18.156 Lecture 20 Notes

Lecture 20 Difference Quotients and Sobolev spaces Define u(x + h · el ) − u(x) , h ∆hi u := Lemma. h 6= 0. Let Ω be a bounded domain in Rn , and u ∈ W 1,p (Ω), for some 1 ≤ p < ∞. Then for ⊂ Ω such that dist(Ω′ , ∂Ω) > h holds any Ω′ ⊂ ||∆hi u||Lp (Ω′ ) ≤ ||Di u||Lp (Ω) . Proof. |∆hi u| u(x + h · el ) − u(x) 1 = ≤ h h Z h Di u(x1 , . . . , xi + ζ, . . . , xn ) dζ 0 Z h Z  h 1 p q 1q · 1 } ≤ { Di u(x1 , . . . , xi + ζ, . . . , xn ) dζ h 0 0 p |∆hi u|p ≤ h q −p · = Z Ω′ |∆hi u|p ≤ = 1 · h Z 1 · h Z 1 h Z Z h p Di u(x1 , . . . , xi + ζ, . . . , xn ) dζ 0 h p Di u(x1 , . . . , xi + ζ, . . . , xn ) dζ 0 Ω′ Z h 0 Di |p dζdx = 1 · h Z h 0 Z ⇒ ′ Ω Di |p dxdζ h 0 ||Di u||Lp (Ω′ ) = ||Di u||Lp (Ω′ ) ≤ ||Di u||Lp (Ω) , 1 1 p ⇒ where we applied Fubini’s Theorem in order to switch order of integration. Conversely we have Lemma. Let u ∈ Lp (Ω) for some 1 ≤ p < ∞ and suppose ∆hi u ∈ Lp (Ω′ ) with ||∆hi u||Lp (Ω′ ) ≤ K ⊂ Ω and 0 < |h| < dist(Ω′ , Ω). Then the weak derivative satisfies ||Di u||Lp (Ω) ≤ K. for all Ω′ ⊂ Consequently if this holds for all i = 1, . . . , n then u ∈ W 1,p (Ω). Proof. We will make use of Alouglou’s Theorem. A bounded sequence in a separable, reflexive Banach space has a weakly convergent subsequence. A topological space is called separable if it contains a countable dense set. A Banach space is called reflexive if (B ⋆ )⋆ = B. A sequence {xn } in a Banach space is said to converge weakly to x when lim F (xn ) → F (x) for n→∞ all linear functionals F ∈ B ⋆ . This is sometimes denoted lim xn ⇀ x. n→∞ 2 n Example: Let ℓ := (a1 , a2 , . . .) : ∞ X i=1 o a2i < ∞ . Consider the sequence {xi := (0, . . . , 0, 1, 0, . . .)} ⊆ ℓ2 . Any bounded linear functional on ℓ2 will be some linear combination of the linear functionals Fj , defined by Fj (a1 , . . .) = aj (each such linear combination corresponds exactly to a point in ℓ2 . That makes sense, indeed by the Riesz Representation Theorem (ℓ2 )⋆ = ℓ2 (note ℓ2 is a Hilbert space not just a Banach space as it has an inner product structure).). For any such F = (a1 , . . .), lim F (xi ) = lim ai = 0. So xi converges to the 0 vector weakly, though certainly not strongly: i→∞ i→∞ by Fourier Theory each point in ℓ2 corresponds to a periodic function on [0, 1], i.e an element of √ L2 (S 1 ), and of course lim exp(n2π −1z) 6→ 0(z). n→∞ 2 We come back to the proof. For the Banach space B = Lp (Ω), B ⋆ = Lq (Ω) with This can be seen directly: If F ∈ (Lp (Ω))⋆ , then exists f such that F (g) = R Ω g · f, 1 1 + = 1. p q ∀g ∈ Lp (Ω), and this will be bounded iff f ∈ Lq (Ω). So we get an identification F ∈ (Lp (Ω))⋆ ∼ = Lq (Ω). By Alouglou’s Theorem there exists a sequence {hm } → 0 with ∆hi m u ⇀ v ∈ Lp (Ω). In other words Z Ω ∆hi m u ψ· → Z Ω ψ · v ∈ Lp (Ω), ∀ ψ ∈ Lq (Ω). And in particular for any ψ ∈ C01 (Ω) (which is dense in Lq (Ω) so will suffice to look at such ψ as will become clear ahead) Z Ω ψ∆hi m u = 1 ψ (u(x + h · el ) − u(x))dx Ω h Z 1 = h = Z Z ψ(x)u(x)dx Ω Z 1 (ψ(x − hei ) − ψ(x))u(x)dx h Z −∆hi ψ(x)u(x)dx Ω = 1 ψ(x − hei )u(x)dx − h Ω Ω h→0 −→ Z −Di ψ(x)u(x)dx Ω since ψ is continuously differetiable. Altogether Z Ω p ψ · v ∈ L (Ω) = Z Ω −Di ψ(x)u(x)dx, which by definition means v is the weak derivative of u in the direction of the xi axis, or simply the undistinctive notation v = Di u. We also get the desired estimate, using the Fatou Lemma Z p Ω |Di u| dx = Z lim inf Ω |∆hi u|p dx i.e ||Di u||Lp (Ω) ≤ K. 3 R ≤ lim inf lim inf ≤ lim inf Z Ω R |∆hi u|p dx ≤ K p , L2 Theory Consider the second order equation in divergence form Lu ≡ L(u) := Di (aij Dj u) + bi Di u + c · u = f, with aij , bi , c ∈ L1 (Ω) (integrable coefficients). We call u ∈ W 1,2 (Ω) a weak solution of the equation if ∀ v ∈ C01 (Ω) − Z aij Dj uDi v + Ω Z (bi Di u + cu)v = Ω Z f v. Ω Elliptic Regularity Let u ∈ W 1,2 (Ω) be a weak solution of Lu = f in Ω, and assume • L strictly elliptic with (aij ) > γ · I, γ > 0 • aij ∈ C 0,1 (Ω) • bi , c ∈ L∞ (Ω) • f ∈ L2 (Ω) ⊂ Ω, u ∈ W 2,2 (Ω′ ) and Then for any Ω′ ⊂
||u||W 2,2 (Ω′ ) ≤ C(||aij ||C 0,1 (Ω) , ||b||C 0 (Ω) , ||c||C 0 (Ω) , λ, Ω′ , Ω, n) · ||u||W 1,2 (Ω) + ||f ||L2 (Ω) . Note L∞ (Ω) stands for bounded functions on Ω while C 0 (Ω) are functions that are also continuous (Ω being bounded). Proof. Start with the definition of u being a solution in the weak sense, ∀ v ∈ C01 (Ω): 4 Z aij Dj uDi v = Z Ω Ω (bi Di u + c − f )v. and take difference quotients, that is replace v with ∆−h v. Z aij Dj uDi (∆−h v) = Z Ω Ω (bi Di u + c − f )(∆−h v). Taking −h is a technicality that will unravel its reason later on, and we really mean ∆−h k v for some k ∈ {1, . . . , n} and then eventually repeat the computation for all k in that range. This will become clear as well. Finally our goal will be to use the Chain Rule and move the difference quotient operator onto u under the integral sign and get uniform bounds on ∆h Du and in this way get a priori W 2,2 (Ω) estimates. The Chain Rule gives ∆h (aij Dj u) =
1 ij a u(x + h · ek )Dj u(x + h · ek ) − {aij (x) − aij (x + h · ek ) + aij (x + h · ek )}Dj u(x) h = aij u(x + h · ek )∆h Dj u − ∆h aij Dj u. And applied to our previous equation, a short calculation verifies that we can ’integrate by part’ wrt ∆h – Z Z Z aij Dj uDi (∆−h v) = Ω Ω ∆h (aij Dj u)Di v Ω Ω ij Z h a u(x + h · ek )∆ Dj uDi v = Z Ω ⇒ −∆h aij Dj uDi v + Z Ω (bi Di u + c − f )(∆−h v) ⇒ aij u(x + h · ek )∆h Dj uDi v ≤ ||∆h aij Dj u||L2 (Ω) ||Di v||L2 (Ω) + + ||bi Di u + cu − f ||L2 (Ω) ||∆−h v||L2 (Ω) , 5 where we have used the Hölder Inequality for p = q = 2. This in turn can be bounded by ≤ ||aij ||C 0,1 (Ω) ||Du||L2 (Ω) ||Dv||L2 (Ω) +
+ ||bi ||L∞ (Ω) ||Du||L2 (Ω) + ||c||L∞ (Ω) ||u||L2 (Ω) + ||f ||L2 (Ω) ||Dv||L2 (Ω) ≤ C(||u||W 1,2 (Ω) + ||f ||L2 (Ω) ) · ||Dv||L2 (Ω) . where we have used the Hölder Inequality for p = 1, q = ∞, i.e a simple bounded integration Z Z
1
1 argument (e.g ||cu||L2 (Ω) = |u|2 2 ), and ∆h aij → Dk aij as aij C 0,1 (Ω). c2 · |u|2 2 ≤ sup |c|2 O Take a cut-off function η ∈ C01 (Ω), 0 ≤ |η| ≤ 1, η Ω′ = 1. We now choose a special v: v := η 2 ∆h u. From uniform ellipticity (aij ζi ζj ≥ λ|ζ|2 ) λ Z h Ω 2 |ηD∆ u| ≤ Z Ω η 2 aij (x + h · ek )Di ∆h uDj ∆h u. Now Di v = 2ηDi η∆h u + η 2 Di ∆h u which we substitute into our previous inequality, Z 2 ij Ω h h η a (x + h · ek )Dj ∆ uDj ∆ u ≤ Z Ω aij (x + h · ek )Dj ∆h u · (Di v − 2ηDi η∆h u) ≤ C(||u||W 1,2 (Ω) + ||f ||L2 (Ω) )||Dv||L2 (Ω) + + C ′ ||(D∆h u)η||L2 (Ω) ||Dη∆h u||L2 (Ω) again by the Hölder Inequality. Now since η ≤ 1
||Di v||L2 (Ω) ≤ C ′′ ||Di η∆h u||L2 (Ω) + ||D∆h u||L2 (Ω) . Combining all the above and again using η ≤ 1, 6 λ Z ′ Ω

|ηD∆h u|2 ≤ C ||u||W 1,2 (Ω) + ||f ||L2 (Ω) · C ′′ ||Dη∆h u||L2 (Ω′ ) + ||D∆h u||L2 (Ω′ ) + C ′ ||(D∆h u)||L2 (Ω′ ) ||Dη∆h u||L2 (Ω′ )
≤ c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + ||Dη∆h u||L2 (Ω′ ) · ||(D∆h u)||L2 (Ω′ )
+ c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) · ||Dη∆h u||L2 (Ω′ ) . Using the AM-GM Inequality ab = r
1 2 1 1 2 a · ǫb2 ≤ a +ǫb2 for the first term and the inequality ǫ 2 ǫ (a + b)c ≤ 12 (a + b + c)2 for the second λ Z Ω′ |ηD∆h u|2 ≤
2 1 2 c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + ||Dη∆h u||L2 (Ω′ ) + ǫ||(D∆h u)||2L2 (Ω′ ) ǫ
2 + c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + ||Dη∆h u||L2 (Ω′ ) . Choose any 0 < ǫ < λ/2. Then subtract the second term on the first line of the rhs from the lhs to get ||ηD∆h u||2L2 (Ω′ ) ≤ c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + ||Dη∆h u||L2 (Ω′ )
2 ||ηD∆h u||L2 (Ω′ ) ≤ c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + ||Dη∆h u||L2 (Ω′ )
⇒ ≤c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) + sup |Dη| · ||∆h u||L2 (Ω′ ) Ω


≤c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) · 1 + sup |Dη| , Ω since ||∆h u||L2 (Ω) ≤ ||Du||L2 (Ω) ≤ ||u||W 1,2 (Ω) ≤ ||u||W 1,2 (Ω) + ||f ||L2 (Ω) where we have applied the first Lemma to u ∈ W 1,2 (Ω). Now we are done as we can choose η such that first η lhs !) and second |Dη| ≤ dist(Ω′ , ∂Ω) (for the rhs ) and so
||1 · D∆h u||2L2 (Ω′ ) ≤ c ||u||W 1,2 (Ω) + ||f ||L2 (Ω) , 7 Ω′ = 1 (for the independently of h. So by our second Lemma the uniform boundedness of the difference quotients of Du in L2 (Ω′ ) implies Du ∈ W 1,2 (Ω′ ) ⇒ u ∈ W 2,2 (Ω′ ) and we have the desired estimate for its W 2,2 (Ω′ ) norm by the above inequality combined with the Lemma. Now that u ∈ W 2,2 (Ω′ ) then the our original equation holds in the usual sense Lu = aij Di ju + Di aij Dj u + bi Di u + c · u = f, a.e ! 8