Intereting Posts

What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?
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lim sup inequality proof – is this the right way to think?
Maps between direct limits
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Dynkin's Theorem, and probability measure approximations
$\epsilon$ – $\delta$ definition of a limit – smaller $\epsilon$ implies smaller $\delta$?
Showing $\sin(\bar{z})$ is not analytic at any point of $\mathbb{C}$
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“What if” math joke: the derivative of $\ln(x)^e$

Let $u\in H^1(\Omega)$, where $\Omega$ is a bounded open set of $\mathbb{R}^n$ with Lipschitz boundary.

We denote the outward unit normal as $n$, defined a.e. on $\partial\Omega$, and the normal derivative of $u$ as

$$

\frac{\partial u}{\partial n}:=\nabla u\cdot n.

$$

- Reconciling several different definitions of Radon measures
- An operator has closed range if and only if the image of some closed subspace of finite codimension is closed.
- Is there an algebraic homomorphism between two Banach algebras which is not continuous?
- Definition of Equivalent Norms
- Poincaré's Inequality on Sobolev Spaces in One Dimension
- Relations between spectrum and quadratic forms in the unbounded case

Which space does the normal derivative belong to?

Is it possible to show $\frac{\partial u}{\partial n}\in L^2(\partial\Omega)$?

I think it’s not possible if we don’t require at least that $u\in H^2(\Omega)$. Indeed it is easy to get

$$

\|\frac{\partial u}{\partial n}\|_{L^2(\partial \Omega)}\le \|\nabla u \|_{L^2(\partial \Omega)}.

$$

By the Trace theorem, we know that $\nabla u \in L^2(\partial\Omega)$ if $\nabla u\in H^1(\Omega)$, i.e. $u\in H^2(\Omega)$.

Note that my notation is quite messy when I deal with the norm of the gradient…

- equivalent norms in Banach spaces of infinite dimension
- Cesàro operator is bounded for $1<p<\infty$
- Uniform boundedness principle for norm convergence
- Fixed point in a continuous map
- Prove that $X^\ast$ separable implies $X$ separable
- Weak limit of an $L^1$ sequence
- Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces.
- Transpose of Volterra operator
- How to prove that $C^k(\Omega)$ is not complete
- Basic Open Problems in Functional Analysis

If $u$ belongs merely to $H^1(\Omega)$, then you cannot define a normal-derivative-trace operator.

Indeed, if $T : H^1(\Omega) \to S(\partial\Omega)$ would be such an operator, where $S(\partial\Omega)$ is some Banach space on the boundary and if $T$ would be linear, you arrive at the following contradiction: if $T$ is reasonably defined, you would have $T \varphi = 0$ for all $\varphi \in C_c^\infty(\Omega)$. By density of $C_c^\infty(\Omega)$ in $H_0^1(\Omega)$ and continuity of $T$, this implies $T u = 0$ for all $u \in H_0^1(\Omega)$. But this is absurd (consider $u \in C^1(\bar\Omega)$).

On the other hand, if $u \in H^1(\Omega)$ and $\Delta u \in L^2(\Omega)$,

you can define the trace of the normal derivative in $H^{-1/2}(\Omega)$ by duality. Indeed, for regular $u$, you have

$$\int_\Omega \Delta u \, v + \nabla u \cdot \nabla v \, \mathrm{d}x = \int_{\partial\Omega} \frac{\partial u}{\partial n} \, v \, \mathrm{d}s.$$

Now, if $v \in H^{1/2}(\partial\Omega)$ is arbitrary (and $\Omega$ possesses some regularity), you find $E v \in H^1(\Omega)$ with $(Ev)|_{\partial\Omega} = v$. Then, you define

$$\langle \frac{\partial u}{\partial n}, v \rangle := \int_\Omega \Delta u \, Ev + \nabla u \cdot \nabla Ev \, \mathrm{d}x.$$

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- Irreducible polynomial of $\mathrm{GF}(2^{16})$
- Probability that sum of rolling a 6-sided die 10 times is divisible by 10?
- Solving $x\sqrt1+x^2\sqrt2+x^3\sqrt3+…+x^n\sqrt{n}+\dots=1$ with $x\in \mathbb{R}$ and $n\in \mathbb{N}$
- A secant inequality for convex functions
- Existence of acyclic coverings for a given sheaf
- Maximum of $\frac{\phi(i)}i$
- Multivariate function interpolation