用户: Yao/PDE Exercises

1. The Sobolev theorem says that if $s>\frac{n}{2}$, it makes sense to evaluate functions in $H^s({\mathbb{R}}^n)$ at a point. For $0\le s\le \frac{n}{2}$, functions in $H^s$ are only defined a.e., but if $s>\frac{k}{2}$ with $k<n$, it makes sense to restrict functions in $H^s$ to subspaces of codimension $k$. More precisely, let us write ${\mathbb{R}}^n={\mathbb{R}}^{n-k}\times {\mathbb{R}}^k,x=(y,z),\xi =(\eta ,\zeta )$ , and define

$$R:\mathcal{S}({\mathbb{R}}^n)\to \mathcal{S}({\mathbb{R}}^{n-k}),Rf(y)=f(y,0).$$

Show that $R$ extends to a bounded map from $H^s({\mathbb{R}}^n)$ to $H^{s-\frac{k}{2}}({\mathbb{R}}^{n-k})$ provided $s>\frac{k}{2}$. Clearly this is a generalization of the trace operator which corresponds to $k=1$.

2. Recall the definition of Hölder space. First $C^0({\mathbb{R}}^n)$ is the space of continuous functions equipped with the norm $$\Vert f{\Vert }_{C^0}=\Vert f{\Vert }_{L^{\infty }\left({\mathbb{R}}^n\right)}.$$For $k\in \mathbb{N},\alpha \in [0,1]$, define the corresponding norm

$$\begin{array}{rl}\Vert f{\Vert }_{C^{0,\alpha }\left({\mathbb{R}}^n\right)}& :=\Vert f{\Vert }_{C^0({\mathbb{R}}^n)}+\underset{x,y\in {\mathbb{R}}^n,x\ne y}{\sup }\frac{|f(x)-f(y)|}{|x-y{|}^{\alpha }}\\ \Vert f{\Vert }_{C^{k,\alpha }\left({\mathbb{R}}^n\right)}& :=\sum _{j=0}^k{\Vert {\nabla }^{j}f\Vert }_{C^{0,\alpha }({\mathbb{R}}^n)}\end{array}$$

Then the $C^{k,\alpha }$ is defined as$$C^{k,\alpha }({\mathbb{R}}^n)=:\left\{f\in C^k\left({\mathbb{R}}^n\right),\Vert f{\Vert }_{C^{k,\alpha }({\mathbb{R}}^n)}<\infty \right\}.$$

Clearly $C^{0,0}=C^0$, and $C^{0,1}({\mathbb{R}}^n)=\mathrm{Lip}({\mathbb{R}}^n)$ (the space of Lipschitz functions).

(1) If $\alpha >1$, show that the $C^{0,\alpha }({\mathbb{R}}^n)$ norm of a function $f$ is finite if and only if $f$ is constant. This explains why we generally restrict the Hölder index $\alpha \le 1$.

(2) Show that $C^{k,\alpha }({\mathbb{R}}^n),k\in \mathbb{N},\alpha \in [0,1]$ is a Banach space.

(3) If $k,l\in \mathbb{N},\alpha ,\beta \in [0,1],f\in C^{k,\alpha }({\mathbb{R}}^n)$ and $g\in C^{l,\beta }({\mathbb{R}}^n)$, and $k+\alpha \le l+\beta$, show that $fg\in C^{k,\alpha }({\mathbb{R}}^n)$, and that the multiplication map is continuous from $C^{k,\alpha }({\mathbb{R}}^n)\times C^{l,\beta }({\mathbb{R}}^n)$ to $C^{k,\alpha }({\mathbb{R}}^n)$.

Remark: As a consequence, any variable coefficient differential operator $L$ of order $m$ with $C^{\infty }({\mathbb{R}}^n)$ coefficients will map $C^{m+k,\alpha }({\mathbb{R}}^n)$ to $C^{k,\alpha }({\mathbb{R}}^n)$ for any $k\in \mathbb{N},\alpha \in [0,1]$.

3. Consider the Poisson equation $\Delta u=f$$$\mathrm{supp}f\in B(0,1),f\in C^{0,\alpha }({\mathbb{R}}^3),\alpha \in (0,1).$$Now the solution is given by the convolution with fundamental solution

$$u(x):=\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{f(y)}{|x-y|}dy$$

(i) Show that $u\in C^0({\mathbb{R}}^3)$.

(ii) Show that $u\in C^1({\mathbb{R}}^3)$, and we have$$\frac{\partial u}{\partial x_j}(x)=-\frac{1}{4\pi }{\int }_{{\mathbb{R}}^3}(x_j-y_j)\frac{f(y)}{|x-y{|}^3}dy$$

(iii) Show that $u\in C^2({\mathbb{R}}^3)$, and

$$\frac{{\partial }^{2}u}{\partial x_i\partial x_j}(x)=\frac{1}{4\pi }\underset{\epsilon \to 0}{\lim }{\int }_{|x-y|\ge \epsilon }\left[\frac{3(x_i-y_i)(x_j-y_j)}{|x-y{|}^5}-\frac{{\delta }_{ij}}{|x-y{|}^3}\right]f(y)dy$$

for $i,j=1,2,3$, where ${\delta }_{ij}$ is the Kronecker delta.

(iv) Show that $u\in C^{2,\alpha }({\mathbb{R}}^3)$, and establish the Schauder estimate$$\Vert u{\Vert }_{C^{2,\alpha }({\mathbb{R}}^3)}\le C_{\alpha }\Vert f{\Vert }_{C^{0,\alpha }\left({\mathbb{R}}^3\right)}$$where $C_{\alpha }$ depends only on $\alpha$.

Remark: Schauder estimate fails for $\alpha =0,1$, for interested reader, please try to prove it or find counterexample. This failure could be one motivation for using Hölder space in the study of elliptic PDE.

4. (a) Assume $v\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n)\cap L_{\mathrm{loc}}^1({\mathbb{R}}^n)$ and $v(\xi )\ge 0$. Show that if $\hat{v}\in L^{\infty }({\mathbb{R}}^n)$, then $v\in L^1({\mathbb{R}}^n)$ and$$(2\pi {)}^{n/2}\Vert \hat{v}{\Vert }_{L^{\infty }}=\Vert v{\Vert }_{L^1}$$(Hint: Consider $v_k(\xi )=\chi (\xi /k)v(\xi )$, with $\chi \in C_0^{\infty }({\mathbb{R}}^n),\chi (0)=1$.)

(b) Now consider$$\hat{u}(\xi )=\frac{⟨\xi {⟩}^{-n}}{1+\log ⟨\xi ⟩}.$$Show that $u\in H^{n/2}({\mathbb{R}}^n),u\notin L^{\infty }({\mathbb{R}}^n)$. This demonstrates the failure of embedding $H^s\hookrightarrow C^0$ at $s=\frac{n}{2}$.

(c) By looking at the following example$$u=\chi (x)\log \log \left(1+\frac{1}{|x|}\right),\chi \in C_c^{\infty }(B(0,1)),\chi (0)=1.$$

5. We know that Rellich compact embedding holds when we have sequence supported in the same compact domain.

(1) Show by examples that the embedding $H^1({\mathbb{R}}^3)\hookrightarrow L^q({\mathbb{R}}^3)$ is not compact, $q\in [2,6)$.

(2) Establish the Radial Sobolev embedding $n\ge 3$ and let $u$ be a radial Schwartz function on ${\mathbb{R}}^n$, prove that$$\Vert |x{|}^{\frac{n}{2}-1}|u|{\Vert }_{L^{\infty }({\mathbb{R}}^n)}⩽C\Vert u{\Vert }_{L^2({\mathbb{R}}^n)}$$for all $x\in {\mathbb{R}}^n$, as well as the variant$$\Vert |x{|}^s|u|{\Vert }_{L^{\infty }(R^n)}⩽C\Vert u{\Vert }_{H^1({\mathbb{R}}^n)},\frac{n}{2}-1\le s\le \frac{n-1}{2}$$(3) Compared with the result in (1), there is improvement in the radial case. Consider the radial Sobolev space$$H_r^1({\mathbb{R}}^3)=\{f\text{ is radial, and }f\in H^1({\mathbb{R}}^3)\}.$$Show that $H_r^1({\mathbb{R}}^3)$ embeds into $L^q({\mathbb{R}}^3)$ compactly, $\forall q\in [2,6)$.

6. Let $f\in C^{\infty }(\mathbb{R})$, and $f(0)=0$. Show that $u\mapsto f(D^{m}u)$, with $D^{m}u=D^{\alpha }u,|\alpha |⩽m$, defines a continuous map $H^{k+m}({\mathbb{R}}^n)\to H^k({\mathbb{R}}^n)$, for $k>\frac{n}{2}$, $k\in \mathbb{Z}$. Show that$$\Vert f(D^{m}u)-f(D^{m}v){\Vert }_{H^k}\le C\Vert u-v{\Vert }_{H^{k+m}}$$with $C=C(f,\Vert D^{m}u{\Vert }_{L^{\infty }},\Vert D^{m}v{\Vert }_{L^{\infty }})$.

Remark: The result holds even if $k\notin \mathbb{Z}$.

7.Let $f\in S_x(\mathbb{R})$ be such that $f(0)=0$. Show that$$\Vert \frac{f(x)}{x}{\Vert }_{H^{s-1}(\mathbb{R})}⩽C\Vert f{\Vert }_{H^s(\mathbb{R})}$$for all $s>1/2$.

(Hint: write $f(x)/x={\int }_0^1{f}^{\prime }(tx)dt$, then take Fourier transforms.)

8. Givens $s\in (0,1),p\in [1,\infty )$, define the ${\tilde{W}}^{s,p}(\Omega )$ norm$$\Vert u{\Vert }_{W^{s,p}(\Omega )}=({\int }_{\Omega }|u(x){|}^{p}dx+{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y)|}{|x-y{|}^{n+sp}}dxdy{)}^p.$$(1) When $\Omega ={\mathbb{R}}^n$, show that $W^{1,p}({\mathbb{R}}^n)\hookrightarrow {\tilde{W}}^{s,p}({\mathbb{R}}^n)$ for all $s\in (0,1)$. Here $W^{1,p}$ is the classical Sobolev space.

(2) Show that we have Poincaré type inequality for smooth bounded domain $\Omega$$$\Vert u-u_{\Omega }{\Vert }_{L^p(\Omega )}^p\le \frac{(\mathrm{diam}\Omega {)}^{n+sp}}{|\Omega |}{\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy$$where $u_{\Omega }=\frac{1}{|\Omega |}{\int }_{\Omega }u(x)dx$.

Remark: a) the definition is only valid for $s<1$, in fact it was proved by Brezis that for $s\ge 1$,$${\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy<\infty \Rightarrow u\text{ is constant}.$$b) This space is one candidate for fractional Sobolev space. There are other options as we had with defining $H^s(\Omega )$, using Fourier transform, or using interpolation. In particular, consider$$\Vert u{\Vert }_{H^{s,p}({\mathbb{R}}^n)}:=\Vert (1+|\xi |{)}^s\hat{f}{\Vert }_{L^p({\mathbb{R}}^n)}.$$Although we expect the two things should be equal, this is not true. In fact for $s\in (0,1)$,$$H^{s,p}={\tilde{W}}^{s,p}\text{ if and only if }p=2.$$