# Tagged Questions

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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### Distributional derivatives for functions that is continuous but nowhere differentiable

It is well known that the Brownian motion is an example of functions that is continuous but nowhere differentiable. In addition, its distributional derivative can be interpreted in the way mentioned ...
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### Are weak (Sobolev) solutions to a linear ODE a classical ones?

Let $\Omega$ be an open subset of $\mathbb{R}$ and let $L$ be the differential operator $$Lf = \sum_{k=0}^{n-1} a_k f^{(k)} + f^{(n)},$$ where $a_k$ are reals. I would like to show that every ...
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### Question concerning the proof of regularity for the Laplacian $f \in H^m(\Omega) \Rightarrow u \in H^{m+2}(\Omega)$

I am stuck at the proof of Theorem 9.25 in Haim Brezis' Sobolev Spaces, Functional Analysis and Partial Differential Equations. This theorem deals with the regularity for the Dirichlet Problem for ...
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### Weak formulation 1-D PDE with non-homogenous robin boundary condition

Question: Question statement Worked Solution: Worked solution I have a few questions I hope you can help me with. Firstly, is my weak formulation correct? Why can't I use a test function in $H^1_0$ ...
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### $u$ continuous and the weak derivative $Du$ continuous $\Rightarrow$ $u \in C^1$?

Supose we have $u \in W^{1,p}$ (i.e $u$ has weak partial derivatives, which we denote by $Du$), and that both $u$ and $Du$ are continuous (More precisely, there is a continuous representative in the ...
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### Weak solution in $\mathbb{R}^{N}$

I'm bit confusing about definition of weak solution. If I have the following problem: $\begin{cases} \tag{P} -\Delta u = f \textrm{ in } \Omega, \\ u = 0 \textrm{ in } \partial\Omega, \end{cases}$ ...
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### Confused about the notation $||\nabla u||_{L^p(\Omega)}$

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, and also Evans' PDE. I am confused about what $||\nabla u||_{W^{1,p}(\Omega)}$ precisely means. In ...
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### 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 ...