Coercivity and the Poincar´e inequality John McCuan April 8, 2014 Coercivity for the bilinear form Z X Z X Z B(u, v) = aij Dj uDi v + bj vDj u + cuv. Ω i,j Ω j Ω associated with the linear partial differential operator X X Lu = − Di (aij Dju) + bj Dj u + cu j i,j is the requirement that for some m > 0, B(u, u) ≥ mkukH 1 . Here we prove carefully the main lemma concerning coercivity for operators of the form L which are elliptic and explain the role played by the Poincar´e inequality. 1 Ellipticity We assume the coefficients aij , bj and c all defined and bounded on the closure of some bounded domain Ω ⊂ Rn . We assume further the condition of uniform ellipticity, namely that for some ǫ0 > 0 X aij ξi ξj ≥ ǫ0 |ξ|2 for all ξ ∈ Rn . Using ellipticity, we get the initial estimate Z Z XZ 2 |u||Dj u| − c¯ |u|2 B(u, u) ≥ ǫ0 |Du| − ¯b 1 where ¯b = sup |bj (x)| and c¯ = sup |c(x)|. j,x∈Ω x∈Ω The last two terms are not in our favor. We only have the (small) ǫ0 kDuk2L2 term on which to rely. To make matters worse, we need to somehow insert an additive term kuk2L2 on the right to get, finally, and H 1 norm on the right. Let us first note that the inequality ab ≤ ǫ2 2 1 a + 2 b2 2 2ǫ can be applied to the second term to preserve at least some of our only help. That is, for any ǫ > 0, Z Z ¯bǫ2 Z ¯b Z 2 2 2 B(u, u) ≥ ǫ0 |Du| − |Du| − 2 |u| − c¯ |u|2. 2 2ǫ In particular, taking ǫ2 < ǫ0 /¯b, we get an inequality Z Z ǫ0 2 B(u, u) ≥ |Du| − M |u|2 2 (1) where M > 0 is some (large) constant. Of course if there were no b and c terms there would be no troublesome Mkuk2L2 term, but we would still have the difficulty of replacing the norm of Du with an H 1 norm of u. We attempt to address this unavoidable difficulty now. 2 Poincar´ e inequality Recall that the H 1 norm may be defined variously by X |Dj u|L2 kukH 1 = |u|L2 + j or kukH 1 = |u|2L2 + X j or even |Dj u|2L2 !1/2 kukH 1 = |u|L2 + max |Dj u|L2 . j 2 In view of our initial estimate above, it looks like we might wish to use the second form of the norm. There are various inequalities which relate/bound norms of a function in terms of norms of its derivative. Perhaps the simplest is the Cc∞ Sobolev inequality: If u ∈ Cc∞ (Rn ) and 1 ≤ p < n, then kukLp∗ ≤ CkDukLp where p∗ = np/(n − p) is the Sobolev exponent. Here C is a positive constant that depends on n and p, but (most importantly) is independent of u. In fact, one only needs u ∈ Cc1 (Rn ) for this result. In this case, we have u ∈ H01 (Ω), so we use the following version called the W01,p Poincar´e inequality: Theorem 1 If Ω is a bounded domain in Rn , n > p ≥ 1, and 1 ≤ q ≤ p∗ , then there is a constant C = C(n, p, q, Ω) such that for all u ∈ W01,p (Ω). kukLq (Ω) ≤ CkDukLp (Ω) Proof: Since Cc∞ is dense in W01,p , there is a sequence of Cc∞ functions uj with kuj − ukW 1,p → 0. Setting u¯j (x) = uj (x), x ∈ Ω 0, xRn \Ω we have u¯j ∈ Cc∞ (Rn ). Therefore, applying the Cc∞ Sobolev inequality, we get uj kLp . k¯ uj kLp∗ ≤ CkD¯ This is precisely the same as kuj kLp∗ (Ω) ≤ CkDuj kLp (Ω) . And we can take a limit to obtain kukLp∗ (Ω) ≤ CkDukLp (Ω) . 3 Finally, we claim that for 1 ≤ q ≤ p∗ there is some C for which kukLq (Ω) ≤ CkukLp∗ (Ω) . To see this, note that since |u|q ∈ Lm where m = p∗ /q ≥ 1, Z ∗ q/p∗ q kukLq (Ω) = |u|p Z ∗ q/p∗ = |u|p χΩ Z q/p∗ m p∗ |Ω| m−1 ≤ |u| = Z |u| p∗ q/p∗ p∗ |Ω| p∗ −q . Thus, p∗ kukLq (Ω) ≤ CkukLp∗ (Ω) 3 with C = |Ω| q(P ∗ −q) . Estimate Returning to (1) and using Theorem 1 in the form kDukLp (Ω) ≥ kukLq (Ω) /C, we obtain Z Z Z ǫ0 ǫ0 2 2 B(u, u) ≥ |Du| + |Du| − M |u|2 4 4 Z Z Z ǫ0 ǫ0 2 2 ≥ |Du| + |u| − M |u|2 4 4C Z 2 ≥ mkukH 1 (Ω) − M |u|2 where ǫ0 o > 0. 4 4C That is essentially the best we can do: m = min nǫ 0 , Lemma 1 (Main coercivity lemma) If L is elliptic, then there is some constants m, M > 0 such that Z 2 B(u, u) ≥ mkukH 1 (Ω) − M |u|2. 4 Corollary 1 If L is elliptic, then there is a constant M > 0 such that ˜ v) = B(u, v) + µhu, viL2 is coercive for each µ ≥ M. B(u, Corollary 2 If L is elliptic, and Z X Z bj uDj u + cu2 ≥ 0 Ω j Ω then B : H01 × H01 → R is coercive. 5 for u ∈ H01 (Ω),