NON-LINEAR ASPECTS OF CALDERÓN-ZYGMUND THEORY

Transcription

1 NON-LINEAR ASPECTS OF CALDERÓN-ZYGMUND THEORY GIUSEPPE MINGIONE University of Parma G. R. Mingione p. 1/11

2 Panorama Overture 1 Gradient integrability theory 2 The subdual range, and measure data problems 3 Bounds via nonlinear potentials 4 Non-local estimates 5 Gradient continuity estimates 6 A fully Fractional approach G. R. Mingione p. 2/11

3 Overture: The standard Consider the model case u = f in R n Then f L q implies D 2 u L q 1 < q < with natural failure in the borderline cases q = 1, As a consequence (Sobolev embedding) Du L nq n q q < n G. R. Mingione p. 3/11

4 The singular integral approach Representation via Green s function u(x) G(x, y)f(y) dy with G(x, y) = x y 2 n if n > 2 log x y if n = 2 Differentiation D 2 u(x) = K(x, y)f(y) dy and K(x, y) is a singular integral kernel, and the conclusion follows G. R. Mingione p. 4/11

5 Singular kernels with cancellations Initial boundedness assumption ˆK L B, where ˆK denotes the Fourier transform of K( ) Hörmander cancelation condition K(x y) K(x) dx B for every y R n. x 2 y G. R. Mingione p. 5/11

6 The fractional integral approach Again differentiating Du(x) I 1 ( f )(x) where I 1 is a fractional integral I β (g)(x) := g(y) dy β [0, n) x y n β and then I β : L q L nq n βq βq < n This is in fact equivalent to the original proof of Sobolev embedding theorem for the case q > 1, which uses that u(x) I 1 ( Du )(x) G. R. Mingione p. 6/11

7 The fractional integral approach Important remark: the theory of fractional integral operators substantially differs from that of singular ones. In fact, while the latter is based on cancelation properties of the kernel, the former only considers the size of the kernel. As a consequence all the estimates related to the operator I β degenerate when β 0. G. R. Mingione p. 7/11

8 Another linear case Higher order right hand side u = div Du = div F Then F L q = Du L q q > 1 just simplify" the divergence operator!! G. R. Mingione p. 8/11

9 Interpolation approach Define the operator T : F T(F) := gradient of the solution to u = div F Then T : L 2 L 2 by testing with the solution, and T : L BMO by regularity estimates (hard part). Campanato-Stampacchia interpolation T : L q L q 1 < q < G. R. Mingione p. 9/11

10 BMO/VMO Define and ω(r) := sup s R (v) Bs := 1 B s A map v belongs to BMO iff A map v belongs to VMO iff 1 B s ω(r) < lim ω(r) = 0 R 0 B s v dx B s v (v) Bs dx G. R. Mingione p. 10/11

11 Part 1: Gradient integrability theory G. R. Mingione p. 11/11

12 First non-linear cases Iwaniec s result (Studia Math. 83) Then it holds that div ( Du p 2 Du) = div ( F p 2 F) F L q = Du L q p q < Di Benedetto & Manfredi (Amer. J. Math. 93) prove this for the p-laplacean system Kinnunen & Zhou (Comm. PDE 99) consider the case div ( c(x)du, Du p 2 2 Du) = div ( F p 2 F) where c(x) is an elliptic matrix with VMO entries G. R. Mingione p. 12/11

13 First non-linear cases The local estimate ( Du q dz B R )1 ( q c Du p dz B 2R ) 1 ( p + c F q dz B 2R )1 q G. R. Mingione p. 13/11

14 General elliptic problems In the same way the non-linear result of Iwaniec extends to all elliptic equations in divergence form of the type div a(du) = div ( F p 2 F) where a( ) is p-monotone in the sense of the previous slides and to all systems with special structure div (g( Du )Du) = div ( F p 2 F) Moreover, VMO-coefficients can be considered too, along the methods of Kinnunen & Zhou div [c(x)a(du)] = div ( F p 2 F) G. R. Mingione p. 14/11

15 General systems - the elliptic case The previous result cannot hold div a(du) = div ( F p 2 F) with a( ) being a general p-monotone in the sense of the previous slide. The failure of the result, which happens already in the case p = 2, can be seen as follows. Consider the homogeneous case div a(du) = 0 The validity of the result would imply Du L q for every q <, and, ultimately, that u L But Sverák & Yan (Proc. Natl. Acad. Sci. USA 02) recently proved the existence of unbounded solutions, even when a( ) is non-degenerate and smooth G. R. Mingione p. 15/11

16 The up-to-a-certain-extent Kristensen & Min. (ARMA 06) - General elliptic systems div a(du) = div ( F p 2 F) for a(du) being a p-monotone vector field and p q < p + 2p n 2 + δ n > 2 Then it holds that F L q = Du L q G. R. Mingione p. 16/11

17 The Dirichlet problem Applications to singular sets estimates Consider the Dirichlet problem { div a(du) = 0 u = v in Ω on Ω Then it holds that Ω Du q dx c Ω ( Dv q + 1) dx G. R. Mingione p. 17/11

18 Discussion about the sharpness - p = 2 For general elliptic systems we have div a(du) = div (F) 2 q n 2 = Then it holds that F L q = Du L q 2n n 2 n > 2 Take F 0 then the example of Sverák & Yan gives the existence of an unbounded solution, while Du L q for some q > 2n/(n 2) would imply that u is bounded 2n n 2 + δ > n n 4 The result is sharp at least when n = 2, 3, 4 G. R. Mingione p. 18/11

19 The parabolic case Kinnunen & Lewis (Duke Math. J. 00) u t div ( Du p 2 Du) = div ( F p 2 F) for p > 2n n + 2 (optimal) Then it holds that F L q = Du L q for p q < p + δ(n, N, p) This is actually available for more general systems, being a result in the spirit of Gehring s lemma The case of higher order systems is treated by Bögelein (Ann. Acad. Sci. Fenn., 08) G. R. Mingione p. 19/11

20 The parabolic case Acerbi & Min. (Duke Math. J. 07) u t div ( Du p 2 Du) = div ( F p 2 F) for p > 2n n + 2 (optimal) Then it holds that F L q = Du L q for p q < The elliptic approach via maximal operators only works in the case p = 2 The result also works for systems, that is when u(x, t) R N, N 1 First Harmonic Analysis free approach to non-linear Calderón-Zygmund estimates G. R. Mingione p. 20/11

21 The parabolic case The result is new already in the case of equations i.e. N = 1, the difficulty being in the lack of homogenous scaling of parabolic problems with p 2, and not being caused by the degeneracy of the problem, but rather by the polynomial growth. The result extends to all parabolic equations of the type u t div a(du) = div ( F p 2 F) with a( ) being a monotone operator with p-growth. More precisely we assume ν(s 2 + z z 2 2 ) p 2 2 z 2 z 1 2 a(z 2 ) a(z 1 ), z 2 z 1 a(z) L(s 2 + z 2 ) p 1 2, G. R. Mingione p. 21/11

22 The parabolic case The result also holds for systems with a special structure (sometimes called Uhlenbeck structure). This means u t div a(du) = div ( F p 2 F) with a( ) being p-monotone in the sense of the previous slide, and satisfying the structure assumption a(du) = g( Du )Du The p-laplacean system is an instance of such a structure G. R. Mingione p. 22/11

23 The parabolic UTCE Duzaar & Min. & Steffen (Mem. AMS, to appear) - General parabolic systems u t div a(du) = div ( F p 2 F) for a(du) being a p-monotone vector field and p q < p + 4 n + δ Then it holds that F L q = Du L q p q < G. R. Mingione p. 23/11

24 Discontinuous coefficients Duzaar & Min. & Steffen (Mem. AMS, to appear) u t div [c(x)a(t, Du)] = div ( F p 2 F) equations or systems (in this last case with limitations on q), or u t div (c(x)b(t) Du p 2 Du) = div ( F p 2 F) we have F L q = Du L q Discontinuous coefficients c(x) VMO b(t) is just measurable This extends recent linear results of Krylov and his students, valid for linear parabolic equations G. R. Mingione p. 24/11

25 Elliptic vs parabolic local estimates Elliptic estimate ( Du q dz B R )1 Parabolic estimate - p 2 ( Du q dz Q R ( q c Du p dz B 2R )1 q [ ( ) 1 ( c Du p p dz + Q 2R ) 1 ( p + c F q dz B 2R Q 2R F q dz Parabolic cylinders Q R B R (t 0 R 2, t 0 + R 2 ) ) 1 q + 1 ] p 2 The exponent p/2 is the scaling deficit of the system )1 q G. R. Mingione p. 25/11

26 Elliptic vs parabolic local estimates Elliptic estimate ( Du q dz B R )1 Parabolic estimate - p 2 ( Du q dz Q R ( q c Du p dz B 2R )1 q [ ( ) 1 ( c Du p p dz + Q 2R ) 1 ( p + c F q dz B 2R Q 2R F q dz Parabolic cylinders Q R B R (t 0 R 2, t 0 + R 2 ) ) 1 q + 1 ] p 2 The exponent p/2 is the scaling deficit of the system )1 q G. R. Mingione p. 26/11

27 Interpolation nature of local estimates Parabolic local estimate - p 2 ( Du q dz Q R ) 1 q [ ( ) 1 ( c(n, N, p) Du p p dz + c(q) Q 2R Taking F = 0 and letting q yields Q 2R F q dz )1 q + 1 ] p 2 sup Q R Du c [ ( ) 1 ] p 2 Du p p dz + 1 Q 2R This is the original sup estimate of DiBenedetto & Friedman (Crelles J. 84) G. R. Mingione p. 27/11

28 The ghost of the maximal operator In the estimate ( Du q dz Q R ) 1 q [ ( ) 1 ( c(n, N, p) Du p p dz + c(q) Q 2R the dependence of c(q) is maximal function like Q 2R F q dz )1 q + 1 ] p 2 c(q) q q p This apparent instability for q ց p is recovered via Gehring s lemma G. R. Mingione p. 28/11

29 The local estimate in the singular case The singular case The local estimate is 2n n + 2 < p < 2 ( Du q dz Q R ) 1 q [ ( ) 1 ( c(n, N, p) Du p p dz + c(q) F q dz Q 2R Q 2R Observe that )1 q + 1 ] 2p p(n+2) 2n 2p p(n + 2) 2n ր when p ց 2n n + 2 G. R. Mingione p. 29/11

30 Elliptic and Parabolic coverings Cover the level sets with balls E(λ) := { Du λ} with usual Euclidean balls B(x 0, R) and study their decay via the well-known formula Du q = λ q 1 E(λ) dλ Take a vitali covering of E(λ), that is {B i } 1 B i B i Du p dx λ p G. R. Mingione p. 30/11

31 Elliptic and Parabolic coverings Use the comparison maps v i p v i = 0 v i = u on B i and their local regularity properties to infer those of the solution u The parabolic case - DiBenedetto s intrinsic geometry Cover the level set E(λ) with intrinsic cubes for p 2, and Q i B(x i, R) (t i λ 2 p R 2, t i + λ 2 p R 2 ) Q i B(x i, λ p 2 2 R) (ti R 2, t i + R 2 ) for p < 2, such that 1 Q i Q i Du p dxdt λ p G. R. Mingione p. 31/11

32 Elliptic and Parabolic coverings These are intrinsic cubes They cannot generate a family with an naturally associated maximal operator Use the comparison maps v i (v i ) t div ( Dv i p 2 Dv i ) = 0 v i = u on par Q i this is like (v i ) t div (λ p 2 Dv i ) = 0 since, in some sense, we still have 1 Q i Q i Dv i p dxdt λ p G. R. Mingione p. 32/11

33 Elliptic and Parabolic coverings Make the change of variables, for p > 2 w(y, s) := v(x i + R 2 y, t i + λ 2 p Rs) (y, s) B 1 ( 1, 1) and then w t w = 0 On intrinsic cylinders solutions behave as caloric functions G. R. Mingione p. 33/11

34 The ghost of the good-λ inequality principle Exit time on the composite quantity 1 Q i Qi Du p dxdt + Mp Q i Q i F p dxdt λ p therefore, while on Q i we approximately have Du λ we have F λ M Large M-inequality principle G. R. Mingione p. 34/11

35 Obstacle problems Results of Bögelein & Duzaar & Min. (Crelles J., to appear) The elliptic case Min Dv p dx in the class Then Local estimate K = {v W 1,p : v(x) Φ(x)} Φ W 1,q = u W 1,q q p ( Du q dz B R ) 1 ( q c Du p dz B 2R ) 1 ( p + c DΦ q dz B 2R )1 q G. R. Mingione p. 35/11

36 Parabolic obstacle problems Parabolic variational inequalities Ω T v t (v u) + a(du), D(v u) dxdt+(1/2) v(, T) 2 L 2 (Ω) 0, in the class K = {v L p ( T, 0; W 1,p 0 (Ω)) L 2 (Ω T ) : v(x, t) Φ(x)} Then, again DΦ L q = Du L q q p The case when Φ depends on time is allowable G. R. Mingione p. 36/11

37 Open problems The full range p 1 < q < Compare with the linear case p = 2. This is the case below the duality exponent, when div ( F p 2 F) W 1,p Iwaniec & Sbordone (Crelle J., 94), Lewis (Comm. PDE, 93) p ε q < ε ε(n, p) Parabolic case: Kinnunen & Lewis (Ark. Math, 02), Bögelein (Ph. D. Thesis, 07) G. R. Mingione p. 37/11

38 Part 2: problems The subdual range, and measure data G. R. Mingione p. 38/11

39 Measure data problems - a review Model case { div a(x, Du) = µ u = 0 in Ω on Ω and µ (Ω) <. Assumptions ν(s 2 + z z 2 2 ) p 2 2 z 2 z 1 2 a(x, z 2 ) a(x, z 1 ), z 2 z 1 a(x, z) L(s 2 + z 2 ) p 1 2, with 2 p n standing assumption Linear case Classical work by Littman & Stampacchia & Weinberger (Ann. SNS Pisa, 1963) G. R. Mingione p. 39/11

40 Notion of solution Distributional a(x, Du)Dϕ dx = Ω Ω ϕ dµ, for every ϕ C c (Ω). Minimal integrability that is u W 1,p 1 Approximations { div a(x, Du k ) = f k L u k = 0 in Ω on Ω f k µ u k u strongly in W 1,p 1 G. R. Mingione p. 40/11

41 Basic model examples The p-laplacean operator Its measurable variations div( Du p 2 Du) = f div[c(x)(s 2 + Du 2 ) p 2 2 Du] = f Its non-autonomous variations div[c(x, u)(s 2 + Du 2 ) p 2 2 Du] = f G. R. Mingione p. 41/11

42 The fundamental solution Consider the problem { div( Du p 2 Du) = δ u = 0 in Ω on Ω And its unique solution (in the approximation class), called the fundamental solution G p (x) x p n p 1 if p n log x if n = p G p (x) can be used to asses the optimality of several regularity estimates G. R. Mingione p. 42/11

43 Regularity estimates, existence Talenti (Ann. SNS Pisa 76, Annali di Matematica 79) Lindqvist (Crelles J. 86) Heinonen & Kilpeläinen & Martio (Oxford book and contributions from the eighties) Boccardo & Gallöuet (J. Funct. Anal. 89) Kilpeläinen & Malý (Ann. SNS Pisa. 92, Acta Math. 94) Greco & Iwaniec & Sbordone (manus. math. 97) Dolzmann & Hungerbühler & Müller (Ann. IHP. 97, Clelles 00) Kilpeläinen-Shanmugalingam-Zhong, Ark. Math. 08 A large number of authors contributed to such estimates, in several different, and interesting ways There exists a solution such that Du p 1 M n n 1 p n The regularity is optimal, as revealed by G p ( ) G. R. Mingione p. 43/11

44 Limiting spaces Marcinkiewicz spaces (weak L γ ) f M γ supλ γ { f > λ} =: f γ M < γ λ>0 Prominent example: potentials Inclusions L log L-spaces 1 x 1 n/γ Mγ x L γ M γ L γ ε ε > 0 f L log L n/γ Lγ f log(e + f ) dx < G. R. Mingione p. 44/11

45 Limiting spaces Definition comes from therefore { f > λ} = 1 λ γ { f >λ} { f >λ} f γ = 1 λ γ f γ L γ f γ λ γ f γ M = supλ γ { f > λ} γ λ>0 f γ L γ G. R. Mingione p. 45/11

46 Function data Boccardo & Gallöuet (Comm. PDE 93) Optimal integrability of Du f L γ p < n Du p 1 L nγ n γ In the case below the duality exponent 1 < γ < np np n + p = (p ) In fact Finally L γ W 1,p when γ (p ) f L log L = Du p 1 L n n 1 G. R. Mingione p. 46/11

47 Function data Kilpeläinen & Li (Diff. Int. Equ. 00) Optimal integrability of Du f M γ p < n Du p 1 M nγ n γ In the case below the duality exponent 1 < γ < np np n + p = (p ) The borderline case γ = (p ) is left by the authors as a tempting open problem, on which I will come back later; progress on this has been made by Zhong (Ph. D. Thesis) This case is interesting since we would have Du M p, almost at the natural integrability G. R. Mingione p. 47/11

48 A density condition Consider the density condition Theorem (Min. Ann SNS 07) µ (B R ) cr n θ θ [p, n] Du p 1 M θ θ 1 with a refinement in Morrey space G. R. Mingione p. 48/11

49 Fractional differentiability Fractional Sobolev spaces We have that v W s,γ with 0 < s < 1 γ 1 iff v(x) v(y) γ x y n+sγ dxdy < Intuitively v(x) v(y) γ x y n+sγ dxdy D s v(x) γ dx Fractional Sobolev embedding W s,γ L nγ n sγ sγ < n G. R. Mingione p. 49/11

50 Differentiability of Du for measure data Differentiability ν(s 2 + z z 2 2 ) p 2 2 z 2 z 1 2 a(x, z 2 ) a(x, z 1 ), z 2 z 1 a(x, z 2 ) a(x, z 1 ) L(s 2 + z z 2 2 ) p 2 2 z 2 z 1 a(x, 0) Ls p 1 Differentiable coefficients a(x, z) a(x 0, z) L x x 0 (s 2 + z 2 ) p 1 2 G. R. Mingione p. 50/11

51 Differentiability of Du for measure data - p = 2 You cannot have Du W 1,1, already in the linear case (otherwise Du L n n 1 ) Full integrability Du M n n 1 vs total lack of differentiability of Du, but almost nothing" is actually lost, in fact we have G. R. Mingione p. 51/11

52 Differentiability of Du for measure data - p = 2 Theorem 1 (Min. Ann. SNS Pisa 07) Du W 1 ε,1 for every ε > 0 G. R. Mingione p. 52/11

53 Caccioppoli s estimate Local estimate B R/2 Du(x) Du(y) B R/2 x y n+1 ε dxdy c R 1 ε ( Du + s) dx + cr ε µ (B R ) B R G. R. Mingione p. 53/11

54 Differentiability of Du for p 2 What should we expect? We have 1 x β W s,γ (B) β < n γ s We apply this fact to the fundamental solution Du 1 x n 1 p 1 with the natural choice γ = p 1, to optimize the differentiability parameter. This yields s < 1 p 1 and we expect Du W s,p 1 s < 1 p 1 G. R. Mingione p. 54/11

55 Indeed... Theorem 2 (Min. Ann. SNS Pisa 07) Du W 1 ε p 1,p 1 for every ε > 0 Recall that we are assuming p 2 so that 1 p 1 1 G. R. Mingione p. 55/11

56 Integrability recovered again Corollary Du p 1 L n n 1 ε since W 1 ε p 1,p 1 L n(p 1) n 1+ε This gives back the first result of Boccardo & Gallöuet (J. Funct. Anal. 89) Theorem 2 is optimal in the scale of fractional Sobolev spaces, i.e. we cannot take ε = 0 otherwise Du p 1 L n n 1 false in general (fundamental solution) G. R. Mingione p. 56/11

57 BV type behavior of Du For p = 2 consider u = µ Switching (which is forbidden for measure or L 1 -data) u with D 2 u, you would have that Du is BV. Nevertheless something survives... Corollary In the general case p 2 consider the singular set Σ u := { x Ω : liminf ρց0 B(x,ρ) or Du(y) (Du) x,ρ p 1 dy > 0 } lim sup ρց0 Then its Hausdorff dimension satisfies dim(σ u ) n 1 an estimate which is independent of p (Du) x,ρ = G. R. Mingione p. 57/11

58 Pointwise behavior of Du This follows from a simple potential theory fact. Define Σ v := { x Ω : liminf ρց0 B(x,ρ) or lim sup ρց0 v(y) (v) x,ρ γ dy > 0 } (v) x,ρ = Then provided sγ < n v W s,γ = dim(σ v ) n sγ G. R. Mingione p. 58/11

59 Two references Regularity of minima: and invitation to the Dark Side of the Calculus of Variations Applications of Mathematics (Prague, 2006) - Lecture notes from Paseky school Nonlinear aspects of Calderón-Zygmund theory - Jahresbericht der Deutschen Mathematiker Vereinigung 2010 G. R. Mingione p. 59/11

60 Part 3: Bounds via non-linear potentials G. R. Mingione p. 60/11

61 The standard Consider the model case u = f in R n Then, if we define we have I β (µ)(x) := R n dµ(y), β (0, n] x y n β u(x) ci 2 ( µ )(x), and Du(x) ci 1 ( µ )(x) G. R. Mingione p. 61/11

62 Integral estimate Basic property of the Riesz potential I β : L q L nq n βq 1 < q < n β and I β : L 1 M n n β Therefore Du L nq n q c I 1( µ ) L nq n q c µ L q and Du n c µ M n 1 L 1 G. R. Mingione p. 62/11

63 Local versions In bounded domains one uses I µ β (x 0, R) := R 0 µ(b(x 0, )) n β d β (0, n] since I µ β (x 0, R) B R (x 0 ) dµ(y) x 0 y n β = I β (µ B(x 0, R))(x 0 ) I β (µ)(x 0 ) for non-negative measures G. R. Mingione p. 63/11

64 What happens in the non-linear case? For instance for non-linear equations with linear growth div a(x, Du) = µ that is equations well posed in W 1,2 (p-growth and p = 2) And degenerate ones like div ( Du p 2 Du) = µ or div (γ(x) Du p 2 Du) = µ G. R. Mingione p. 64/11

65 Non-linear potentials The non-linear Wolff potential is W µ β,p (x 0, R) := R 0 ( µ (B(x0, )) n βp ) 1 p 1 d β (0, n/p] which for p = 2 reduces to the usual Riesz potential I µ β (x 0, R) := R 0 µ(b(x 0, )) n β d β (0, n] The non-linear Wolff potential plays in non-linear potential theory the same role the Riesz potential plays in the linear one G. R. Mingione p. 65/11

66 A fundamental estimate For solutions to div ( Du p 2 Du) = µ p n we have ( u(x 0 ) c B(x 0,R) u p 1 dx ) 1 p 1 + cw µ 1,p (x 0, 2R) Kilpeläinen-Malý (Acta Math. 94) - Trudinger & Wang (Amer. J. Math. 02) Recall W µ 1,p (x 0, 2R) := 2R 0 ( µ (B(x0, )) n p ) 1 p 1 d For p = 2 we have W µ 1,p = Iµ 2 G. R. Mingione p. 66/11

67 Integral estimates follow via Wolff inequalities We have µ L q = W µ β,p L nq(p 1) n qpβ q (1, n) with related explicit estimates, also in Marcinkiewicz spaces Such a property allows to reduce the study of integrability of solutions to that of non-linear potentials The key is the following inequality 0 ( µ (B(x0, )) n βp ) 1 p 1 d ci β { [I β ( µ )] 1 p 1 } (x 0 ) the last quantity is called Havin-Maz ja potential G. R. Mingione p. 67/11

68 The setting We consider equations under the assumptions div a(x, Du) = µ a(x, z) + a z (x, z) ( z 2 + s 2 ) 1 2 L( z 2 + s 2 ) p 1 2 ν 1 ( z 2 + s 2 ) p 2 2 λ 2 a z (x, z)λ, λ a(x, z) a(x 0, z) L 1 ω( x x 0 )( z 2 + s 2 ) p 1 2 with p 2 and the Dini continuity condition of the coefficients x R 0 ω( ) d < G. R. Mingione p. 68/11

69 The potential gradient estimate for p = 2 Theorem (Min. JEMS, to appear) D ξ u(x 0 ) c D ξ u dx + ci µ 1 (x 0, 2R) B(x 0,R) holds for almost every point x 0 and ξ {1,...,n} G. R. Mingione p. 69/11

70 The general potential gradient estimate Theorem (Duzaar & Min. Amer. J. Math., to appear) Du(x 0 ) c ( Du + s) dx + cw µ 1/p,p (x 0, 2R) B(x 0,R) holds for almost every point x 0 This means 2R Du(x 0 ) c ( Du + s) dx + c B(x 0,R) 0 ( µ (B(x0, )) n 1 ) 1 p 1 d G. R. Mingione p. 70/11

71 First remarks Since µ L q = W µ 1/p,p L nq(p 1) n q q (1, n) therefore, for instance nq(p 1) µ L q n q = Du Lloc (Ω) q (1, n) G. R. Mingione p. 71/11

72 Integral estimates follow as a corollary For the model case equation div ( Du p 2 Du) = µ the previous estimate implies for instance all those included in the papers Iwaniec, Studia Math. 83 Di Benedetto & Manfredi, Amer. J. Math. 93 Boccardo & Gallöuet, J. Funct. Anal. 87, Comm PDE 92 Talenti, Ann. SNS Pisa 76 Kilpelainen & Li, Diff. Int. Equ. 00 Dolzmann & Hungerbühler & Müller, Crelle J. 00 Alvino & Ferone & Trombetti, Ann. Mat. Pura Appl. 00 Boccardo, Ann. Mat. Pura Appl. 08 Moreover the borderline cases which appeared as open problems in these papers also follow G. R. Mingione p. 72/11

73 Sharpness of Dini continuity The Dini continuity assumption R 0 ω( ) d < cannot be weakened Recent example by Jin & Mazya & Van Shaftingen reveals even for the linear case div(a(x)du) = 0 with A(x) being continuous-not-dini continuous, it may happen that Du L although it must be Du L q for every q < Indeed, it happens that Du BMO G. R. Mingione p. 73/11

74 Lorentz spaces: Basics Rearrangement of the function µ µ (s) := sup {t 0 : {x Ω : µ(x) > t} > s}. it follows the equimeasurability property { µ > t} = {µ > t} A function µ belongs to L(γ, q), for γ [1, ) and q (0, ) iff g L(γ,q)(Ω) = ( q γ 0 ( µ ( ) 1/γ) ) 1 q q d < while all the previous inclusions are strict. This is equivalent (Fubini) to require that µ L(γ,q)(Ω) = [ q 0 (λ γ {x Ω : µ(x) > λ} ) q γ dλ λ ] 1 q < G. R. Mingione p. 74/11

75 Lorentz-Marcinkiewicz spaces g belongs to L(γ, )(Ω) M γ iff µ γ L(γ, )(Ω) := sup λ λ γ {x Ω : µ(x) > λ} < where 1 γ < The second index tunes the first one and for 0 < q < t < r we have L r L(r, r) L(t, q) L(t, t) L(t, r) L(q, q) L q G. R. Mingione p. 75/11

76 Lorentz spaces are not bizarre Recall that 1 x n γ M γ L(γ, ) Then improving the integrability in a logarithmic factor gives 1 x n γ log β x L(γ, q) q > 1 β so we conclude that Lorentz spaces are even less fine than we would like!!! G. R. Mingione p. 76/11

77 Maximal characterization by Hunt Define the averaging µ (s) := 1 s s 0 µ (t) dt and, accordingly, µ γ,q := ( 0 ( µ ( ) 1/γ) ) 1 q q d Then for γ > 1 it holds that [µ] γ,q µ γ,q c(γ, q)[µ] γ,q G. R. Mingione p. 77/11

78 Selected application 1: Lorentz regularity For the model case div ( Du p 2 Du) = µ then ( ) np µ L np n + p, q This answers to problems posed in Boccardo, CRAS. 87 Kilpelainen & Li, Diff. Int. Equ. 00 = Du L(p, q(p 1)) Alvino & Ferone & Trombetti, Ann. Mat. Pura Appl. 00 Boccardo, Ann. Mat. Pura Appl. 08 G. R. Mingione p. 78/11

79 Selected application 2: Talenti s estimates for Du Cianchi (Non-linear potentials, elliptic equation and rearrangements, Ann SNS Pisa) ( Du Ω ) (s) c Du L 1 (Ω ) ( +c s n 1 n ks ks 0 µ (r) dr ) ( ) Ω 1 +c r n(p 1) (µ (r)) 1 p 1 dr whenever s [0, Ω ], for some absolute constant k k(n, p, Ω) This is the long awaited gradient version of the famous paper of Talenti (Ann. SNS 76) G. R. Mingione p. 79/11

80 Selected application 3: Borderline spaces Corollary - Lorentz spaces characterization of Lipschitz continuity µ L(n, 1/(p 1)) = Du L G. R. Mingione p. 80/11

81 Part 4: Non-local estimates G. R. Mingione p. 81/11

82 A Non-local estimate Consider again diva(x, Du) = µ x a(x, Du) is only measurable G. R. Mingione p. 82/11

83 A Non-local estimate Theorem (Min., Math. Ann. 10) {M( Du ) Tλ} T (p+δ) {M( Du ) λ} for every T >> 1 some δ > 0 and every λ + {W µ 1/p,p (x, R) λ} G. R. Mingione p. 83/11

84 A Non-local estimate Theorem (Min., Math. Ann. 10) {M( Du ) Tλ} T (p+δ) {M( Du ) λ} for every T >> 1 some δ > 0 and every λ + {W µ 1/p,p (x, R) λ} Here M(Du) is the Hardy-Littlewood maximal operator of Du G. R. Mingione p. 84/11

85 A Non-local estimate Theorem (Min., Math. Ann. 10) {M( Du ) Tλ} c 1 T (p+δ) {M( Du ) λ} for every T >> 1 some δ > 0 and every λ +c(t) {W µ 1/p,p (x, R) λ} Here M(Du) is the Hardy-Littlewood maximal operator of Du Again Borderline regularity follows: ( ) np µ L np n + p, q = Du L(p, q(p 1)) G. R. Mingione p. 85/11

86 A Non-local estimate integration on leve sets yields (Tλ) q 1 {M( Du ) Tλ} d(tλ) c 1 (T q p δ ) λ q 1 {M( Du ) λ} dλ +c(t)t q λ q 1 {W µ 1/p,p (x, R) λ} dλ This means, with q < p + δ [M(Du)] q c 1 (T q p δ ) [M(Du)] q +c(t)t q [W µ 1/p,p (x, R)]q now reabsorb the second integral in the first by choosing T large G. R. Mingione p. 86/11

87 Morrey spaces A density condition Norm for measures µ (B R ) R n θ θ [0, n] µ L 1,θ (Ω) := sup R θ n µ (B R ) B R Ω Functions B R v γ dx R n θ v γ L γ,θ (Ω) := sup R θ n v B R Ω B γ dx R G. R. Mingione p. 87/11

88 Be careful with Morrey spaces In fact we have L γ,n L γ L γ,0 L but, for instance L 1,θ L log L whenever θ is close to zero. Therefore this is an integrability scale orthogonal to the Lebesgue one. Also: Morrey spaces are neither rearrangement invariant spaces (obvious) nor interpolation spaces (Stein, Zygmund, Blasco, Ruiz, Vega) G. R. Mingione p. 88/11

89 Morrey spaces Morrey-Marcinkiewicz spaces (Stampacchia, Adams, Lewis) and set supλ γ B R { v > λ} R n θ λ>0 v γ := sup supλ γ R θ n B M γ,θ (Ω) R { v > λ} B R Ω λ>0 := sup R θ n v γ M(B R ) B R Ω G. R. Mingione p. 89/11

90 Lorentz spaces g belongs to L(γ, q)(ω) iff g L(γ,q)(Ω) := [ γ 0 (λ γ {x Ω : g(x) > λ} ) q γ dλ λ ] 1 q < where 1 γ < 0 < q For q = we have L(γ, ) = M γ The second index tunes the first one and for 0 < q < t < r we have L r L(r, r) L(t, q) L(t, t) L(t, r) L(q, q) L q G. R. Mingione p. 90/11

91 Lorentz-Morrey spaces g belongs to L θ (γ, q)(ω) iff whenever g L(γ,q)(BR ) R n θ γ 1 γ < 0 < q < θ [0, n] Morreyzation of the Lorentz norm g Lθ (γ,q)(ω) := sup R θ n γ g L(γ,q)(BR ) B R Ω G. R. Mingione p. 91/11

92 Adams theorem (Duke Math. J. 75) Morrey regularization f L γ,θ = I β (f) L θγ θ γβ,θ γβ < θ The case γ = 1 µ, f L 1,θ = I β (µ, f) M θ θ β,θ β < θ Borderline f L 1,θ L log L = I β (f) L θ θ β β < θ G. R. Mingione p. 92/11

93 Adams theorem (Duke Math. J. 75) Corollary, for u = f (take β = 1) it holds that f L γ,θ = Du L θγ θ γ,θ 1 < γ < θ L 1,θ -data Borderline f L 1,θ = Du M θ θ 1,θ f L 1,θ L log L = Du L θ θ 1 G. R. Mingione p. 93/11

94 Significant case Another theorem of Adams Therefore µ (B R ) R n θ, Du L p θ < p = µ W 1,p maximal regularity holds for solutions to div (c(x, u) Du p 2 Du) = µ Significant case µ (B R ) R n θ p θ n G. R. Mingione p. 94/11

95 The non-linear case div a(x,u,du) = µ,f When θ = n f L γ and γ < np np n + p = (p ) When θ < n, we take f L γ,θ and γ θp θp θ + p =: p(θ) This is initially suggested by the fact when dealing with Morrey spaces θ plays the role of the dimension n. G. R. Mingione p. 95/11

96 Non-linear Adams theorem Theorem (Min., Math. Ann. 10) For div a(x, u, Du) = µ, f it holds that f L γ,θ = Du p 1 L θγ θ γ,θ 1 < γ p(θ) L 1,θ -case µ, f L 1,θ = Du p 1 M θ θ 1,θ Bordeline µ, f L 1,θ L log L = Du p 1 L θ θ 1 µ, f L log L θ = Du p 1 L θ θ 1,θ G. R. Mingione p. 96/11

97 Morrey-Gehring regularity Results for γ > θp θp θ + p = p(θ) For solutions to div a(x, u, Du) = f it holds that if γ > θp θp θ + p and p < θ n then f L γ,θ = Du L h,θ h > p G. R. Mingione p. 97/11

98 A theorem of Adams & Lewis (Studia Math. 82) Optimal ( θγ f L θ (γ, q) = I β (f) L θ θ βγ, ) θq θ βγ γβ < θ The standard case θ = n, via interpolation f L(γ, θ) = I β (f) L ( ) nγ n βγ, q γβ < θ G. R. Mingione p. 98/11

99 A discontinuity phenomenon A discontinuity phenomenon L θ ( θγ θ βγ, ) θq θ βγ L ( θγ θ βγ, ) θq θ βγ L ( ) nγ n βγ, q but L ( ) nγ n βγ, q L ( nγ n βγ, ) nq n βγ Therefore while lim θ n Lθ (γ, q) = L(γ, q) we do not recover the sharp embedding result form the Lorentz-Morrey one letting θ n G. R. Mingione p. 99/11

100 The non-linear Adams & Lewis theorem Theorem (Min., Math. Ann. 10) For solutions to div a(x, Du) = f L θ (γ, q) we have ( ) θγ Du p 1 L θ θ γ, θq θ γ locally in Ω. G. R. Mingione p. 100/11

101 Particular cases For p = 2 and θ = n (no Morrey) we recover the linear results in Lorentz spaces Taking p = 2 and γ = q gives back Adams theorem Taking p < n, θ = n and γ = q (no Morrey, no Lorentz), gives back the results of Boccardo & Gallöuet Taking p < n, θ = n (no Morrey) and q = gives back the results of Kilpeläinen & Li, plus borderline cases Taking p < n, θ = n (no Morrey) gives back the results of Alvino & Ferone & Trombetti, plus borderline cases G. R. Mingione p. 101/11

102 The local estimate Optimal local estimates are available Du p 1 Lθ ( θγ θ γ, θq θ γ)(b R/2 ) cr θ γ γ n ( Du + s) p 1 L1 (B R ) +c f L θ (γ,q)(b R ) G. R. Mingione p. 102/11

103 Part 5: Oscillation estimates G. R. Mingione p. 103/11

104 Gradient continuity estimates A borderline space. If then but if we only have µ L n+ε Du is Hölder continuous Du L n then the gradient can be unbounded even in the case of the standard Poisson equation u = µ (see Cianchi - J. Geom. Anal. 92). In the case u = µ then Cianchi (ibid.) proved that µ L(n, 1) = Du L Du L if µ L(n, q) with q > 1 G. R. Mingione p. 104/11

105 Gradient estimates We have seen W µ 1/p,p (x, R) L = Du L for some R > 0 Observe that when W µ 1/p,p (x, R) L then R lim R 0 Wµ 1/p,p (x, R) = lim R 0 0 ( µ (B(x, )) n 1 ) 1 p 1 d = 0 almost everywhere in Ω and are uniformly bounded w.r.t. x Something is in between... G. R. Mingione p. 105/11

106 Gradient continuity Theorem (Duzaar & Min., Calc. Var., to appear) Assume lim R 0 Wµ 1/p,p (x, R) = 0 uniformly in Ω Then Du is continuous in Ω The threshold between C 0,1 and C 1 regularity of solutions is established by the different kinds of convergence (pointwise and uniform) of non-linear Wolff potentials G. R. Mingione p. 106/11

107 Link to Lorentz spaces Now we note that sup x Ω c(n)r n W µ 1/p,p (x, R) c But recall that if µ L(n, 1/(p 1)) then 0 (µ ( ) 1/n) 1 p 1 d 0 (µ ( ) 1/n) 1 p 1 d < As a consequence, we have that if µ L(n, 1/(p 1)) then lim R 0 Wµ 1/p,p (x, R) = 0 uniformly in Ω G. R. Mingione p. 107/11

108 Corollary Theorem (Duzaar & Min., Calc. Var., to appear) µ L(n, 1/(p 1)) = Du is continuous in Ω G. R. Mingione p. 108/11

109 Yet another borderline case: Density conditions Theorem (Lieberman 93) Assume that µ (B ) n 1+ε ε > 0 then Du is Hölder continuous Theorem (Duzaar & Min., Calc. Var., to appear) Assume that µ (B ) n 1 h( ) and 0 [h( )] 1 d p 1 < then Du is continuous G. R. Mingione p. 109/11

110 The case of systems Consider the p-laplacean system div( Du p 2 Du) = µ, and u is now vector valued u: Ω R N Theorem (Duzaar & Min., Calc. Var., to appear) µ L(n, 1/(p 1)) = Du is continuous in Ω G. R. Mingione p. 110/11

111 Part 6: A fully fractional approach G. R. Mingione p. 111/11

112 The setting for p = 2 We consider equations div a(du) = µ under the assumptions { a(z) + az (z) z L z ν 1 λ 2 a z (z)λ, λ G. R. Mingione p. 112/11

113 The setting for p = 2 Theorem (Min. 08) D ξ u(x 0 ) c D ξ u dx + ci µ 1 (x 0, 2R) B(x 0,R) for every ξ {1,...,n}, where I µ β (x 0, R) := R 0 µ(b(x 0, )) n 1 d G. R. Mingione p. 113/11

114 Classical Gradient estimates Consider energy solutions to diva(du) = 0 for p = 2 First prove Du W 1,2 Then use that v = D ξ u solves div(a(x)dv) = 0 A(x) := a z (Du(x)) The boundedness of D ξ u follows by Standard DeGiorgi s theory This is a consequence of Caccioppoli s inequalities of the type B R/2 D(D ξ u k) + 2 dx c R 2 B R (D ξ u k) + 2 dx where (D ξ u k) + := max{d ξ u k, 0} G. R. Mingione p. 114/11

115 Recall the definition We have v W σ,q (Ω ) iff v L q (Ω ) and [v] σ,1;ω = ( Ω Ω v(x) v(y) q ) 1 q dxdy < x y n+σq G. R. Mingione p. 115/11

116 There is differentiability problem For solutions to but nevertheless it holds diva(du) = µ in general Du W 1,1 Theorem (Min. Ann. SNS Pisa 07) Du W 1 ε,1 loc (Ω, R n ) for every ε (0, 1) This means that [Du] 1 ε,1;ω = Ω Ω Du(x) Du(y) x y n+1 ε dxdy < holds for every ε (0, 1), and every subdomain Ω Ω G. R. Mingione p. 116/11

117 Step 1: A non-local Caccioppoli inequality Theorem (Min. JEMS, to appear) Let w = D ξ u with diva(du) = µ where ξ {1,...,n} then [( w k) + ] σ,1;br/2 c R σ ( w k) + dx + cr µ (B R) B R R σ holds for every σ < 1/2 Compare with the usual one for diva(du) = 0 [(w k) + ] 2 1,2;B R/2 D(w k) + 2 dx B R/2 c R 2 B R (w k) 2 + dx G. R. Mingione p. 117/11

118 Step 1: A non-local Caccioppoli inequality This is the most important step, revealing robustness of energy inequalities, which hold below the natural growth exponent 2, and for fractional order of differentiability, although the equation has integer order Classical VS fractional classical fractional spaces L 2 L 2 L 1 L 1 differentiability σ G. R. Mingione p. 118/11

119 Step 2: Fractional De Giorgi s iteration Theorem (Min. JEMS, to appear) Let w be an L 1 -function w satisfying the fractional Caccioppoli s inequality [( w k) + ] σ,1;br/2 L R σ ( w k) + dx + LR µ (B R) B R R σ for some σ > 0 and every k 0. Then it holds that w(x 0 ) c w dx + ci µ 1 (x 0, 2R) B(x 0,R) for every Lebesgue point x 0 of w Proof of the gradient potential estimate: apply the previous result to w D ξ u G. R. Mingione p. 119/11