The Mollifier Theorem

Transcription

1 The Mollifier Theorem Definition of the Mollifier The function Tx Kexp 1 if x x 0 if x 1, x R n where the constant K is chosen such that R n Txdx 1, is a test function on Rn. Note that Tx vanishes, together with all its derivatives as x 1, so Tx is infinitely differentiable and has compact support. The graph of Tx is sketched in the following figure..for n 1 and 0, let The Mollifier Function S x 1 T x and P x T x. Then S x 0 and P x 0 for all x S x 0 and P x 0 for x R S x dx 1 0, S 0 as 0, R P x dx 0 as 0, P 0 K/e 0, Evidently, S x becomes thinner and higher as tends to zero but the area under the graph is constantly equal to one. On the other hand, P x has constant height but grows thinner as tends to zero. These test functions can be used as the seeds from which an infinite variety of other test functions can be constructed by using a technique called regularization which we will now describe. For n 1 we have S x 1 n T x and P x T x. 1

2 For U a bounded open set in R n, and for u L 1 loc U, define for any 0 and any x U x U : distx,u, J ux S x y uy dy z S z ux z dz z 1 S 1z ux z dz. 1.1a 1.1b 1.1c We refer to J ux as the mollified ux. This mollified function, J ux, is a smoothed version of the original function, ux. Properties of the Mollifier Note first that J ux is infinitely differentiable; i.e., for any 0 and any x U, it is clear from (1.1a) that i.e., J ux e i J ux S x e i y S x y uy dy J ux e i J ux xi S x y uy dy as 0. Since S x is infinitely differentiable, it follows that J ux is infinitely differentiable on the open set U. It is evident from (1.1a) that for 1 p, 0, and x U, J ux S x y 11/p S x y 1/p uy dy. Then, using Holder s inequality, we get J ux p S x y dy p1 S x y uy p dy and since R S x dx 1 0, J ux p dx S x y uy p dy dx W uy p S x ydx dy W uy p dy for open sets W U, and V W. This result is just that assertion that J u LpV u L pw for V W U 1.2 Next, use (1.1c) to write J ux ux z 1 S 1zux z uz dz. If the function u ux is, in fact, continuous on U, then this last result shows that max V J ux ux max V ux z uz 0 as 0; 1.3 i.e., J ux converges uniformly to ux for x when ux is continuous on.u. For u L p loc U, W U, and arbitrary 0, use the fact that the continuous functions 2

3 are dense in L pw to choose v CW such that uv LpW. Then for V W, J uu LpV J uj v LpV J v v LpV v u L pv uv LpW J v v LpV v u L pw 2 J v v LpV It follows now from (1.3) that for u L p loc U, V U, J uu L p V 0 as We can summarize these results in the following, Theorem (Local Approximation) Suppose U is open and bounded in R n, 1 p, and for 0, let U denote the subset x U : distx,u. (a) For every 0, u L p loc U implies J u C U (b) (i) u CU implies u converges to u uniformly on compact subsets of U; i.e., J uu CV max V J ux ux 0 for all V U (ii) u converges to u in L p loc U; i.e., u L p loc U implies that for all V W U, J u L p V u L p W (c) u converges to u in W k,p loc U; Result (c) follows from (b) by induction. and J uu L p V 0 as 0 Corollary (Global Approximation) Suppose U has a smooth boundary, and 1 p. (a) For every 0, (b) u L p U implies that J u L p U u L p U u L p U implies J u C U L p U. and J uu L p U 0 as 0 (c) u W k,p U implies that there exists functions m C U W k,p U such that m u k,p 0 as m. The proof of the corollary makes use of a partition of unity (see theorem 2 pg 251 in Evans). Weak Equals Strong For U a bounded open set in R n, we define v vx to be the weak derivative of order, of u ux, x U if U ux x dx 1 U vxx dx for all C c U 3

4 Similarly, we define v vx to be the strong L p derivative of order, of u ux, x U if for any V U, there exists a sequence n C c U such that n u p dx 0 and n v p dx 0, as n. Using mollifiers, we can show that these two notions are equivalent. Suppose first that v vx is the weak derivative of order, of u ux. Then, since S C c U, J ux x S x y uy dy 1 y S x y uy dy S x y vy dy J vx (by definition of weak derivative) Now apply 1.4 to write J uu p dx 0 and J uv p dx J v v p dx 0, as n. Thus every weak derivative is a strong L p derivative. Conversely, suppose v vx is the strong L p derivative of order, of u ux with n u p dx 0 and n v p dx 0, as n, for arbitrary V U, and n C c U. Then for any C c U, u n dx u dx n dx u dx 1 n dx u dx 1 vdx 1 v n dx Then it follows that u dx 1 vdx C 1 n u p dx C 2 n v p dx which implies that every strong L p derivative is a weak derivative. 4

5 Weyl s Lemma Weyl s lemma is a famous result that asserts that for U a bounded open set in R n, if u ux is harmonic in U, (i.e., u C 2 U and 2 ux 0, x U ) then ux is infinitely differentiable in U. To see why this result is true, recall that every harmonic function has the mean value property. That is, x U, r, ux uy dŝy 1 Brx uy dsy. nr n1 A n Brx Then J ux S x y uy dy 1 n T x y uy dy 1 T r n 0 uy dsy dr ux Brx 0 ux S B0 y dy ux. na n n T r r n1 dr But this says that 0, x U, J ux ux. Since J ux is infinitely differentiable on U, it follows that ux is infinitely differentiable on U although u need not even be continuous on the closure, Ū. 5