Schauder estimate 与 Sobolev inequality：椭圆方程正则性的两种语言

旧博客原文

原题：Schauder estimate and Sobelov inequality

In this note we discuss the Schauder theory for uniformly elliptic linear equations and Sobelov inequality.

the three main topics ars a priori estimate in Holder norms,regularity of arbitrary solutions and the solvability of the Dirichlet problem.Among these topics,a priori estimates are the most fundamental and the basis of the follows two.we will discuss both the interior Schauder estimate and global Schauder estimate.

-Schauder Theory-

1. Interior Schauder Theory

${\Omega}$ be a domain in ${R^n}$ ,bounded most of the time.
${a_{ij},b_i,c}$ be defined in ${\Omega}$ ,with ${a_{ij}=a_{ji}}$ .where ${1\leq i,j\leq n}$ .
we consider the operator ${L}$ given by,

$\displaystyle Lu=a_{ij}\partial_{ij}u+b_i\partial_iu+c,in \ \Omega.$

easy to see ${Lu}$ is defined for any ${u\in C^2(\Omega)}$ .
the operator ${L}$ is always be assumed to be strictly elliptic in ${\Omega}$ ;namely,
$\displaystyle a{ij}\xi_i\xi_j \geq \lambda|\xi|^2$

for any ${\xi\in R^n,x\in \Omega}$ ,where ${\lambda}$ is a positive constant.
1.1. Interior Schauder Estimate

define the weighted ${C^{k,\alpha}}$ norm,

$\displaystyle |u|^*_{C^{k,\alpha}(B_R)}=\sum_{i=0}^k R^i|D^iu|_{L^{\infty}(B_R)}+R^{k+\alpha}[D^ku]_{C^{\alpha}(B_R)}$

easy to see ${R}$ come from a scaling.
consider the PDE.
$\displaystyle Lu=a_{ij}\partial_{ij}u+b_i\partial_iu+c=f,in \ \Omega.$

we want to proof this type estimate,
$\displaystyle |u|_{C^{2,\alpha}(A)} \leq C(|u|_{L^{\infty}(\Omega)}+|f|_{C^{\alpha}(\Omega)})$

where ${A\subset \Omega }$
we first deal with a easy case, ${a_{ij}}$ is constant. in this case we proof the estimate:

Lemma 1 ${f \in C^{\alpha}(B_R)}$ ,for some ${\alpha \in (0,1)}$ ,and ${(a_{ij})}$ be a constant symmetric ${n\times n}$ matrix satisfying
$\displaystyle \lambda |\xi|^2 \leq a_{ij}\xi_i\xi_j \leq \Lambda|\xi|^2$

${\exists \lambda,\Lambda >0,\forall \xi \in R^n}$ . suppose ${u\in C^2(B_R)}$ satisfies:
$\displaystyle a_{ij}\partial_{ij}u=f, in \ B_R$

then , ${u \in C^{2,\alpha}(B_{\frac{R}{2}})}$ ,moreover,
$\displaystyle |u|^*_{C^{2,\alpha}(B_{\frac{R}{2}})}\leq C[|u|_{L^{\infty}(B_R)}+R^2|f|^*_{C^{\alpha}(B_R)}]$

Proof: $\Box$ to continue,we prove an interpolation inequality for Holder continuous functions.

Lemma 2 Let ${\alpha,\mu \in (0,1)}$ and ${B_R}$ be a ball of radius ${R}$ in ${R^n}$ ,then, (1)for any ${u\in C^{1,\alpha}(\overline B_R)}$ ,
$\displaystyle \mu^{\alpha}R^{\alpha}[u]_{C^{\alpha}(B_R)}\leq C[\mu R |\nabla u|_{L^{\infty}(B_R)}+|u|_{L^{\infty}(B_1)}]$

(2)for any ${u\in C^{1,\alpha}(\overline B_R)}$ ,
$\displaystyle \mu R |\nabla u|_{L^{\infty}(B_R)} \leq C[\mu^{1+\alpha}R^{1+\alpha}|\nabla u|_{C^{\alpha}(B_R)}+|u|_{L^{\infty}(B_1)}]$

(3)for any ${u\in C^2(\overline B_R)}$ ,
$\displaystyle \mu R|\nabla u|_{L^{\infty}(B_R)}\leq C[\mu^2R^2|\nabla^2 u|_{L^{\infty}(B_R)}+|u|_{L^{\infty}(B_R)}]$

where ${C}$ is a positive constant depending on n and ${\alpha}$ .
Proof: $\Box$

Corollary 3 Let ${\alpha,\mu\in (0,1)}$ and ${B_R}$ be a ball of radius ${R}$ in ${R^n}$ .Then,for any ${u\in C^{2,\alpha}(\overline B_R)}$ ,
$\displaystyle \sum_{i=0}^2(\mu R)^i|\nabla^i u|_{L^{\infty}(B_R)}+\sum_{i=0}^1(\mu R)^{i+\alpha}[\nabla^i u]_{C^{\alpha}(B_R)}\leq C[(\mu R)^{2+\alpha}[\nabla^2 u]_{C^{\alpha}(B_R)}+|u|_{L^{\infty}(B_R)}]$

Proof: $\Box$

Now we are ready to prove an interior estimate for ${C^{2,\alpha}}$ -norms of solutions of uniformly elliptic equations.The trick is to freeze coefficients.

Lemma 4
2. Global Schauder Theory

-Sobelov inequality-

Theorem 5
$\displaystyle W_0^{1,p}(\Omega)\longrightarrow L^{\frac{np}{n-p}}(\Omega),1\leq p <n$

moreover,we have: ${\exists C=C(n,p)}$ , ${\forall u\in W^{1,p}_0(\Omega)}$ ,
$\displaystyle ||u||_{\frac{np}{n-p}} \leq C||Du||_p,1\leq p<n$

Proof:

$\displaystyle p=1$

suffice to proof:
$\displaystyle ||u||_{\frac{n}{n-1}}\leq C||Du||_1$

obvious we have:
$\displaystyle |u(x)|\leq \int_{-\infty}^{\infty}|Du(x)|dx$

so ${\int_{\Omega} |u|^{\frac{n}{n-1}}\leq \int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu(x)|dx)^{\frac{1}{n-1}}}$ .
so ${||u||_{\frac{n}{n-1}}\leq (\int_{\Omega}\Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)^{\frac{1}{n-1}})^{\frac{n-1}{n}}\leq \int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)^{\frac{1}{n}} \leq \int_{\Omega} \frac{1}{n} \sum_{i=1}^n(\int_{-\infty}^{\infty}|D_iu|)\leq C||Du||_1}$
$\displaystyle 1<p<n$

use the similar argument as ${p=1}$ to prove the situation ${1<p<n}$ .
suffice to prove ${||u||_{\frac{np}{n-p}}\leq C||Du||_p}$ .
obvious we have: ${|u(x)|^p\leq \int_{-\infty}^{\infty}p|u|^{p-1}|Du|}$ .
${(\int_{\Omega}|u(x)|^{\frac{np}{n-p}})^{\frac{n-p}{np}}}$
${\leq (\int_{\Omega} \Pi_{i=1}(\int_{-\infty}^{\infty} p|u|^{p-1}|D_iu| )^{\frac{1}{n-p}})^{\frac{n-p}{np}} }$
${\leq C\int_{\Omega} \Pi_{i=1}^n(\int_{-\infty}^{\infty}p|u|^{p-1}|D_iu|)^{\frac{1}{np}}}$
${\leq\frac{c}{n}\sum_{i=1}^n\int_{\Omega}(\int_{-\infty}^{\infty}p|u|^{p-1}|D_iu|)^{\frac{1}{p}}}$
${\leq \frac{c}{n}\sum_{i=1}^n\tilde C p[(\int_{\Omega} (|u|^{p-1})^{\frac{p}{p-1}})^{\frac{p-1}{p}}+(\int_{\Omega} |D_iu|^p)^{\frac{1}{p}}]^{\frac{1}{p}} }$
${\leq C||Du||_p}$ . Q.E.D. $\Box$
4.

$\displaystyle W_0^{1,p}(\Omega)\longrightarrow C(\bar\Omega),n<p$

moreover,we have: ${\exists C=C(n,p)}$ , ${\forall u\in W^{1,p}_0(\Omega)}$ ,
$\displaystyle sup_{\Omega}|u| \leq C|\Omega|^{\frac{1}{n}-\frac{1}{p}}||Du||_p,p>n$

${\mu\in (0,1]}$ ,

$\displaystyle (V_{\mu}f)(x)=\int_{\Omega}|x-y|^{n(\mu-1)}f(y)dy$

then ${V_{\mu}: L^1(\Omega) \longrightarrow L^1(\Omega) }$ is well-defined by the following lemma:
Lemma 6 ${V_{\mu}:L^p \longrightarrow L^q}$ continously for any q, ${1\leq q \leq \infty}$ satisfy ${0\leq \delta=\delta(p,q)=\frac{1}{p}-\frac{1}{q} \leq \mu}$ .
furthermore,for any ${f\in L^p(\Omega)}$
$\displaystyle ||V_{\mu}f||_q \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta}w_n^{1-\mu}|\Omega|^{\mu-\delta}||f||_p$

Proof: ${h(x-y)=|x-y|^n(\mu-1)}$ directly calculate follows that :

$\displaystyle ||h||_r \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta} w_n^{1-\mu}|\Omega|^{\mu-\delta}$

now follows young inequality and this priori estimate we have:
${||V_{\mu}f||_q=(\int_{\Omega}(\int_{\Omega}|x-y|^{n(\mu-1)}f(y)dy)dx)^{\frac{1}{q}}}$
${\leq (\int_{\Omega}(\int_{\Omega}h^{\frac{r}{q}}h^{r(1-\frac{1}{p})}|f|^{\frac{p}{q}}|f|^{p\delta})^qdx)^{\frac{1}{q}}}$
${\leq (\int_{\Omega}(\int (h^r|f|^p)^{\frac{1}{q}}(\int h^r)^{1-\frac{1}{p}}(\int f^p)^{\delta})^{q})^{\frac{1}{q}}}$
${\Longrightarrow}$
$\displaystyle ||V_{\mu}f||_q \leq sup_{x \in \Omega} \{\int h^r(x-y)dy\}^{\frac{1}{r}}||f||_p$

and by the priori estimate,we have:
$\displaystyle ||V_{\mu}f||_q \leq (\frac{1-\delta}{\mu -\delta})^{1-\delta}w_n^{1-\mu}|\Omega|^{\mu-\delta}||f||_p$

Q.E.D. $\Box$
Lemma 7 ${f\in L^p(\Omega)}$ , ${g=V_{\mu}f}$ .
${\Longrightarrow}$ ${\exists c_1,c_2}$ constant depend only on ${n,p}$ ,such that
$\displaystyle \int_{\Omega} exp[\frac{g}{c_1||f||_p}]^{p^`}dx\leq c_2|\Omega|,p^`=\frac{p}{p-1}$

Proof: we have

$\displaystyle ||g||_q \leq q^{1-\frac{1}{p}+\frac{1}{q}}w_n^{1-\frac{1}{p}}|\Omega|^{\frac{1}{q}}||f||_p$

${\Longrightarrow}$
$\displaystyle \int_{\Omega} |g|^{p^`q}dx \leq p^`q(w_np^`q||f||_p^{p^`})^q|\Omega|$

${\Longrightarrow}$
$\displaystyle \int_{\Omega}\sum_{N_0}^{N}\frac{1}{k!}(\frac{|g|}{c_1||f||_p})^{p^`k}\leq p^`|\Omega|\sum(\frac{p^`w_n}{c_1^p})^k\frac{k^k}{(k-1)!}$

then take ${c_1,c_2}$ suffice large. Q.E.D. $\Box$
Lemma 8 let ${u\in W^{1,1}_0(\Omega)}$
$\displaystyle u(x)=\frac{1}{nw_n} \int_{\Omega} \frac{(x_i-y_i)D_iu(y)}{|x-y|^n}$

a.e. in ${\Omega}$ .
Proof: frist zero extended ${u}$ to whole space.and we have ${u(x)=\int_{-\infty}^xD_iu(x)}$ .

$\displaystyle u(x)=\int_0^{\infty}D_ru(x+rw)dr$

forall ${w\in \partial B_1(0)}$ ,so
$\displaystyle u(x)=-\frac{1}{nw_n}\int_0^{\infty}\int_{|w|=1}D_ru(x+rw)drdw=\frac{1}{nw_n}\int_{\Omega}\frac{(x_i-y_i)D_iu(y))}{|x-y|^ndy}$

Q.E.D. $\Box$
Theorem 9 let ${u\in W^{1,n}_0(\Omega)}$ ,then there exists constant ${c_1,c_2}$ such that
$\displaystyle \int_{\Omega}exp[\frac{|u|}{c_1||Du||_n}]^{\frac{n}{n-1}}dx\leq c_2|\Omega|$

Proof: a $\Box$

Theorem 10 ${u\in W_0^{1,p}(\Omega),p>n}$ ,then ${u\in C^{\gamma}(\Omega)}$ , ${\gamma=1-\frac{n}{p}}$ .
moreover ${\forall ball B=B_R}$
$\displaystyle osc_{\Omega \cap B_R}u \leq C R^{\gamma} ||Du||_p$

Proof: a $\Box$

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Schauder 理论和 Sobolev 理论是椭圆方程正则性的两套基本语言。前者追踪 Holder 范数，适合系数和右端足够连续的情形；后者追踪积分可积性，适合弱解和变分方法。

Schauder estimate 与 Sobolev inequality：椭圆方程正则性的两种语言 — Schauder 估计追踪 Holder 正则性，Sobolev 不等式追踪积分正则性，两者共同构成椭圆方程的基本正则性工具。

1. 线性一致椭圆算子

考虑

$$Lu=a^{ij}(x)\partial_{ij}u+b^i(x)\partial_i u+c(x)u=f.$$

一致椭圆性是指存在 $\lambda>0$，使

$$a^{ij}(x)\xi_i\xi_j\ge \lambda|\xi|^2.$$

这是所有先验估计的起点。

2. Interior Schauder estimate

若系数和 $f$ 都在 $C^\alpha$ 中，则局部有

$$\|u\|_{C^{2,\alpha}(B_{1/2})}\le C\bigl(\|u\|_{C^0(B_1)}+\|f\|_{C^\alpha(B_1)}\bigr).$$

常系数情形可以先通过 Newton potential 或 Fourier 方法得到，再用冻结系数和 perturbation 推广到变系数情形。

3. Scaling 与 weighted norms

Schauder 估计的形状由 scaling 决定。若把球 $B_r$ 缩放到单位球，二阶导数带来 $r^{-2}$，Holder seminorm 还会多出 $r^{-\alpha}$。weighted norm 正是为了把这些尺度因子记录清楚。

4. Sobolev inequality

Sobolev 不等式给出

$$\|u\|_{L^{p^\ast}}\le C\|\nabla u\|_{L^p},\qquad p^\ast=\frac{np}{n-p}.$$

它不直接给出经典二阶 Holder 正则性，但能建立弱解存在性、能量估计和 bootstrapping。

5. 两种理论的关系

Schauder 理论适合光滑数据的 classical solution，Sobolev 理论适合弱解和变分框架。椭圆正则性常常先用 Sobolev 得到弱解，再通过 De Giorgi-Nash-Moser 或 Schauder 估计提升正则性。

Schauder estimate 与 Sobolev inequality：椭圆方程正则性的两种语言

旧博客原文

-Schauder Theory-

-Sobelov inequality-

补充说明

1. 线性一致椭圆算子

2. Interior Schauder estimate

3. Scaling 与 weighted norms

4. Sobolev inequality

5. 两种理论的关系

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