Sobolev functions on puncture ball

Let us first see the prove of sobolev embedding of {W^{1,1}\rightarrow L^2} on the plane.

Lemma: Suppose {f\in W^{1,1}(\mathbb{R}^2)} with compact support. Then

\displaystyle ||f||_{L^2}\leq ||\nabla f||_{L^1}

Proof: Let us suppose {f\in C^\infty_c(\mathbb{R}^2)}, the general case can be proved by approximation. Since {f} has compact support, then

\displaystyle |f(x,y)|=\left|\int_{-\infty}^x\frac{\partial f}{\partial x}(t,y)dt\right|\leq \int_{-\infty}^{\infty}|\nabla f|(t,y)dt=F(y)

\displaystyle |f(x,y)|=\left|\int_{-\infty}^y\frac{\partial f}{\partial y}(x,s)ds\right|\leq \int_{-\infty}^{\infty}|\nabla f|(x,s)ds=G(x)

Then

\displaystyle \int_{\mathbb{R}^2}f^2(x,y)dxdy\leq \int_{-\infty}^\infty\int_{-\infty}^\infty F(y)G(x)dxdy

\displaystyle =\int_{-\infty}^\infty F(y)dy\int_{-\infty}^\infty G(x)dx\\ =\left(\int_{-\infty}^\infty\int_{-\infty}^\infty|\nabla f|dxdy\right)^2

\Box

Suppose we have a function {u\in W^{1,1}_{loc}(D\backslash\{0\})}, where {D} is the unit disc in {\mathbb{R}^2}, {u} can blow up wildly near the origin. However if we know {\nabla u\in L^1(D)}, then actually {u\in L^2(D)} and {u\in W^{1,1}(D)}.

Proof: Because the bad thing happened only at origin, we can suppose {u} has spt inside {\frac{1}{4}D} or {D_{1/4}}. Put a substantially large square box {B_\epsilon} with length {\frac 12} inside the left half of the disc {D^-}whose distance to the origin is {\epsilon} see the picture.

puncture disc

puncture disc

Then on the three sides, {l_1}, {l_2}, {l_3}, {u=0}. Using the proof of the above, one can prove

\displaystyle ||u||_{L^2(B_\epsilon)}\leq ||\nabla u||_{L^1(B_\epsilon)}\leq ||\nabla u||_{L^1(D)}

Letting {\epsilon\rightarrow 0}, we get {u\in L^2(D^-)}. The same proof works for the right part {D^+}. Finally {u\in L^2(D)}. Choose a cut off function {\zeta_\epsilon=\zeta(x/\epsilon)}. Then

\displaystyle u\zeta_\epsilon\rightarrow u\text{ in }L^1

\displaystyle \nabla(u\zeta_\epsilon)\rightarrow \nabla u\text{ in }L^1

So {u\in W^{1,1}(D)}. \Box

Remark: This is called the removable singularity. There is a more general theorem related to this. Assume {n\geq 2}, {K\subset\subset\Omega} such that {\mathcal{H}^{n-2}(K)=0}. Suppose {u\in W^{1,1}_{loc}(\Omega\backslash K)} and {\int_{\Omega\backslash K}|\nabla u|dx<\infty}. Then {u\in W^{1,1}(\Omega)}.

I learned this from Prof. Brezis’s class. Also see his book: Sobolev maps with values into the circle.

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