Ngô Quốc Anh

July 21, 2010

Regularity Lifting by Contracting Operators

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 4:47

I just read this method in a book due to W.X. Chen and C.M. Li published by American Institute of Mathematical Sciences this year 2010.

Let V be a topological vector space. Suppose there are two extended norms (i.e. the norm of an element in V might be infinity) defined on V

\|\cdot\|_X, \|\cdot\|_Y :V\to [0,\infty].

Let

X:=\{v \in V: \|v\|_X <\infty\}

and

Y:=\{v \in V: \|v\|_Y <\infty\}.

Theorem (Regularity Lifting I). Let T be a contraction map from X into itself and from Y into itself. Assume that f \in X, and that there exits a function g \in Z := X \cap Y such that f = Tf + g in X. Then f also belongs to Z.

Proof. Firstly, let

\sqrt{\|\cdot\|_X^2+\|\cdot\|_Y^2}

be a norm on Z. We first show that T : Z \to Z is a contraction. Since T is a contraction on X, there exists a constant \theta_1, 0 < \theta_1 < 1 such that

\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_X} \leqslant {\theta _1}{\left\| {{h_1} - {h_2}} \right\|_X},\quad\forall {h_1},{h_2} \in X.

Similarly, we can find a constant \theta_2, 0 < \theta_2 < 1 such that

\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_Y} \leqslant  {\theta _2}{\left\| {{h_1} - {h_2}} \right\|_Y},\quad\forall {h_1},{h_2}  \in Y.

Let \theta = \max\{\theta_1,\theta_2\}. Then, for any h_1, h_2 \in Z

\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_Z} = \sqrt {\left\| {T{h_1} - T{h_2}} \right\|_X^2 + \left\| {T{h_1} - T{h_2}} \right\|_Y^2} \leqslant \theta {\left\| {{h_1} - {h_2}} \right\|_Z}.

Since T : Z \to Z is a contraction, given g \in Z, we can find a solution h \in Z such that h = Th + g. Notice that T : X \to X is also a contraction and g \in Z \subset X, the solution of the equation x = Tx + g must be unique in X. Because both h and f are solutions of the same equation x = Tx + g in X, we deduce that f = h \in Z.

Remark. In practice, we usually choose V to be the space of distributions, and X and Y to be function spaces, for instance, X = L^p(\Omega) and Y = W^{1,q}(\Omega). We start from a function f in a lower regularity space X, if we can show that T is a contraction from X to itself and from Y to itself, then we can lift the regularity of f to be in

Z = X \cap Y = L^p(\Omega)\cap W^{1,q}(\Omega).

Applications to PDEs. Now, we explain how the Regularity Lifting Theorem proved in the previous subsection can be used to boost the regularity of week solutions involving critical exponent

\displaystyle -\Delta u =u^\frac{n+2}{n-2}.

Still assume that ­ \Omega is a smooth bounded domain in \mathbb R^n with n \geqslant 3. Let
u \in H^1_0(\Omega) be a weak solution of the above PDE. Then by Sobolev embedding

\displaystyle u \in L^\frac{2n}{n-2}(\Omega).

We can split the right hand side of the PDE in two parts

\displaystyle u^\frac{n+2}{n-2}=u^\frac{4}{n-2}u=a(x)u.

Then obviously a \in L^\frac{n}{2}(\Omega). Hence, more generally, we consider the regularity of the weak solution of the following equation

\displaystyle -\Delta u =a(x)u+b(x).

Theorem. Assume that a,b \in L^\frac{n}{2}(\Omega). Let u \in H_0^1(\Omega) be any weak solution of the foregoing PDE. Then u \in L^p(\Omega) for any 1 \leqslant p<\infty.

The proof of this theorem can be found in the book mentioned above.

Let us go back to the first PDE. Assume that u is a H^1_0(\Omega) weak solution. From the above theorem, we first conclude that u is in L^q(\Omega) for any 1 < q < \infty. Then by a standard regularity result known as the W^{2,p}-regularity for a second order uniformly elliptic operator in divergence form, u is in W^{2,q}(\Omega). This implies that u \in C^{1,\alpha}(\Omega) via Sobolev embedding. Finally, by repeated applications of the Schauder estimates, we derive that u \in C^\infty(\Omega).

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