Questions tagged [weak-derivatives]

For question about weak derivatives, a notion which extends the classical notion of derivative and allows us to consider derivatives of distributions rather than functions.

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How to prove that extensions of Sobolev functions if Sobolev?

Problem: Let us introduce this definition: Definition: given an open set $\Omega \subset \mathbb{R}^d$ and $u \in L_{loc}^1(\Omega)$ we say $v \in L_{loc}^1(\Omega)$ is $\frac{\partial}{\partial x_i}u$...
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Do the sum of some function belong to specific spaces

Consider the following function $f(x)$ on $[a,c]$ for $a,c \in \mathbb{R}$ \begin{cases} h (x) &a< x<b \\ g(x) & b\leq x<c \end{cases} Assume h and g are $C^1$ on ...
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Strong and weak derivatives

Let $\Omega \subset \Bbb{R}^n$ be a connected open subset with $\partial \Omega$ Lipschitz. If a function $u$ is infinitely weakly differentiable within $\Omega$ wrt Lebesgue measure $\lambda$, i.e. ...
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If $g_n\in C^\infty_c((0,\infty)\times (0,1))\to\mathbf{1}_{(-\infty,T]}(t) e^{-x}$ strongly in $L^1$, what happens to $\partial_tg_n, \partial_xg_n$?

Let $g_n \in C^\infty_c((0,T)\times (0,1))$. Suppose that the sequence $g_n$ converges strongly in $L^1$ to $\mathbf{1}_{(-\infty,T]}(t) e^{-x}$. What do $\partial_t g_n$ and $\partial_x g_n$ converge ...
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Find a weak solution of the ODE

Find a weak solution to the following ODE: $u' + u = H_0(x)$ where $H_0(x) = \begin{cases} 0 & x < 0 \\ 1 & x \geq 0 \end{cases}$ My professor advised that we try to guess the solution and ...
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Find weak derivative of sign-like function

Let $f:\mathbb{R}^2\to \mathbb{R}$ be a function defined as follows. $$f(x,y)=\begin{cases} 1, &\text{if } x>y\\ -1,&\text{if } x<y. \end{cases}$$ To compute the weak derivative $f_x$ of ...
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Convergence of Galerkin methods with shape functions satisfying boundary conditions

I have seen multiple references here in Math Stack Exchange and online to solution of PDEs using functions that satisfy a given set of boundary conditions. However, the convergence of those, is rarely ...
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Check if a function is in H2

I want to check if the Function $u(x)=|x-1|$ is in $H^{2}(\Omega) , \Omega = (0,2)$ My idea: I want to check if the function has a weak derivative. But that would only proof $L^{1}$ what should I do ...
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Showing that $u(x,y)=xy\ln|\ln\sqrt{x^2+y^2}|$ belongs to $H^2$, but not to $C^2$.

I want to show that that $u(x,y)=xy\ln|\ln\sqrt{x^2+y^2}|$ is $C^1(\Omega)$ and $H^2(\Omega)$, but not $C^2(\Omega)$, where $\Omega=B_{\frac{1}{2}}(0)$ is the ball centered at $0$ and with radius $1/2$...
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Weak derivative for sobolev spaces in manifolds and vector bundles

In these lecture notes https://www3.nd.edu/~lnicolae/Lectures.pdf I read how one can define sobolev spaces between a riemannian manifold and a vector bundle over it. First I don't understand the ...
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Let $\Omega\subset \mathbb{R}^n$ be open and bounded. Is it true that $C^{0,1}(\Omega) \subset W^{1,\infty}(\Omega)$?

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. If $\partial \Omega \subset C^1$, the result follows from Theorem 4 on page 279 of "Partial Differential Equations" by Evans. Without ...
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Approximate $f\in \text{Lip}_K(\Omega)$ with $f\geq 0,f=0$ on $\partial \Omega$ by $f_n\in C_c(\Omega)\cap \text{Lip}_K(\Omega)$ with $f_n\geq 0$.

Let $\Omega\subset \mathbb{R}^n$ be open and bounded. For $0<K\leq \infty$, $\text{Lip}_K(\Omega)$ denotes the set of lipschitz-continuous functions on $\Omega$ with lipschitzconstant less than or ...
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Why do we write $H_0^1(\Omega) \cap H^2$ instead of only $H^2_0(\Omega)$?

I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain $\Omega$, we write that the regularity of $u$ is $H^2 \cap H_0^1(\Omega)$. ...
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Sobolev space of maps with higher dimensional target?

Consider the space $W^{2,2}(\Omega)$ where $\Omega = [0,1]^3$. This space contains continuous functions of the form $f:\Omega \to \mathbb{R}$. I am curious, can a Sobolev space be constructed by maps ...
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Can I extend a weak derivative to the whole $\mathbb R$?

Let $\rho(x,t)>0$ be a Holder-continuous function on $\mathbb R^d\times(0,\infty)$. I know that: there exists (in weak sense) $\partial_t\rho\in L^1_\text{loc}(\mathbb R^d\times(0,\infty))$; there ...
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Show that D'Alembert solution gives distributional solution to the wave-equation

I am given the following question: Let $\phi, \psi\in C^0(\mathbb{R})$ and $$u(x,t)=\frac{1}{2}(\phi(x+t)+\phi(x-t)+\frac{1}{2}\int_{x-t}^{x+t}\psi(y)dy$$ Show that $u$ is a solution to the wave-...
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