Ngô Quốc Anh

August 29, 2010

Achieving regularity results via bootstrap argument, 4

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 2:36

Let us consider the following equation

\displaystyle u(x) = \int_{{\mathbb{R}^n}} {\frac{{u{{(y)}^{\frac{{n + \alpha }}{{n - \alpha }}}}}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} , \quad x \in {\mathbb{R}^n}

for n\geqslant 1 and 0<\alpha<n. In this entry, by using boothstrap argument, we show that

Theorem. If positive function u \in L_{loc}^\frac{2n}{n-\alpha}(\mathbb R^n) solves the equation, then u \in C^\infty(\mathbb R^n).

In the process of proving the result, we need the following result

Proposition. Let V \in L^\frac{n}{\alpha}(B_3) be a non-negative function and set

\displaystyle \delta(V)=\|V\|_{L^\frac{n}{\alpha}(B_3)}.

For \nu >r>\frac{n}{n-\alpha}, there exist positive constants \overline \delta<1 and C \geqslant 1 depending only on n, \alpha, r and \nu such that for any 0 \leqslant V \in L^\frac{n}{\alpha}(B_3) with \delta(V) \leqslant \overline \delta, h \in L^\nu(B_2) and 0 \leqslant u \in L^r(B_3) satisfying

\displaystyle u(x) \leqslant \int_{{B_3}} {\frac{{V(y)u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} + h(x), \quad x \in {B_2}

we have

\displaystyle {\left\| u \right\|_{{L^\nu }({B_{1/2}})}} \leqslant C\left( {{{\left\| u \right\|}_{{L^r}({B_3})}} + {{\left\| h \right\|}_{{L^\nu }({B_2})}}} \right).

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June 13, 2010

Achieving regularity results via bootstrap argument, 2

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 1:37

Today, we shall discuss a very strong tool in the theory of elliptic PDEs in order to achieve the smoothness of solution. The tool we just mentioned is known as the Calderón-Zygmund L^p estimates or the Calderón-Zygmund inequality. Precisely,

Theorem (Calderón-Zygmund). Let 1<p<\infty and f \in L^p(\Omega) (\Omega is open and bounded). Let u be the weak solution of the following PDE

\displaystyle \Delta u = f.

Then u\in W^{2,p}(\Omega') for any \Omega' \Subset \Omega.

Let us consider the regularity of solution of

\displaystyle \Delta u +\Gamma(u)|\nabla u|^2=0

with a smooth \Gamma. We also require that \Gamma is bounded.

Motivation. The above PDE occurs as the Euler-Lagrange equation of the variational problem

\displaystyle I(u)=\int_\Omega g(u(x))|\nabla u(x)|^2dx \to {\rm min}

with a smooth g with is bounded and bounded away from zero. Moreover, g' is bounded.

In fact, to derive the Euler-Lagrange equation, we consider

\displaystyle I(u + t\varphi ) = \int_\Omega {g(u + t\varphi ){{\left| {\nabla (u + t\varphi )} \right|}^2}dx}

where \varphi \in H_0^{1,2}(\Omega). In that case

\displaystyle \frac{d}{{dt}}I(u + t\varphi ) = \int_\Omega {\left[ { - 2g(u)\Delta u - g'(u){{\left| {\nabla u} \right|}^2}} \right]\varphi dx}

after integrating by parts and assuming for the moment u \in C^2. Thus, the minimizer will verify

\displaystyle - 2g(u)\Delta u - g'(u){\left| {\nabla u} \right|^2} = 0

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May 8, 2010

Achieving regularity results via bootstrap argument

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:35

Let us now consider a very important technique in PDEs called the  bootstrap method or bootstrap argument. Actually, this method is a part of the proof of the Schauder estimates, etc. I will follow the recent paper due to Li, Strohmer and Wang published in Proc. Amer. Math. Soc. in 2009.

The equation considered here is the following

\displaystyle u(x) = A\int_\Omega {\frac{{|u|^p (y)}}{{|x - y{|^{n - \alpha }}}}dy} + B

together with

u\big|_{\partial\Omega}=\beta

where \Omega is a C^1 bounded domain. We also assume

p,A >0, \quad \beta, B \geqslant 0, \quad 1<\alpha <n.

Theorem. If u \in L^q(\Omega) is a solution to the PDE for some q>1 then u \in C(\overline \Omega).

In order to run the bootstrap argument, we need the following auxiliary result

Lemma. Suppose w\in L^r(\Omega) with 1\leqslant r<\infty and

\displaystyle v(x) = A\int_\Omega {\frac{{|w| (y)}}{{|x -  y{|^{n - \alpha }}}}dy} + B

then v \in W^{1,s}(\Omega) where

\begin{cases}\frac{1}{s}+\frac{\alpha-1}{n}=\frac{1}{r}, & {\rm if }\; 1 \leqslant r <\frac{n}{\alpha-1},\\{\rm any} \; s \geqslant 1, & {\rm if } \; r \geqslant \frac{n}{\alpha-1}.\end{cases}

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