Mollifiers. From R. Showalter's book on PDEs. (i) Mollifiers are C01 approximations of the delta-function. (ii) Mollification is an operator of convolution with a mollifier: n f"(x) = Z f(x y)'"(y) dy ; x R : Rn − 2 n (iii) Mollification is a smoothening operator: f"(x) C (R ). 2 1 (iv) Mollification is a bounded linear operator with norm 1. ≤ p (v) Any function in L (G), p = or C0(G) can be approximated by its 6 1 p mollification. Hence C01(G) is dense in L (G) and C0(G). We shall begin with some elementary results concerning the approxima- tion of functions by very smooth functions. The \strong inclusion" K G between subsets of Euclidean space Rn means K is compact, G is⊂⊂ open, and K G. If A and B are sets, their Cartesian product is given by A B =⊂ [a; b]: a A; b B . If A and B are subsets of Kn (or any other× vectorf space) their2 set2 sumg is A + B = a + b : a A; b B . If G is not open, this definition can be extended iff there exists2 and2 openg set O such that O¯ = G¯, by saying that K G if K O. For each " > 0, let n ⊂⊂ ⊂⊂ '" C (R ) be given with the properties 2 01 n '" 0 ; supp('") x R : x " ; Z '" = 1 : ≥ ⊂ f 2 j j ≤ g Such functions are called mollifiers and can be constructed, for example, by taking an appropriate multiple of 2 2 1 exp( x " )− ; x < " , "(x) = 0 ; j j − jxj " . j j ≥ Let f L1(G), where G is open in Rn, and suppose that the support of f satisfies supp(2 f) G. Then the distance from supp(f) to @G is a positive number δ. We extend⊂⊂ f as zero on the complement of G and denote the extension in L1(Rn) also by f. Define for each " > 0 the mollified function n f"(x) = Z f(x y)'"(y) dy ; x R : (0.1) Rn − 2 1 Lemma 0.1 For each " > 0, supp(f") supp(f) + y : y " and n ⊂ f j j ≤ g f" C (R ). 2 1 Proof : The second result follows from Leibniz' rule and the representation f"(x) = Z f(s)'"(x s) ds : − The first follows from the observation that f"(x) = 0 only if x supp(f)+ y : y " . Since supp(f) is closed and y : y 6 " is compact,2 it follows f j j ≤ g f j j ≤ g that the indicated set sum is closed and, hence, contains supp(f"). p Lemma 0.2 If f C0(G), then f" f uniformly on G. If f L (G), 2 ! p 2 1 p < , then f" p f p and f" f in L (G). ≤ 1 k kL (G) ≤ k kL (G) ! Proof : The first result follows from the estimate f"(x) f(x) Z f(x y) f(x) '"(y) dy j − j ≤ j − − j sup f(x y) f(x) : x supp(f) ; y " ≤ fj − − j 2 j j ≤ g and the uniform continuity of f on its support. For the case p = 1 we obtain f" 1 ZZ f(x y) '"(y) dy dx = Z '" Z f k kL (G) ≤ j − j · j j by Fubini's theorem, since f(x y) dx = f for each y Rn and this R j − j R j j 2 gives the desired estimate. If p = 2 we have for each C0(G) 2 Z f"(x) (x) dx ZZ f(x y) (x) dx '"(y) dy ≤ j − j Z f 2 2 '"(y) dy = f 2 2 ≤ k kL (G)k kL (G) k kL (G)k kL (G) by computations similar to the above, and the result follows since C0(G) is dense in L2(G). (the result for p = 1 or 2 is proved as above but using the H¨olderinequality in place of Cauchy-Schwartz.)6 Finally we verify the claim of convergence in Lp(G). If η > 0 we have a g C0(G) with f g Lp η=3. The above shows f" g" Lp η=3 and we2 obtain k − k ≤ k − k ≤ f" f Lp f" g" Lp + g" g Lp + g f Lp k − k ≤ k − k k − k k − k 2η=3 + g" g Lp : ≤ k − k 2 For " sufficiently small, the support of g" g is bounded (uniformly) and − g" g uniformly, so the last term converges to zero as " 0. !The preceding results imply the following. ! p Theorem 0.3 C01(G) is dense in L (G). Theorem 0.4 For every K G there is a ' C01(G) such that 0 '(x) 1, x G, and '(x) =⊂⊂ 1 for all x in some neighborhood2 of K. ≤ ≤ 2 Proof : Let δ be the distance from K to @G and 0 < " < " + "0 < δ. Let f(x) = 1 if dist(x; K) "0 and f(x) = 0 otherwise. Then f" has its support within x : dist(x; K) ≤ " + " and it equals 1 on x : dist(x; K) " " , f ≤ 0g f ≤ 0 − g so the result follows if " < "0. 3 Distributions. From R. Showalter's book on PDEs. (i) Distributions are linear functionals on C01(G). (ii) For a `nice' function distributions are defined as integrals Tf (φ) = fφ RG (iii) A fundamental example of a distribution is the delta-function. (iv) Distributions are not bounded functionals, but they are closed func- tionals. (v) A derivative of a distribution is a distribution, well-defined by integra- tion by parts. (vi) A derivative of a distribution differs from from a regular derivative. Recall Cm(G) = f C(G): Dαf C(G) for all α m ; f 2 2 j j ≤ g α C1(G) = f C(G): D f C(G) for all α ; f 2 2 g C1(G) = f C1(G) : supp(f) is compact : 0 f 2 g All of them are linear spaces. Cm(G) is also a Banach space (check) with the norm α f Cm(G) = max D f C(G); f C(G) = sup f(x) : jj jj α jj jj jj jj x G j j 2 C1(G) is not a Banach space, but it is a complete metric space with the metric 1 1 f g Cn(G) ρ(f; g) = n jj − jj : X 2 1 + f g n n=1 jj − jjC (G) C1(G) is not even metrizable, but there is a (nonconstructive) `locally com- pact topological vector space' topology induced by the metric like C1(G), that makes it complete. 4 0.1 A distribution on G is defined to be a conjugate-linear functional on C01(G). That is, C01(G)∗ is the linear space of distributions on G, and we also denote it by (G). D∗ 1 1 Example. The space Lloc(G) = L (K): K G of locally integrable functions on G can be identified\f with a subspace⊂⊂ ofg distributions on G as in the Example of I.1.5. That is, f L1 (G) is assigned the distribution 2 loc Tf C (G) defined by 2 01 ∗ Tf (') = Z f';'¯ C01(G) ; (0.2) G 2 where the Lebesgue integral (over the support of ') is used. Theorem 0.3 1 shows that T : Lloc(G) C01(G)∗ is an injection. In particular, the (equiv- alence classes of) functions! in either of L1(G) or L2(G) will be identified with a subspace of (G). D∗ 0.2 We shall define the derivative of a distribution in such a way that it agrees with the usual notion of derivative on those distributions which arise from α continuously differentiable functions. That is, we want to define @ : ∗(G) (G) so that D ! D∗ α m @ (Tf ) = TDαf ; α m ; f C (G) : j j ≤ 2 But a computation with integration-by-parts gives α α TDαf (') = ( 1)j jTf (D ') ;' C1(G) ; − 2 0 and this identity suggests the following. Definition. The αth partial derivative of the distribution T is the distri- bution @αT defined by α α α @ T (') = ( 1)j jT (D ') ;' C1(G) : (0.3) − 2 0 α α Since D L(C01(G);C01(G)), it follows that @ T is linear. Every distri- bution has2 derivatives of all orders and so also then does every function, 1 e.g., in Lloc(G), when it is identified as a distribution. Furthermore, by the very definition of the derivative @α it is clear that @α and Dα are compatible with the identification of C (G) in (G). 1 D∗ 5 0.3 We give some examples of distributions on R. Since we do not distinguish 1 the function f L (R) from the functional Tf , we have the identity 2 loc 1 f(') = Z f(x)'(x) dx ; ' C1(R) : 2 0 −∞ (a) If f C1(R), then 2 @f(') = f(D') = Z f(D'¯ ) = Z (Df)' ¯ = Df(') ; (0.4) − − where the third equality follows by an integration-by-parts and all others are definitions. Thus, @f = Df, which is no surprise since the definition of derivative of distributions was rigged to make this so. (b) Let the ramp and Heaviside functions be given respectively by x ; x > 0 1 ; x > 0 r(x) = H(x) = 0 ; x 0 , 0 ; x < 0 . ≤ Then we have 1 1 @r(') = Z xD'¯(x) dx = Z H(x)' ¯(x) dx = H(') ;' C01(G) ; − 0 2 −∞ so we have @r = H, although Dr(0) does not exist. (c) The derivative of the non-continuous H is given by 1 @H(') = Z D'¯ =' ¯(0) = δ(') ;' C01(G); − 0 2 that is, @H = δ, the Dirac functional.