Introduction to Elliptic Regularity Theory I.

Transcription

1 Introduction to Elliptic Regularity Theory I. ELTE 2013 Introduction to Elliptic Regularity Theory I.

2 Topics in Elliptic PDE Theory Analytical solutions Potential Theory Classical theory Sobolev functions Existence + Uniqueness (Nonexistence + nonuniqueness) Maximum principles Harnack-type inequalities Regularity for nondivergence/fully nonlinear problems Schauder Theory Regularity for weak formulations L p Perturbative Nonperturbative Regularity for energy functionals Partial regularity Introduction to Elliptic Regularity Theory I.

3 Function Spaces I. (with is C 1 and jj < C1) 1 q p 1 H) L p./,! L q./ k > ` H) W k;p./,!,! W `;p./ 1 p < n; W 1;p./,! L p./ and 1 q < p H) W 1;p./,!,! L q./ 1 p D 1 p 1 n ; or p D np.> p/ n p n < p < 1 H) W 1;p./,! C 0;1 n=p./ 1 kp < n; 1 q < p H) W k;p./,!,! L q./ 1 p D 1 p k n ; or p D np.> p/ n kp n D kp; p > 1; p q < 1 H) W k;p./,! L q./ n < kp < 1 H) W k;p./,! C k Œn=p 1;./, where 2.0; 1/ if Œn=p D n=p and D Œn=p n=p C 1 otherwise. Introduction to Elliptic Regularity Theory I.

4 Function Spaces II. Local Hölder seminorm 8f 2 C./ 8 2.0; 1/ 8x 2 W jf j x; D sup y2 jf.x/ kx f.y/j yk Hölder seminorm 8f 2 C./ 8 2.0; 1/ W jf j D sup x;y2 jf.x/ kx f.y/j yk The Hölder space C r;./ is a Banach space with the norm kf k C r;./ D kf k C r./ C max jd f j : jjdr Obviously C r;./ C r./. 0 < < ˇ < 1 H) C 0;ˇ./,!,! C 0;./ (from Arzela Ascoli) Introduction to Elliptic Regularity Theory I.

5 Interior L 2 regularity Trivial: kuk L 2./ C k uk L 2./ C kuk W 2;2./ L 2 regularity establishes the opposite direction. Theorem (Interior L 2 regularity for the Poisson equation) Let u 2 W 1;2./ be a weak solution of u D f 2 L 2./. Then 8 0 b W kuk W 2;2. 0 / C n;; 0.kuk L 2./ C kf k L 2.//: Moreover u D f a.e. on. Where are the boundary conditions? Conditions on the boundary? Why interior? Folklore: k uk 2 and kuk 2 are enough to control kuk 2;2 By embedding, kd 2 uk 2 already controls kduk 2 Introduction to Elliptic Regularity Theory I.

6 Interior L 2 regularity proof sketch One proof using Nirenberg s difference quotients:.d h k u/.x/ D u.x C he k/ u.x/ : h (Note that kd h uk p kduk p, partial integration-like rule, product rule,...) Different proof using a Calderón ygmund-like inequality. Introduction to Elliptic Regularity Theory I.

7 Sidetrack: Calderón ygmund inequality Consider the Newton potential, w.x/ D G.x; y/f.y/ dy; where G.x; y/ kx yk nc2 if n 3 and G.x; y/ log jx yj if n D 2. (See singular integrals in harmonic analysis) Theorem (C inequality) Let 1 < p < 1, R n a bounded domain, f 2 L p./. Then w 2 W 2;p./, w D f a.e. on and kd 2 wk L p./ C n;p kf k L p./: Note that Dw is a Riesz potential, but D 2 w is a C operator. Corollary If u 2 W 2;p 0./, then kd 2 uk L p./ C n;p k uk L p./. Introduction to Elliptic Regularity Theory I.

8 Interior L p regularity Consider the divergence form uniformly elliptic operator with bounded coefficents Lu D div.a.x/du/ Strong problem: Lu D f, u D g Weak problem: ' 2 Cc 1./.A.x/Du; D'/ dx D f ' dx Theorem Let u 2 W 1;2./ be a weak solution to Lu D f 2 W k;2./. Suppose that a ij 2 C kc1./. Then for all 0 b, u 2 W kc2;2. 0 /. Moreover, kuk W kc2;2. 0 / C.kuk L 2./ C kf k W k;2./ /. Introduction to Elliptic Regularity Theory I.

9 Boundary L p regularity Under suitable smoothness assumptions, interior regularity may be extended to the boundary. Theorem Let be of class C kc1, u a weak solution to Lu D f 2 W k;2./, u D g 2 W kc2;2./ Suppose that a ij 2 C kc1./. Then u 2 W kc2;2./ and kuk W kc2;2./ C.kf k W k;2./ C kgk W kc2;2./ /: Nonconvex polyhedral domains? Introduction to Elliptic Regularity Theory I.

10 Hölder spaces I. Some nice results related to Hölder continuous functions. Theorem (Campanato) Let p 1, n < n C p, and suppose that satisfies 9ı > 0 8x 0 2 8r > 0 W jb.x 0 ; r/ \ j ır n : Then for u 2 L p./ we have u 2 C 0;./ for D. n/=p, if and only if 9K > 0 8x 0 2 8r > 0 W ˇ ˇ ˇp K p r B.x 0 ;r/ ˇu ux0 ;r Variance, L p -approximation by means on balls. Only if is easy. Introduction to Elliptic Regularity Theory I.

11 Hölder spaces II. Corollary Suppose that u 2 L 2./ satisfies 9 2.0; 1/ 8x 0 2 8r > 0 W B.x 0 ;r/ ju u x0 ;rj 2 K 2 r nc2 Then u 2 C 0;./ and 8 0 b W kuk C 0;./ C.M C kuk L 2.//: Combine with Theorem (Poincaré inequality) Let be convex, u 2 W 1;p./. Then 8E ; jej > 0, ku jj 1=n diam./ n u E k L p./ C n kduk L jej p./ Introduction to Elliptic Regularity Theory I.

12 Hölder spaces III. To get Theorem (Morrey) Suppose that u 2 W 1;2./, and 8x 0 2 8r > 0 W B.x 0 ;r/ kduk 2 M 2 r n 2C2 : Then u 2 C 0;./. Another result of Morrey involving the oscillation... Theorem (Morrey) Suppose that u 2 W 1;p./, n < p. Then u 2 C 0;1 in fact n=p./, and 8x 0 2 8r > 0 W osc \B.x0 ;r/ u C n;p r 1 n=p kduk L p./: Introduction to Elliptic Regularity Theory I.

13 Interior C 0; regularity I. Let u be the Newton potential of f, 2.0; 1/. f 2 L 1./ H) kuk C 1;./ C n; ;jj kf k L 1./ f 2 C 0;./ H) kuk C 2;./ C n; ;jj kf k C 0;./ Proof: regularize, formally differentiate the integral, justify. Introduction to Elliptic Regularity Theory I.

14 Interior C 0; regularity II. Theorem Let u be a weak solution of u D f (on ). f 2 C 0./ H) u 2 C 1;./; and 8 0 b W kuk C 1;. 0 / C.kf k C 0./ C kuk L 2.// f 2 C 0;./ H) u 2 C 2;./; and 8 0 b W kuk C 2;. 0 / C.kf k C 0;./ C kuk L 2.// No uniform estimate on kuk C 1;./. Corollary (Hölder trick) Let u be a weak solution of u D f (on ), 0 b and n < p. f 2 L p./ H) u 2 C 1;./; and 8 0 b W kuk C 1;. 0 / C.kf k L p./ C kuk L 2.// Introduction to Elliptic Regularity Theory I.

15 Interior C 0; regularity III. Corollary (Inductive generalization) Let u 2 W 1;2./ be a weak solution of u D f (on ), 0 b. f 2 C k;./ H) kuk C kc2;. 0 / C.kf k C k;./ C kuk L 2.// Moreover, if f 2 C 1./ then u 2 C 1./. Introduction to Elliptic Regularity Theory I.

16 Schauder theory I. J. Schauder studied the regularity of elliptic problems of nondivergence form. Lu.x/ D X a ij.x/d ij u C X b i.x/d i u C c.x/u 1i;j n 1in Assume that a ij.x/ is symmetric and uniformly elliptic, and that ka ij k C./; kb i k C./; kck C./ K. Theorem (Interior Schauder estimates) Let f 2 C 0;./ and u 2 C 2;./, such that Lu D f. Then 8 0 b W kuk C 2;. 0 / C ;0 ; ;n;;k.kf k C 0;./Ckuk L 2.// Introduction to Elliptic Regularity Theory I.

17 Schauder theory II. Theorem (Global Schauder estimates) Let be a C 2; -domain, f 2 C 0;./, g 2 C 2;./ and u 2 C 2;./, such that Lu D f on and u D g Then kuk C 2;./ C ; ;n;;k.kf k C 0;./ C kgk C 2;./ C kuk L 2.// Using the maximum principle, we get Corollary If c.x/ 0 then kuk C 2;./ C ; ;n;;k.kf k C 0;./ C kgk C 2;.//. Introduction to Elliptic Regularity Theory I.

18 Schauder theory III. Theorem Let be a C 1 domain, f 2 C 0;./, g 2 C 2;./, c.x/ 0. The problem Lu D f (in ), u D g admits a unique solution in C 2;./. Proof: First, for, approximate everything with C 1./ functions. Use operator interpolation the Schauder estimates control the operator norms. Introduction to Elliptic Regularity Theory I.

19 References J. Jost. Partial Differential Equations, 2007 L. C. Evans. Partial Differential Equations, 2010 P. Grisvard. Elliptic Problems in Nonsmooth Domains, 1985 Q. Han and F. Lin. Elliptic Partial Differential Equations, 1997 M. Giaquinta. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, 1983 F. Demengel and G. Demengel. Functional Spaces for the Theory of Elliptic Partial Differential Equations, 2012 Introduction to Elliptic Regularity Theory I.

20 ELTE 2014

21 Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russel

22 Nonperturbation methods: Nonsmooth coefficients Setting. D B 1. The coefficient functions a ij 2 L 1 and c 2 L q (q > n=2) satisfy structural conditions (with and ƒ) in Lu D X D j.a ij.x/d i u/ C c.x/u i;j Given f 2 L q, u 2 W 1;2 is called a subsolution, if X 8' 2 W 1;2 0 ; ' 0 W a ij D i ud j ' C cu' i;j f ' Local boundedness. Prove that a subsolution is bounded from above. It follows that a homogeneous weak solution is bounded form above and below. Hölder continuity. Prove that a weak solution is (locally) Hölder continuous.

23 Local boundedness of subsolutions I. For all 0 < < 1 and p > 0, sup u C C n;;ƒ;p;q.1 / n=p B B 1.u C / p! 1=p C B 1 jf j q! 1=q! Proof. Let D 1=2, p D 2. General case by scaling argument. 1 Choose a clever test function to... 2 Arrive at a reverse Hölder s inequality of the form 8r < R 8p 1 < p 2 W B r u p 2! 1=p2 C r;r;p1;p2 B R u p 1! 1=p1 3 Apply Moser s iteration to carefully choose a sequence of r s, R s and p 1 s, p 2 s to obtain the result.

24 Local boundedness of subsolutions II.

25 Local boundedness of subsolutions III. Fix k; m > 0, ˇ 0, and let v D u C C k ; w D ( v; if u < m k C m; if u m and ' D 2.wˇ v kˇc1 / Then D' D 2D.wˇ v kˇc1 / C 2ˇwˇ 1 vdw C 2 wˇ Dv D 2D.wˇ v kˇc1 / C 2 wˇ.ˇdw C Dv/ since Dw D 0 on fu mg (and also on fu < 0g). Furthermore ' D 0 and D' D 0 on fu 0g, and Dv D Du.on fu > 0g/:

26 Local boundedness of subsolutions IV. Using the structural conditons and Young s inequality, jf j 2 wˇ v C jcju 2 wˇ v f ' 2ƒ 2ƒ2 X cu' a ij D i ud j ' D wˇ i;j C ˇ 2 wˇ X.kDvk/.vkDk/ ƒ i;j 4ƒ 2 kdvk 2 C ƒ v2 kdk 2 wˇ v 2 kdk 2 C ˇ 2.wˇ v a ij D i wd j w C Cˇ wˇ 2 kdwk 2 C 2 kˇc1 / X i;j a ij D i vd j X 2 wˇ a ij D i vd j v i;j wˇ 2 kdwk 2 C wˇ 2 kdvk 2 wˇ 2 kdvk 2

27 Local boundedness of subsolutions V. In summary ˇ wˇ 2 kdwk 2 C wˇ 2 kdvk 2 C jf j 2 wˇ v C jcj 2 wˇ v 2 C C wˇ v 2 kdk 2 C c 0 2 wˇ v 2 C C wˇ v 2 kdk 2 ; where c 0 D jf j k C jcj: Let k D kf k q... Introduce the notation z D wˇ=2 v, to get ˇ 2 wˇ kdwk 2 C 2 wˇ kdvk 2 C c 0 z 2 2 CC z 2 kdk 2

28 Local boundedness of subsolutions VI. Furthermore Dz D ˇ 2 w ˇ 2 1 vdw C w ˇ ˇ 2 Dv D 2 w ˇ ˇ 2 Dw C w 2 Dv ˇ kdzk 2 2wˇ kdwk 2 C wˇ kdvk 2 2.ˇ C 1/.ˇwˇ kdwk 2 C wˇ kdvk 2 / hence 2 kdzk 2 C.1 C ˇ/ c 0 z 2 2 C C.1 C ˇ/ z 2 kdk 2 ; and since kd.z/k 2 C 2 kdzk 2 C C z 2 kdk 2 the left-hand side integrand may be replaced by kd.z/k 2.

29 Local boundedness of subsolutions VII. c 0 2 L q H) c 0 z 2 2 kc 0 k q ƒ 1Cƒ.z/ 2q q 1! 1 1=q : This last integral is estimated as follows. Since q > n=2, we have Using z 2 L 2, kzk 2q q 1 2 D 2n n 2 > 2q q 1 > 2: kzk 2 C.1 /kzk 2 kzk 2 C.1 /C kd.z/k 2 ; by interpolation. In particular! 2 q 1.z/ 2q 2q q 1 2 C.z/ 2 C.1 / 2 C kd.z/k 2

30 Local boundedness of subsolutions VIII. In summary,! 2=2.z/ 2 C kd.z/k 2 C.1 C ˇ/ D C.1 C ˇ/ z 2 kdk 2 C kdk 2 C 2 z 2 The desired reverse Hölder inequality follows by choosing an appropriate cutoff. Let 0 < r < R 1 and 2 Cc 1.B R/ such that, 1.on B r /; kdk 2 R r : Then! 2=2 z 2 C 1 C ˇ B r.r r/ 2 z 2 B R.z/ 2!

31 Local boundedness of subsolutions IX. Recall that z D wˇ=2 v, so! 2 w ˇ2 2 v 2 2 C 1 C ˇ B r.r r/ 2 Let D ˇ C 2, and recall that w v, hence B R wˇ v 2 : w 2 2 B r! C 1 C ˇ.R r/ 2 v : B R Finally, by letting m! C1 v I L 2 2.Br / C 1 C ˇ 1= v I L.R r/ 2.B R /

32 Local boundedness of subsolutions X. Moser s iteration Iterate the reverse Hölder inequality by choosing 2 i D 2 2 i r i D 1 2 C 2 i 1 Note that 0 D 2, r 0 D 1 and i! 1, r i! 1=2. Also hence i D 2 2 i 1; and r i 1 r i D 2 i 1 ; kv I L i.b ri /k C C i C ˇ 1=i v I L i 1.r i r i 1 / 2.B ri 1 / i v I L i 1.B / ri 1

33 Local boundedness of subsolutions XI. Moser s iteration Iterating, we get kv I L i.b ri /k C P i 2 2 i v I L 2.B 1 / : By passing to the limit as i! 1, and recalling that v D u C C k, we get sup B 1=2 u C C.ku C k L 2.B 1 / C k/

34 Local Boundedness of homogeneous solutions Theorem Let u 2 W 1;2./ satisfy Lu 0 weakly. Then for all 0, u is bounded from above on 0. Therefore if Lu D 0, then u is bounded form above and below on every 0. Proof. Consider the cutoff sequence v k D u _ k (k > 0). Then v k > 0 and Lv k 0, so the previous result applies. Extend this to this to the subdomain 0.

35 De Giorgi Moser Nash theorem From now on, assume that kk 2.A.x/; / ƒkk 2. Theorem Suppose that Lu D 0 in B 1 weakly. Then there exists 0 < < 1, such that ku I C 0;.B 1=2 /k C n;;ƒ ku I L 2.B 1 /k Which can be extended by perturbation arguments to Theorem Suppose that Lu D f in B 1 weakly, and c 2 L n. Then there exists 0 < < 1, such that u 2 C 0;.B 1 /. More precisely, there exists R > 0 such that for all x 2 B 1=2 and r < R, we have kduk 2 C r n 2 2.kf k 2 L q.b 1 / C kuk2 W 1;2.B 1 / /: B.x;r/

36 Classical Harnack inequality I. Setting. Let u 2 C.B 1 / \ C 2.B 1 /, u 0 and u D 0 in B 1. From Green s representation formula for the ball (a.k.a. Poisson integral), 80 < r < 1 8x 2 B r W 1 r 1 C r u.0/ u.x/.1 C r/ n 1.1 r/ n 1 u.0/: More generally, let 0, then there exists C 0 > 0, such that sup u.x/ C 0 inf u.x/ x2 0 x2 0 Consequences. Liouville s theorem, removable singularity theorem (! 9 unsolvable Laplace BVP), Harnack s convergence theorems,...

37 Classical Harnack inequality II.

38 Moser s Weak Harnack inequality Theorem Let u 2 H 1./, u 0 a weak supersolution with c 0. Then for any B R, 0 < p < 2 =2 and 0 < < < 1, we have inf u C R 2 B R n=q kf k L q.b R / CR n=p B R u p! 1=p : Compare this with the boundedness result: if u 0 is a weak subsolution, then for any B 3r, p > 0, we have sup u C r n=p B r B 2r u p! 1=p C r 2 n=q kf k Lq.B2r /!

39 Moser s Strong Harnack inequality Let u 0 be a weak solution. Choose R WD 3r, WD 2=3 and WD 1=6 to get sup u C inf u C r 2 n=q kf k L B r B q.b r / ; r=2 where (again) the constant is independent of u! :)

40 Proof of Moser s Weak Harnack inequality I. Proof. Assume R D 1. First, we prove that the inequality holds for some p > 0. Let k > 0, and v D u C k ; and w D 1=v Note that Dv D Du, and so Dw D Dv=v 2 D Du=v 2. Substitute 0 '=v 2 2 H0 1.B 1/ as a test function in the supersolution condition to get X D i u aij v 2 D ' j ' 2a ij v 3 D iud j v f 'v 2 hence by letting F D f =v, we have X aij D i wd j ' C F w' 0

41 Proof of Moser s Weak Harnack inequality II. Let k D kf k q... Note that kf k q 1. The local boundedness result is applicable: we have for all 2.; 1/ and p > 0 that 1 sup B v D sup B 1=p w C w p D C B B v p! 1=p But sup 1 v D 1 inf v, hence inf v C B B v p! 1=p D B v p B v p ƒ Estimate me from above!! 1=p B v p! 1=p

42 John Nirenberg inequality I. Theorem Suppose that f 2 L 1./ satisfies 8B.x; r/ W jf B.x;r/ f x;r j M (BMO): Then there exists p 0 > 0 and C n > 0, such that 8B.x; r/ W e p 0 M jf f x;r j C n : B.x;r/ Proof. A tricky application of Calderón ygmund decomposition.

43 John Nirenberg inequality II. As a corollary, we have e p 0 M f e p 0 M f D B.x;r/ B.x;r/ B.x;r/ e p 0 M.f f x;r / B.x;r/ e p 0 M.f f x;r / C 2 n r2n Hence choosing f WD log w and r D, we get the desired estimate for the exponent p D p 0 =M. [Compare this with (Hölder) Cn q rqn f q f q ; where f > 0 a.e. on B.x; r/.] B.x;r/ B.x;r/

44 John Nirenberg inequality III. It remains to check that log w 2 BMO. It is easy to show that 8 2 Cc 1.B 1/ W kd log vk 2 2 C B 1 kdk 2 B 1 [Hint: choose '=v as a test function, D log v D Dv=v.] Then kd log vk 2 C r n 2 : B.y;r/ Hölder and Poincaré yields! 1=2 r n j log v.log v/ y;r j r n=2 j log v.log v/ y;r j 2 B.y;r/ B.y;r/ r n=2c1 B.x;r/ kd log vk 2! 1=2 C

45 Proof of Moser s Weak Harnack inequality III. Extension to a general exponent 0 < p < 2 =2 follows by an argument similar to the proof of the reverse Hölder inequality.

46 Hölder continuity I. Theorem Let u 2 W 1;2./ satisfy Lu D 0 weakly. Then for all 0, 90 < < 1 9C > 0 8x; y 2 0 W ju.x/ u.y/j C kx yk : Proof. Let B R D B.x; R/. Then m.r/ D inf BR u and M.R/ D sup BR u are finite by the previous theorem. Let then for all y 2 S.x; r/!.r/ D M.R/ m.r/ ; u.x/ u.y/!.r/: Hence, an estimate of the form r!.r/ C!.R/; R where 0 < < 1 is sufficient.

47 Hölder continuity II. Proof cont d. Observe that u m.r/ > 0 and M.R/ u > 0 and Lu D 0 weakly in B R. Apply Moser s Harnack inequality to get M.R/ m.r=2/ D sup B R=2.M.R/ C.M.R/ u/ C inf B R=2.M.R/ u/ M.R=2// and M.R=2/ m.r/ D sup B R=2.u C.m.R=2/ m.r// C inf B R=2.u m.r//: m.r// Hence by changing to C 0 WD max.c; 1/, M.R=2/ m.r=2/ C 0 1 C 0 C 1.M.R/ in other words!.r=2/!.r/ with 0 < < 1. m.r//;

48 Hölder continuity III. Proof cont d. Inductively, we get!.r=2 n / n!.r/. Given r < R=2 there exists n 2 N, such that R 2 nc1 r R 2 n : For some 0 < < 1, such that 2, we get!.r/!.r=2 n / n!.r/.1=2 n /!.R/ C.r=R/!.R/ [Fun fact: From this last recursion an easy proof of Liouville s theorem is possible.]

49 Application: Regularity of energy functionals I. Theorem (Hilbert s 19th problem, de Giorgi Nash) Let F 2 C 1.R n /, such that kdf./k C kk and D 2 F is uniformly positive definite. We have u 2 C 1./, where 8' 2 H0 1./ W F.Du/ F.Du C D'/ Proof sketch. Since x 7! '.x he k / 2 H0 1./ for sufficiently small h > 0, X A i.du.x C he k //D i ' D 0: hence X i i A i.du.x C he k // A i.du.x// D i ' D 0:

50 Application: Regularity of energy functionals II. Newton Leibniz trick: A i.du.x C he k // A i.du.x// D 1 i tdu.x C he k / j 0 Let thus j a h ij.x/ D 1 i tdu.x C he k / j X aij h.x/d j. h;k u/d i ' D 0 i;j t/du.x/ D j u.x C he k / u.x/ dt t/du.x/ dt; hence in the limit h! 0, by letting v D D k u, we have X a ij.x/d j vd i ' D 0 i;j

51 Application: Regularity of energy functionals III. Therefore the earlier regularity result yields u 2 C 1;. Then a ij 2 C 0;, but we need more theory to deduce higher regularity, since the Schauder theory handles nondivergence operators...

52 Extension: Nonlinear divergence form problems Trudinger and Serrin extended Moser s result to problems of the form 8' 2 W 1;p X 0./ W a i.; u; Du/D i ' C b.x; u; Du/' dx D 0 where u 2 W 1;p./ and loc X a i.x; ; / i c 1 kk p c 2 i i X ja i.x; ; /j C jb.x; ; /j c 3.1 C kk p=p0 / i

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54 Csirik Miha ly

55 Introduction to Elliptic Regularity Theory III. ELTE 2014 Introduction to Elliptic Regularity Theory III.