C^2$ function

Let $\Omega$ be an open set of $R^n$ of classe $C^2$, let $f \in {C^2}({\bar \Omega } )$ which satisfies the following properties: $f>0$ on $\Omega$ and $|\nabla f|>0$ on $\bar \Omega$. and $f=...
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1answer
147 views

analytic semigroups and norm continuous semigroups

Are every analytic semigroups norm continuous? Are there counterexamples otherwise? What would make analytic semigroup norm continuous as well? (My apology, frankly, I am not sure if norm continuous ...