Tagged Questions

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Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ make sense?

If $u \in L^2(0,T;H^1)$ has a distributional derivative $u' \in L^2(0,T;H^{-1})$, does $(u^+)' \in L^2(0,T;H^{-1})$ exist, i.e., is $u^+$ weakly differentiable in time? By $u^+$ I mean the positive ...
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For which real values of $\alpha$ PDE $\Delta u(x,y)+2u(x,y)=x-\alpha$ has at least one weak solution?

Problem. Consider boundary value problem: \begin{cases} \Delta u(x,y)+2u(x,y)=x-\alpha, & \text{in $\Omega$,} \\ u(x,y)=0, & \text{on $\partial\Omega$,} \\ \end{cases} where $\alpha$ is ...
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$C_c^{\infty}$ 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 ...
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norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$\|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ?$$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
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Steklov Average in time and spatial gradient can be interchanged?

I'm trying to understand a proof on Steklov average and weak derivatives. Let $\Omega$ be a bounded domain of $\mathbb{R}^n$, $T>0$, $h\neq 0$ and $u\in L^2(0,T;H^1(\Omega))$ and extend $u$ by ...
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Weakly differentiable but classically nowhere differentiable

Is there any example of a function which is weakly differentiable but none of its versions are classically differentiable (or differentiable only on a set of measure 0) ? Thanks
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Weak Differentiability of Holder functions

Is it true that every Holder function is weakly differentiable? If not please give counterexample. Thanks
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When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).