Different types of domains $\Omega \subset \mathbb{R}^n$ in PDEs

In PDEs I often read things like:

Let $$\Omega$$ be a bounded

• Lipschitz or
• $$C^1$$ or
• $$C^2$$ or
• $$C^\infty$$

domain

But I have no clue what this means in real life. I understand a $$C^0$$ domain has no corners, so no 90 degree angles. What do the above different types of domains look like intuitively? What should a self-respecting PDE person know about these things?

I read somewhere that you can approximate any domain with a $$C^\infty$$ domain. Is this true? More details would be appreciated.

2 Answers

In 2D, some heuristic examples:

$C^{\infty}$-domain: disk

Lipschitz/$C^{0,1}$-domain: Pacman, star-shaped nonconvex polygon, convex polygon

$C^{1}$-domain: consider a domain, when locally viewed, some part of boundary looks like the shape of gluing the graph of $x^2$ and $-x^2$ together at the origin. The rest of the boundary is just smooth. As you can see the derivative of $x^2$ and $-x^2$ agree at the origin, but not the second derivative.

$C^{2}$-domain: similar with $C^{1}$-domain, just gluing the graph $x^3$ and $-x^3$ together at the origin, the curve there is $C^2$.

The regularity of the domain plays an important role for the elliptic regularity for a Sobolev function: $$\|u\|_{W^{2,p}(\Omega)}\leq \| \Delta u\|_{L^p(\Omega)}$$ when $\Omega$ is $C^{1,1}/C^2$ or convex. In other words, the Poisson equation $-\Delta u = f$ boundary value problem can get a regularity lifting of 2 from the data $f$, i.e., the weak differentiability goes up by 2.

• A counterexample in nonconvex domain: the regularity lifting is not true anymore. Consider in the Pacman domain $\Omega$ parametrized using polar coordinates: $$\Omega = \{(r,\theta): 0<r<1, 0<\theta<\pi/\alpha\}$$ with $1/2<a<1$ then $$u = (1-r^2)r^{\alpha}\sin(\alpha \theta)$$ solves the homogeneous Dirichlet boundary problem: \begin{aligned} -\Delta u &= (4\alpha+4)r^{\alpha}\sin(\alpha \theta)\quad \text{ in } \Omega \\ u &= 0\quad \text{ on } \partial \Omega \end{aligned} The right hand side data is in $L^p$, but $u\notin W^{2,p}$ for its second derivative shows strong singular behavior near the origin. If we extend the Pacman to the full disk, such solution won't be there anymore for the value $$\lim_{\theta\to 0^+} u(r,\theta) = \lim_{\theta\to 2\pi^-} u(r,\theta)$$

I doubt we could approximate ANY open set by $C^{\infty}$-smooth domain, since we simply introduce the definition of the boundary of this open set in the Lebesgue sense. For domain that is at least Lipschitz I believe the approximation is true, if we are talking about pointwise limit.

• What does the notation $C^{0,1}, C^{1,1}$ mean?
– TCL
Apr 29 '13 at 19:52
• @TCL Oh I just got lazy and didn't mention that they are domains of which the boundary locally can be viewed as the graph of Hölder continuous functions. en.wikipedia.org/wiki/H%C3%B6lder_condition Apr 29 '13 at 19:58
• Thank you, great stuff. Do you have any description of $C^{1/2}$ surface, or $C^s$ for $s$ non-integer? Apr 30 '13 at 20:54
• @maximumtag The natural definition of fractional coefficients boundary would be using Hölder continuity, i.e., the boundary locally can be mapped to the graph of a Hölder continuous function $C^{k,\alpha}$. Wikipedia has some examples: en.wikipedia.org/wiki/H%C3%B6lder_condition . And $C^{1/2}$ is just $C^{0,1/2}$. May 1 '13 at 1:07
• Hello, could you help me if you know some about my questions? Thank you very much. math.stackexchange.com/questions/4006669/… Jan 31 at 10:30

These refer to the smoothness of the boundary, $\partial \Omega$.

If $\Omega \subset \mathbb{R}^n$ is open and bounded, we say $\partial \Omega$ is $C^k$ if for each point $\xi \in \partial \Omega$, $\exists r > 0$ and a $C^k$ function $\psi : \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ such that

$$\Omega \cap B_r(\xi) = \{ x \in B_r({\xi}) \, | \, x_n > \psi(x_1,\ldots,x_{n-1}) \}$$

For $C^{\infty}$ or Lipschitz domains, replace these with $C^k$ in the definition.

Intuitively, you take a ball around any point on the boundary and transform the part of the domain in the ball to the upper half plane with a $C^k$ function.