Geometric intuition of mean value property of nonlinear elliptic equation

I wish to gain some understanding of the MVP of nonlinear elliptic equation by geometric intuition.

Linear elliptic equation case

First of all, I have a very good geometric explain of the MVP of Laplace equation, i.e.

MVP of laplace equation

$\Delta u=0$ in $\Omega$ , $\forall B(x_0,r)\subset \Omega$ is a Ball, we have following identity:

$\frac{1}{\mu(\partial(B))}\int_{\partial B}u(x)dx=u(x_0)$

I need to point out first, this property is not difficult to proof by standard integral by part method, but the following method have more geometric intuition. And in some sense explain why this property holds.

The proof is not very difficult to explain by mathematic formula, but I wish to divide the proof into two part, explain one part by graph and literal interpretation.

part 1 of the proof:

we consider a 1-parameter group of foliation, and consider the integral identity with this foliation.

$\int_{v\in S_{n-1}}\int_{\gamma_v} \frac{\partial \partial_{v}u}{\partial t}dt=\int_{B(x_0,r)}\partial_{n}u-\int_{S_{n-1}}\partial_{v}udv=\int_{B(x_0,r)}\partial_{n}u$

Part 2 of the proof:

and we have:

$\int_{v\in S_{n-1}}\int_{\gamma_v} \frac{\partial \partial_{v}u}{\partial t}dt=0$

by the pointwise equation $\Delta u=0$, one key point is $\partial_{-v-v}u=\partial_{vv}u, \forall v\in S_{n-1}$.

This approach cloud easily to transform to the general elliptic equation case and it seems a little difficult to transform to Possion equation, the non-hemomorphism case.

Nonlinear elliptic equation case

A-B-P estimate for general nonlinear uniformly elliptic equation

ABP estimate is the most basic estimate in fully nonliear elliptic equation.
The ABP maximum principle states (roughly) that, if

$a^{ij} \partial _i \partial _j u \geq f, in \ \Omega \subset \mathbb{R}^n (a^{ij} \geq C Id >0),$

Then (assuming sufficient regularity of the coefficients),

\sup _{\Omega} u \leq \sup _{\partial \Omega} u + C (\int _{\Omega} \vert f \vert^n )^{1/n} ………. (*)

I will give an intuitive explanation of the proof of (*) .
Usually, in order to prove maximum principles, the key idea is to use that at a local max the second derivative is negative-definite, then choose a good basis and get some identity of 1-order drivative and inequality for 2-order’s. This process is used in such like the proof of the Hopf lemma, and some inter gradient estimate, consider some flexiable function like $e^{Au}$ or sometihng else anyway.

But in the proof of ABP we need more geometric intution and more trick.

First we do a rescaling:
if $a^{ij} \partial _i \partial _j u \geq 1, in B_1 \subset \mathbb{R}^n (a^{ij} \geq C Id >0),u|_{\partial B_1}\geq 0.$
then:

$|inf_{B_1}u| \leq C |A|^{1/n} ....(**)$

And then We explain what is the contact set. It is the subset $\Gamma^{+} of \Omega$ such that u agree with it convex envelop. i.e. $\Gamma^{+}=\{x|u=convex \ evolap \ of \ u\ at \ x\}$ . The geometric meaning is it has at least one lower support plane. So what is $\Gamma^{+}$ it is just the set that u is very low on it. Or in another way of view you consider $-u$ as a lot of mountains then $\Gamma^+$ is the place near the tops of which mountain can see every thing (locally).
Then we look at every point in $\Gamma^{+}$ , then determination of the hessian matrix $det(u_{ij})$ at this point have a control due to the PDE $a^{ij} \partial _i \partial _j u \geq 1$, and the uniformly elliptic property.

The determination of hessian matrix could be view as a determination of Jacobe matrix of the map $(u_1,...,u_n)\to (e_1,...,e_n)$ .and by Area formula we have:

$\int_{\Phi(\Omega)} f( \Phi^{-1}(y)) dy =\int_{\Omega}f(x)|J({\Phi(x)})| dx,$

It is easy to see for a constant $c$ ,$B_{c|sup_{\Omega}|u||}(0)\subset \Phi(\Omega)$ (Base on the PDE on every point, the geometric intution is just the function u could not be very narrow cone at every point). So we have , take $f=\chi_{\Gamma^{+}}$ ,

$|B_{c\sup_{\Omega}|u|}(0)|^{n}\leq \int_{\Gamma^{+}}\chi_{\Gamma_{+}}(x)|J_{\Phi}(x)|dx$

so we have:

$|B_{c\sup_{\Omega}|u|}(0)|\lesssim |{\Gamma_{+}}|^{\frac{1}{n}}...(***)$

and the classical matrix inequality for every positive definite matrix A we have
:

$det(AB)\leq (\frac{tr(AB)}{n})^n...(****) .$

combine (***),(****) ,we have:

$sup_{\Omega}|u|\lesssim ||\frac{a^{ij}u_{ij}}{D^*}||_{L^n({\Gamma^+})} .$

graph

General approach to get MVP for elliptic equation which is come from geometry

MVP for K-hessian equation, with geometric explanation

MVP for K-curvature equation, with geometric explanation

MVP for p-Laplace equation, with geometric explanation

A discret to continuous approach to the Dirichlet principle.

Direchlet principle:
$\Omega \subset R^n$ is a compact set with $C^1$ boundary. then there exists unique solution $f$ satisfied $\Delta f=0$ in $\Omega$, $f=g$ on $\partial \Omega$.

Perron lifting and barrier function

We know the standard approach of the Dirichlet principle is perron lifting and construction of barrier function on the boundary.

The key point is if we define the variation energy $E(u)=\int_{\Omega}|\nabla u|^2$, then it is easy to see for $u_1,u_2$ is in perron set, $E(sup (u_1,u_2))\geq \max\{E(u_1),E(u_2)\}$. So we can begin from a maximization sequence to construct a Cauchy sequence by perron lifting and by the involve of barrier function to make the solution compatible with the boundary condition then arrive a proof.

But when I was a freshman in undergraduate school and I do not know the method of perron lifting I try something I name it from discret to continuous approach to try to solved the problem. It is always a puzzle in my mind iff we can solve the Dirichlet principle in this way, roughly speaking, it is divid into two part:

1. Investigate the discretization of harmonic function in smaller and smaller scale. The discretization I consider is just $\Omega\cap \epsilon \mathbb Z^2$ i.e. the $\epsilon$-latties in $\Omega$,and discretization Laplace operator $\Delta_{\epsilon}u(x_1,...,x_n)=\sum_{i_1,...,i_n\in\{-1,1\}}\frac{u(x_1+i_1,...,x_n+i_n)}{2^n}$. Some result is much easier to arrive with the discretization thing, you know ,such as the existence of solution is just come from simple linear algebra. and we can deduce harneck inequality, gradient estimate, even green function. So we get a solution $\hat f_{\epsilon}$ of $\epsilon$ discretization and we do a extension $\Omega\cap \epsilon \mathbb Z^2$ to $\Omega$ by take value of a small tube by the center of the tube, where the value have a definition by $\hat f_{\epsilon}$, and now we get $f_{\epsilon}$.

2. The second step is to proof the solution $f_{\epsilon}$ with $\epsilon$-discretization problem will coverage to the solution of original problem;i.e. we want to proof a $L^{\infty}$ estimate;i.e. $\forall \delta>0$, $\exists \epsilon>0, \forall 0<\epsilon_1,\epsilon_2<\epsilon$ we have $\forall x\in \Omega$, $|f_{\epsilon_1}(x)-f_{\epsilon_2}(x)|<\delta$. and by Albano-Ascoli theorem to construct $f$. Then we need to proof $f$ is the harmonic function we find, to verify this information we use the mean-value property. So we need to prove $f$ satisfied mean-value property for every ball in $\Omega$.

Here is my first question,
> **Question 1:** How to prove the $L^{\infty}$ estimate and the MVP of $\epsilon$-discretization will coverage to the MVP in $R^n$ case occor in second step?

My attempt to the $L^{\infty}$ estimate is by renomelazation which seems could work, but the annoying thing is to proof the mean-value property will coverage to the real one, I try to use some result of random walk, but it seem not works…

My second question is:
> **Question 2:** Are this approach a universal phenomenon? At least could we use this approach to establish the existence of solution for linear elliptic and parabolic equation?

The Third question is:
> **Question 3:** If we consider some inverse problem, that is to say, form a MVP instead of a PDE to derive a solution, could this always be possible? some example is, if we change the mean value property for harmonic function from the average of ball to cube or triangle or elliptic or something else, what happen? Is there always a solution satisfied the news MVP point-wise? If not, Is there some counterexample? on another hand, if yes, are them came from some PDE?

Gromov’s idea applicate to parabolic equation

Gromov’s idea applicate to parabolic equation.

Key point:

1.rescaling+renormalization.

2.analysis it on every scale.

k-hessian equation and k-curvature equation

here is the problem, how to understand k-hessian equation and k-curvature equation.

k-hessian equation

k-hessian equation is:

$H_k(u)=\sigma_k(D^2(u))=f$ (*)

where u is admissible, i.e. $\forall 1\leq i\leq k$, $\sigma_i(D^2(u))\geq 0$. this is just the condition to make (*) be a elliptic equation.

The most important result is the following three:

1.sovable (*) with direchlet boundary condition.

This is mainly the contribution of Caffaralli in 90’s. According flexible function and maximum principle we can establish the $C^{1,\alpha}$ estimate and $C^{2,\alpha}$ estimate in the inter. And the $C^{2,\alpha}$ estimate near the boundary is establish according to the conformation invariant and some perbutation of the solution of k-hessian equation after special rescaling.

2.Hessian measure.

This is mainly the work of X.J.Wang and Trudinger. they proved:

in the meaning of viscosity solution, if $\sigma_k(D^2(u))=f$. then we can associate a measure $\mu$ with $u$,and the following is right:

when $u\in C^2(\Omega)$, $\mu(B_r(x))=\int_{B_r{x}}\sigma_k(D^2(u))$.

if $u_1,..,u_n,...$ coverage to $u$. then $\mu_1,...,\mu_n,...$ coverage to $\mu$ in weak sense.

this is merely depend on a priori estimate on $u$

3.pointwise estimate corresponding wolff potential.

the Wolff potential is:

$W^{\mu}_{k}(x,r)= \int_{0}^r(\frac{\mu(B_t(x))}{t^{n-2k}})^{\frac{1}{k}}\frac{1}{t}dt$

We can easily use rescaling to understand the reasonable of this potential, and use this potential Lubutin establish the following pointwise estimate:

$u\in \Phi_k(B_{4R}(x))$, $u\leq 0$, then we have:

$W^{\mu}_k(x,\frac{R}{2})\leq |u(0)| \leq W^{\mu}_k(x,2R)-sup_{B_{2R}}|u|$

the RHS could look as a corollary of classical A-B-P estimate. the LHS need combine several observation. mean-value property and some else.

This result could use to establish some result on singularity point can be removable.

k-curvature equation

1.sovable (*) with direchlet boundary condition.

This is also established by cafferalli.

2.curvature measure.

This is established very recently. mean curvature equation in 2014, by perron lift and modified, general case in 2016 by more complex calculate and method.

3.pointwise estimate corresponding wolff potential.

This still do not established, and is the main thing I focus on. Due to we can look as k-curvature as a “projection” of k-hessian equation, Calderon-Zegmund decomposition and the estimate of k-hessian equation maybe useful.

My ideas

look is as “average” of “loop space”, “surface space”.

1.Grassmannian bundle

n algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
${\displaystyle p:G_{d}(E)\to X}$
such that the fiber

${\displaystyle p^{-1}(x)=G_{d}(E_{x})}$ is the Grassmannian of the d-dimensional vector subspaces of $E_x$. For example,

${\displaystyle G_{1}(E)=\mathbb {P} (E)}$ is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into

${\displaystyle 0\to S\to p^{*}E\to Q\to 0}$.
Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank $r = rk(E)$ and

${\displaystyle \wedge ^{r}S}$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection

${\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)}$ corresponds to the morphism over X:
${\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)}$,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle $T Gd(E)/X$ of $Gd(E)$ is given by[1]
${\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}$
which is morally given by the second fundamental form. In particular, when d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:
${\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0}$,
which is the relative version of the Euler sequence.

2.Explain of the fully nonlinear elliptic equation

Now, we could consider the determination $\sum_{i_1,...,i_k\in\{1,...,n\}}det(u_{ij})_{i,j\in \{i_1,...,i_k\}\times\{i_1,...,i_k\}}$ as the determination of transform: $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$.

Now we need to understand $\sigma_k(D^2(u))=f$ at a point $x_0$ as the average of determination of transform matrix of $(u_{i_1},...,u_{i_k}) \longrightarrow (e_{i_1},...,e_{i_k})$ on Grassmannian manifold $G_k(x_0)$ is equal to $f(x_0)$, i.e.:

$\int_{G_k(x_0)} det(\frac{\partial u_{i_a}}{\partial e_{i_b}}) d\mu=f(x_0)$

where $\mu$ is the natural haar measure on $G_k(x_0) \simeq G_k$.

But the difficult to make the argument rigorous is that $u_i$ is scale and $e_i$ is vector.

Regularity of k-curvature equation

this is a note after reading the article”” of Cafferalli.

in his article,a large type of fully nonlinear elliptic equation has been established.in particular,including the k-curvature equation.and use the continue method,we just need to establish a ingredient estimate,$C^2$ estimate in the interior and $C^2$ estimate near the boundary.we establish these estimate step by step,base on construct special flexible function and use the maximum principle to establish the first and second estimate,for the $C^2$ estimate near the boundary we need to investigate the influence of permutation on the boundary carefully.

Heat flow and the zero of polynomial-a approach to Riemann Hypesis

this is a note after reading the blog:Heat flow and the zero of polynomial.

1.instead of consider the original version:

$\partial_{zz}f(z,t)=\partial_tf(z,t)$.

consider the corresponding “equidistribution version” is also interesting:

$\partial_{zz}f(z,t)=\theta(z,t)\partial_tf(z,t)$,especially $\theta(z,t)=e^{2\pi i\alpha t},\alpha\in R-Q$.

2.

where $f(z)=z^n+a_{n-1}z^{n-1}+...+a_1z+a_0$.

$f(z,t)=\sum_{k=1}^n\sum_{0\leq m\leq k-2,2|k-m}\frac{k!}{m!(k-m)!}z^mt^{k-m}.$

$=\sum_{k=1}^m\sum_{0\leq m\leq k-2,2|k-m}C_k^mt^{k-m})z^mt^{k-m}$

$\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_k^mt^{k-m})z^m$.

rescaling:

$F_t:(z_1(t),...,z_n(t))\longrightarrow (\frac{z_1(t)}{t},...,\frac{z_n(t)}{t})$.

$F_t\cdot f(z,t)=\sum_{m=0}^{n-2}(\sum_{k=m,2|k-m}^nC_{k}^mt^{k-n})z^m$.

$\lim_{t\to \infty}F_t\cdot f(z,t)=\sum_{m=0,2|n-m}^{n-2}C_n^mz^m$.(*)

even term $\longrightarrow$ constant.(after renormelization)

odd term $\longrightarrow$ 0(invariant).so at least the sum zeros of is invarient.

by the algebraic fundamental theorem,we have n zero $\{z_1,...,z_n\}$of (*).

until now,we already now if the n zeros is distinct,then because the energy is the energy is the same and the entropy is increase so $\exists T>>0,\forall t_i,t_j>T$,$\{t>T|z_i(t)\} \cap \{t>T|z_j(t)\}=\emptyset$.$\lim_{t\to \infty}|z_i(t)|=\infty$ and $\lim_{t\to \infty}arg(z_i(t))=z_i$.

but how to know the information of the change of direction at “blow up” time?

1.change direction only at $blow up$.

2.energy invariant $\sum_{1\leq i\neq j\leq n}\frac{1}{|x_i-x_j|^2}$.

3.general philosophy

deformation some function under some evolution equation, such like heat equation,wave equation,shrodinger equation.and there is some conversion thing under the equation,and some quantity that could calculate directly such like the trace of spectral.

4.difficultis

this philosophy could generate to the analytic function case,but to make the limit case(I only know how ti deal with this now)coverage.we need very good control on the coefficient.

and to investigate the change of direction at blow up point maybe we need some knowledge about the burid group.

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

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

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

use the similar argument as ${p=1}$ to prove the situation ${1.
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

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 $\Box$