Articles of weak derivatives

function a.e. differentiable and it's weak derivative

Note – I am just starting to learn about theory of distributions, so this may be a trivial question, if so I’d be grateful for a reference, nevertheless the question is the following: suppose I have a function $f$ such that $f \in L^1$ and $f$ is differentiable almost everywhere (in the strong sense) and […]

Does convergence in H1 imply pointwise convergence?

I’m trying to figure out if convergence in $H^1(a,b)$ implies pointwise convergence (by the way: what is the usual name of this space?). It is defined to be Hilbert space of absolutely continuous functions on some (possibly infinite) interval such that both function and its (weak) derivative are square integrable. Scalar product in this space […]

Definition of weak time derivative

My quesion involves the weak time derivative. In the book: ‘Partial Differential Equations’ by Evans the time derivative $u’$ of a function $u: [0,T] \rightarrow H^1_0(U)$ is defined by an element $u_t \in L^2(0,T;H^{-1}(U))$ such that $$ \forall \phi \in C_0^{\infty}([0,T]): \int_{[0,T]} u \phi’ dt = -\int_{[0,T]} u’ \phi dt $$ Where $u’\phi$ and $u […]

How do I show that a distribution is locally p-integrable?

Let $\Omega \subseteq \mathbb{R}^n$ (open) and $u \in \mathcal{D}’$ be a distribution, that has a distributional derivative which is in $L^p(\Omega)$ (for some $p \geq 1$). Show that $u \in L^p_{loc}(\Omega)$ …where $L^p(\Omega)$ is defined as in https://en.wikipedia.org/wiki/Lp_space#Lp_spaces and $L^p_{loc}(\Omega)$ is defined as in: https://en.wikipedia.org/wiki/Locally_integrable_function#Generalization:_locally_p-integrable_functions My thoughts: 1) For notational simplicity let’s call $Du = […]

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are also other types of generalizations that change the space like Frechet or Gateaux derivatives in Banach spaces.) One is that we take any […]

Why is it useful to show the existence and uniqueness of solution for a PDE?

Don’t get me wrong, I understand that it is important in mathematics to qualitatively study the problems given. But I would like to know to what extent this helps for example, to actually solve the problem. I am reading books that deal with variational approach for elliptic PDEs like the Laplacian. Apart from transforming the […]

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 $\Omega \subset \mathbb{R}^n$ be an open set of class $C^2$ with $\partial \Omega$ bounded. Let $f \in L^2(\Omega)$ and $u \in H_0^1(\Omega)$ satisfies the weak formulation […]

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$?

How to prove if $u\in W^{1,p}$, then $|u|\in W^{1,p}$? Since $|u|\in L_p$, I only need to show weak derivative of $|u|$ exists and $D|u| \in L_p$. Can anyone give me some hint? Thanks!

$C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book I found that in order to prove it, we need the following claim: Claim: Let $\displaystyle{\zeta \in C^{\infty} ([0, \infty)) }$ such that $\zeta (t) =1 […]

Weak derivative zero implies constant function

Let $u\in W^{1,p}(U)$ such that $Du=0$ a.e. on $U$. I have to prove that $u$ is constant a.e. on $U$. Take $(\rho_{\varepsilon})_{\varepsilon>0}$ mollifiers. I know that $D(u\ast\rho_{\varepsilon})=Du\ast\rho_{\varepsilon}$, so $u\ast\rho_{\varepsilon}(x)=c $ for every $x\in U$, since it is a smooth function. How can I conclude?