Regularity for Partial Differential Equations

Feature Article

Regularity for Partial Diﬀerential Equations: from De Giorgi-Nash-Moser Theory to Intrinsic Scaling

by Jos´eMiguel Urbano Departamento de Matem´atica - CMUC Universidade de Coimbra

1 A beautiful problem smoothness of L, the minimizer (assuming it exists) is also smooth. In the academic year 1956-1957, John Nash had a visit- Problems of this type are related to elliptic equations in ing position at the Institute for Advanced Study (IAS) that a minimizer u is a weak solution of the associated in Princeton, on a sabbatical leave from MIT, but he Euler-Lagrange equation actually lived in New York City. The IAS at the time “was known to be about the dullest place you could n 1 X ∂ find” and Nash used to hang around the Courant In- L (∇u(x)) = 0 in Ω . ∂x ξi stitute which was close to home and full of activity. i=1 i That’s how he came across a problem that mathematicians had been trying to solve for quite a while. The This equation can be differentiated with respect to xk, story goes that Louis Nirenberg, at the time a young to give that, for any k = 1, 2, . . . , n, the partial deriva- ∂u professor at Courant, was the person responsible for the tive := vk satisfies a linear PDE of the form ∂xk unveiling: “...it was a problem that I was interested in and tried to solve. I knew lots of people interested in n X ∂ ∂vk this problem, so I might have suggested it to him, but a (x) = 0 , (1) ∂x ij ∂x I’m not absolutely sure”, said Nirenberg recently in an i,j=1 i j interview to the Notices of the AMS (cf. [19]). with coefficients a (x) := L (∇u(x)). The PDE is As so many other great questions of 20th century math- ij ξiξj elliptic provided L is assumed to be convex. ematics, it all started with one of Hilbert’s problems presented on the occasion of the 1900 International In the 1950’s, regularity theory for elliptic equations Congress of Mathematicians in Paris, namely the 19th was essentially based on Schauder’s estimates which, problem: Are the solutions of regular problems in the k,α roughly speaking, guarantee that if aij ∈ C then the calculus of variations always necessarily analytic? A solutions of (1) are of class Ck+1,α, for k = 0, 1,... simple example of such a problem is, in modern termi- 1,α So if it could be shown that u ∈ C then aij(x) := nology, the problem of minimizing a functional 0,α 1,α Lξiξj (∇u(x)) would belong to C , v to C and u to Z C2,α; a bootstrap argument would then solve Hilbert’s min L(∇w(x)) dx 19th problem. w∈A Ω where Ω ⊂ Rn is a bounded and smooth domain, the Meanwhile, the existence theory had been developed Lagrangian L(ξ) is a smooth (possibly nonlinear) scalar through the use of direct methods: the minimization function defined on Rn and A is a set of admissible problem has a unique solution provided L, apart from functions (typically the elements of a certain function satisfying natural growth conditions like space satisfying a boundary condition like w = g on ∂Ω, for a given g). The question is to prove that, given the |L(ξ)| ≤ C |ξ|p ,

1Cathleen Morawetz, quoted in [15].

8 is also coercive and uniformly convex. The notion of and to satisfy the uniform ellipticity condition (for solution had to be conveniently extended and the ad- λ > 0) missible set A taken to be the set of functions that, n together with their first weak derivatives, belong to Lp, X a ξ ξ ≥ λ|ξ|2 , ∀x ∈ Ω, ∀ξ ∈ Rn . i.e., that belong to the Sobolev space W 1,p. ij i j i,j=1 So the existence theory gave a minimizer u ∈ W 1,p and A weak solution of equation (2) is a function u ∈ the missing step for the regularity problem to be solved W 1,2(Ω) which satisfies the integral identity was 1,p 1,α n u ∈ W =⇒ u ∈ C X Z ∂u ∂ϕ aij = 0 (3) p ∂x ∂x i.e., from first derivatives in L to Höldercontinuous i,j=1 Ω i j first derivatives. In terms of the elliptic PDE (1), reg- 1,2 1,2 ularity theory worked if the leading coefficients were for all test functions ϕ ∈ W0 (Ω) (the elements of W already somewhat regular (at least continuous) since which vanish on the boundary ∂Ω in a suitable weak it was based on perturbation arguments and compari- sense). son of the solutions with harmonic functions. Assum- To simplify the writing we assume from now on that ing only the measurability and the boundedness of the coefficients (together with the essential structural as- n n o Ω = B1 := x ∈ R : |x| < 1 . sumption of ellipticity) was insufficient, and nothing was known about the regularity of the solutions in this case. Theorem 1 Every weak solution of (2) is locally bounded. The problem was solved by C.B. Morrey in 1938 for the special case n = 2 but the techniques he employed Proof. Let k ≥ 0 and η be a smooth function with were typically two dimensional, involving complex anal- compact support in B . Put v = (u − k)+ and take ysis and quasi-conformal mappings. The n-dimensional 1 ϕ = vη2 as test function in (3). The use of the assump- problem remained open until the late 50’s and that’s tions and Young’s inequality give exactly what Nirenberg told Nash about. Z 2 Z 2 2 4Λ 2 2 |∇v| η ≤ 2 |∇η| v . (4) B1 λ B1 2 De Giorgi’s breakthrough These Cacciopoli inequalities on level sets of u will be the building blocks of the whole theory and once they The problem wouldn’t resist the genius of John Nash are obtained the PDE can be forgotten: the problem and Ennio De Giorgi. The two men worked totally un- becomes purely analytic. aware of each other’s progress and solved the problem Next, by Hölderand Sobolev’s inequalities (with 2∗ = using entirely different methods. 2n/(n − 2) being the Sobolev exponent),

It was De Giorgi who did it first (actually for p = 2; 2 ∗ Z Z ∗ 2 2 the result would later be extended to any p ∈ (1, ∞)) 1− ∗ (vη)2 ≤ (vη)2 |{vη 6= 0}| 2 and it is his proof that will now be analyzed. To re- B1 B1 ally understand in full depth De Giorgi’s ideas there 2 Z 2 is no way around the technicalities. In what follows I ≤ c(n) |{vη 6= 0}| n |∇(vη)| B did my best to explain things in a clear way but the 1 reader should not expect everything to be trivial or im- and since, due to (4), mediately understandable; so please grab a pencil and Z 2 Z a piece of paper and be prepared to struggle a bit with 2 4Λ 2 2 |∇(vη)| ≤ 2 + 1 |∇η| v inequalities and iterations. B1 λ B1 Consider the equation we arrive at Z Z n 2 2 2 2 X ∂ ∂u (vη) ≤ c(n, λ, Λ) |{vη 6= 0}| n |∇η| v . a (x) = 0 in Ω (2) ∂x ij ∂x B1 B1 i,j=1 i j Now for fixed 0 < r < R < 1, choose the cut-off func- n ∞ where Ω ⊂ R is a smooth bounded domain and the tion η ∈ C0 (BR) such that 0 ≤ η ≤ 1, η ≡ 1 in Br and 2 coefficients aij are only assumed to be measurable and |∇η| ≤ R−r . Putting, for ρ > 0, bounded, with n o kaijkL∞ ≤ Λ , A(k, ρ) = x ∈ Bρ : u(x) > k ,

9 we obtain (with C ≡ c(n, λ, Λ)) From (6) we obtain

Z C 2 Z 2 m(1+ 2 ) 2 n 2 1+ 2 ψ(k0, r0) n 2 n ψ(k0, r0) (u − k) ≤ |A(k, R)| (u − k) . ψ(k , r ) ≤ 2Cγ n (R − r)2 m m 2 2m m A(k,r) A(k,R) k n γ n γ

For h > k and 0 < ρ < 1, 2 1+ 2 and choose ﬁrst γ > 1 such that γ n = 2 n and then Z Z k large enough so that (u − h)2 ≤ (u − k)2 A(h,ρ) A(k,ρ) 2 2 ψ(k0, r0) n 1+ n ∗ 2Cγ 2 ≤ 1 ⇐ k = C ψ(k0, r0) and k n Z 2 2 (h − k) |A(h, ρ)| ≤ (u − k) where C∗ ≡ C∗(n, λ, Λ). A(k,ρ) 1 so we have Finally let m → ∞ in (7) to get ψ(k, 2 ) ≤ 0, i.e.,

+ 2 k(u − k) kL (B 1 ) = 0 . Z 2 (u − h)2 A(h,r) Hence + ∗ + 2 sup u ≤ C ku kL (B1) . C 2 Z B 1 ≤ |A(h, R)| n (u − h)2 2 (R − r)2 A(h,R) Using a dilation argument, this estimate can be reﬁned; 1+ 2 Z ! n indeed, for any θ ∈ (0, 1) and p > 1, it holds C 1 2 ≤ 2 4 (u − k) (R − r) (h − k) n A(k,R) + C(n, λ, Λ) + sup u ≤ ku k p . n/p L (B1) + Bθ (1 − θ) 2 or, equivalently, with ψ(s, ρ) = k(u − s) kL (Bρ) The same type of reasoning gives similar conclusions C 1 2 − 1+ n concerning u and the result follows. ψ(h, r) ≤ 2 ψ(k, R) , (5) R − r (h − k) n for any h > k > 0 and 0 < r < R < 1. The basic general idea to obtain results concerning the continuity of a solution of a PDE at a point consists in We are now ready to use the brilliant iteration scheme estimating its oscillation in a nested sequence of con- devised by De Giorgi. Deﬁne, for m = 0, 1, 2,... centric balls (cylinders in the parabolic case), centered 1 at the point, and showing that it converges to zero as k = k (1 − ) m 2m the balls shrink to the point. If this can be measured 1 1 quantitatively it gives a modulus of continuity. rm = (1 + m ) 2 2 Denote the oscillation of a function u in Br by osc(u, r). A further analysis, which uses the previous theorem, where k is to be determined later. Due to (5), we then leads to have, for m = 0, 1, 2,...,

m+1+ 2m 2 n 2 1 1+ n Theorem 2 Let u ∈ H (B2) be a weak solution of (2) ψ(km, rm) ≤ C 2 ψ(km−1, rm−1) (6) k n in B2. There exists a constant γ = γ(n, λ, Λ) ∈ (0, 1) such that and can prove, by induction, that, for some γ > 1, osc (u, 1/2) ≤ γ osc (u, 1) .

ψ(k0, r0) ψ(km, rm) ≤ , ∀ m = 0, 1, 2,... (7) γm Thus (see below), there exists some constant α ∈ (0, 1) such that, for 0 < r < R < 1, if k is chosen suﬃciently large. In fact, it is trivial that it holds for m = 0; now suppose it holds for m − 1 and r α osc (u, r) ≤ C osc (u, R) , write R

2 1+ n 1+ 2 ψ(k0, r0) which gives a H¨oldermodulus of continuity and ψ(k , r ) n ≤ m−1 m−1 γm−1 2 ψ(k , r ) n ψ(k , r ) Theorem 3 (De Giorgi - Nash) Every weak solu- = 0 0 0 0 . 2m −(1+ 2 ) m tion of (2) is H¨older continuous. γ n n γ

10 3 Three papers... and a correc- Now, for ρ < r, we can take k such that 2−k−1 r < ρ ≤ −k tion 2 r to obtain ρβ osc (u, ρ) ≤ C osc (u, r) In a series of three fundamental papers (and a correc- r tion) published in Communications on Pure and Ap- with β = − log α > 0, and as a consequence the Hölder plied Mathematics, the journal of the Courant Insti- log 2 continuity of the function u. tute, JürgenMoser made significant contributions to the theory. He first gave in [11] a new proof of De Gior- Moser extended his results to parabolic equations, ob- gi’s theorem, using the simple general principle that the taining a Harnack inequality for nonnegative solutions estimates (4) hold for any convex function f(u) of a so- of the parabolic analogue of (2), assuming only the lution u; the results were obtained by applying such boundedness of the coefficients and the condition cor- estimates to powers f(u) = |u|p, p ≥ 1, and to the log- responding to ellipticity (cf. [13] and [14]). Again his arithmic function log+ u−1. Then he proved Harnack’s approach was essential nonlinear and contrasted dra- inequality for elliptic equations (cf. [12]): matically with the approach via fundamental solutions that had been used by Hadamard and Pini to obtain Theorem 4 (Moser’s Harnack inequality) If u is Harnack estimates for solutions of the heat equation. a positive weak solution of (2) and K is a compact sub- set of Ω, then 4 When Stanley met Living- max u ≤ C min u , K K stone where C ≡ C(Ω, K, λ, Λ). In 1958 the ICM would take place in Edinburgh and the The proof of the Harnack inequality made no use of deliberations on the Fields medalists were concluded the Hölder continuity of the solutions, which in turn is early that year (the two medals were eventually award- a simple consequence of that fact, as Moser showed in ed to Thom and Roth). Solving the regularity problem would probably be worth a medal. Nash in his own the paper. In fact, assume again that Ω = B1. Let, for 0 < r < 1, words [18]: “It seems conceivable that if either De Gior- gi or Nash had failed on this problem (...) then the lone M(r) = max u , m(r) = min u , climber reaching the peak would have been recognized Br Br with mathematics’ Fields medal.”Nash solved the problem in the spring of 1957 using a nonlinear approach and apply Harnack’s inequality to the domains Br and to attack linear equations. The main results would be B r to get 2 announced in a note to the Proceedings of the Nation- al Academy of Sciences ([17]), submitted by Marston max (M(r) − u) = M(r) − m(r/2) B r Morse of the IAS on October 6, 1957. By then, Nash 2 had already found out, in late spring, about De Giorgi’s ≤ C M(r) − M(r/2) proof: “(...) although I did succeed in solving the problem, I ran into some bad luck since, without my being = C min (M(r) − u) B r sufficiently informed on what other people were doing in 2 and the area, it happened that I was working in parallel with Ennio De Giorgi of Pisa, Italy. And De Giorgi was first M(r/2) − m(r) ≤ C m(r/2) − m(r) actually to achieve the ascent of the summit (...).”In fact, the seminal paper of De Giorgi was presented by since M(r) − u and u − m(r) are positive solutions in Mauro Picone on April 24, 1957 to the Academy of Sci- ences of Torino and the results had been announced at Br. Adding these two inequalities, we obtain the Congress of the Unione Matematica Italiana, which C − 1 took place in Pavia in October, 1955. Some say that M(r/2) − m(r/2) ≤ M(r) − m(r) C + 1 Nash was devastated when he learned about De Giorgi. That summer De Giorgi visited the Courant Institute C−1 or, with α = C+1 < 1, and Peter Lax would say later about the meeting of the two men: “It was like Stanley meeting Livingstone.” osc (u, r/2) ≤ α osc (u, r) . The approach of Nash to the problem was totally dif- By induction, ferent from De Giorgi’s and some people think that his ideas were never fully understood (cf. [4]). He treated osc (u, 2−k r) ≤ αk osc (u, r); k = 1, 2,... parabolic equations directly and obtained the results for

11 elliptic equations as corollaries. The essence of his rea- In the elliptic case, this fact allowed for the extension soning consisted of obtaining control of the properties to quasilinear equations of the type of fundamental solutions of linear parabolic equations with variable coefficients. The crucial estimate is the ∇ · a(x, u, ∇u) = b(x, u, ∇u) in Ω , moment bound where the principal part a satisfies the growth assump- √ Z √ tion k1 t ≤ |x| T (x, t) dx ≤ k2 t , |a(x, u, ∇u)| ≤ Λ |∇u|p−1 + ϕ(x) and the ellipticity condition which controls the moment of a fundamental solution T . About this result Nash wrote in [16]: “(...) it opens a(x, u, ∇u) · ∇u ≥ λ |∇u|p − ϕ(x) , the door to the other results. We had to work hard to get (the bound), then the rest followed quickly.” for constants 0 < λ ≤ Λ and a bounded ϕ ≥ 0; the prototype is the p-Laplacian equation Although the problem was “morally”solved, writing the paper proved to be technically very hard and Nash con- ∇ · (|∇u|p−2∇u) = 0 . tinued to work on it when he went back to MIT in the summer of 1957. A few steps in the proof were not Notice the nonlinear dependence on the partial deri- clear and only a joint effort with such people as Lennart vatives and the nonlinear growth with respect to the Carleson (who was visiting MIT on leave from Uppsala) gradient. The equation is degenerate if p > 2 and and Elias Stein, both explicitly credited in the paper for singular if 1 < p < 2, since its modulus of ellipticity p−2 some of the proofs, eventually led to his famous paper |∇u| vanishes or blows up, respectively, at points published in 1958 (Nash writes as an acknowledgement: where |∇u| = 0. Ladyzhenskaya and Ural’tzeva estab- “We are indebted to several persons”and then names lished the Hölder continuity of weak solutions (cf. [9], eleven colleagues). There’s a petite histoire about the the bible of elliptic equations), extending De Giorgi’s publishing of the paper (cf. [15]): Nash first submitted results, and Serrin [20] and Trudinger [21] obtained the it to Acta Mathematica through Carleson, who was an Harnack inequality for nonnegative solutions following editor there, and made him know that he wanted the Moser’s ideas. paper to be refereed quickly. Carleson gave it to Lars Surprisingly enough, the theory didn’t succeed so well Hörmander (later a Fields medalist, together with John in the parabolic case Milnor, in the Stockholm ICM in 1962) who did the job in two months and recommended the paper for pub- ut − ∇ · a(x, u, ∇u) = b(x, u, ∇u) in Ω × [0,T ) lication. But Nash withdrew the paper, which would appear later in the American Journal of Mathematics. and Moser’s proof could only be extended (by Aron- The reason for this might have been that, after “loos- son, Serrin and Trudinger) for the case p = 2, which ing”the Fields, Nash wanted the paper to be eligible for corresponds to principal parts with a linear growth on the AMS Bôcher Prize (awarded for a notable research |∇u|. The same happened with the methods of De Gior- memoir in analysis published during the previous five gi, which the Russian school extended to the parabolic years in a recognized North American journal). The case (always for p = 2), thus rediscovering Nash’s re- 1959 prize would be awarded to Louis Nirenberg for his sults concerning the Hölder continuity (of solutions of work on partial differential equations. parabolic equations) by entirely different methods. So, unlike the elliptic case, degenerate or singular equations Whether Nash’s ideas were ever understood in full like p−2 depth by anyone except himself remains unclear. The ut − ∇ · (|∇u| ∇u) = 0 , fact is that his work, although profusely cited, didn’t for which the principal part of the equation grows non- give rise to much subsequent research. It was the more linearly with |∇u|, seemed to behave differently, and the understandable approach of De Giorgi and Moser that questions of regularity remained open until the 1980’s. the PDE community adopted and developed to full ex- tent. To understand the difficulty, consider a parabolic cylinder Q(τ, R) := BR × (0, τ) . 5 Intrinsic scaling The use of Cacciopoli inequalities on level sets leads in this case to expressions of the form (compare with (4)) Z Z τ Z The work of De Giorgi, Moser and Nash concerned lin- 2 p p p ear PDE’s but the approach was essentially nonlinear sup v η + ∇v η 0

12 The iterative argument of De Giorgi, as adapted by the The inclusion Russian school to the parabolic case, required the equa- γ(u) − ∇ · (|∇u|p−2∇u) 3 0 , p > 2, tion to be nondegenerate (p = 2) so that the integral t norms appearing in these estimates were homogeneous. where γ is a maximal monotone graph with a singular- This is not the case in the inequality above: the pres- ity at the origin, occurs as a model for the well-known ence of the power p jeopardizes the homogeneity in the two-phase Stefan problem when a nonlinear law of diffu- estimates and the recursive process itself. The key idea sion is considered, u being in that case the temperature to overcome the difficulty presented by the inhomogene- and γ(u) the enthalpy. As before, the equation is de- ity was introduced by DiDenedetto (cf. [3] for an ac- generate in the space part, but now it is also singular in count of the theory and an extensive list of references) the time part since “γ0(0) = ∞”. In this case a further and consists essentially of looking at the equation in power appears in the energy estimates (the power one, its own geometry, i.e., in a geometry dictated by its which is due to estimating the singular term) and no degenerate structure. This amounts to re-scaling the re-scaling permits the compatibility of the three pow- standard parabolic cylinders by a factor depending on ers involved. The proof of the regularity in [22] uses the oscillation of the solution. This procedure of in- the geometry of the nonsingular case to deal with the trinsic scaling, which somehow is an accommodation of degeneracy but the price of a dependence on the oscilla- the degeneracy, allows the recovering of the homogene- tion in the various constants that are determined along ity in the energy estimates, written over these re-scaled the proof has to be paid. Owing to this fact, it is no cylinders, and the proof then follows more or less easily. longer possible to exhibit a modulus of continuity for One can say heuristically that the equation behaves in the solution of the problem but only to define it implic- its own geometry like the heat equation. itly. This is enough to obtain the continuity but the Hölder continuity, which holds in the nonsingular case, Let’s briefly describe the procedure for the degenerate is lost. case p > 2. Consider R > 0 such that Q(Rp−1, 2R) ⊂ Ω × [0,T ), define Another example concerns the parabolic equation with two degeneracies ω := osc (u, Q(Rp−1, 2R)) ∂tu − ∇ · (α(u)∇u) = 0 , and construct the cylinder where u ∈ [0, 1] and α(u) degenerates for u = 0 and u = 1. An equation of this type is physically rele- ω 2−p Q(a Rp,R) , with a = vant since it shows up in a model describing the flow 0 0 A of two immiscible fluids through a porous medium and where A depends only on the data. Note that for p = 2, also in polymer chemistry and combustion. In [23] it is i.e., in the nondegenerate case, we have a0 = 1 and shown that u is locally Hölder continuous if α decays these are the standard parabolic cylinders that reflect like a power at both degeneracies. A fine analysis of the natural homogeneity of the space and time vari- what happens near the two degeneracies leads to the ables. Assume, without loss of generality, that ω < 1 construction of the cylinders used in the iterative pro- and also that cess, with the appropriate geometry being once again 1 ω p−2 dictated by the structure of the PDE. = > R a0 A p p−1 which implies that Q(a0R ,R) ⊂ Q(R , 2R) and the Acknowledgements. I would like to thank Prof. João relation Queirófor his encouragement and interest. p osc (u, Q(a0R ,R)) ≤ ω . (8) This is in general not true for a given cylinder since its dimensions would have to be intrinsically defined in Bibliografia terms of the oscillation of the function within it; it is the starting point of the iteration process, in which the [1] H. Brezis and F. Browder, Partial differential equations difficulties coming from the degenerate structure of the in the 20th century, Adv. Math. 135 (1998), 76-144. problem are overcome through the use of the re-scaled [2] E. De Giorgi, Sulla differenziabilitàe l’analiticitàdelle cylinders. The details are extremely technical and the estremali degli integrali multipli regolari, Mem. Accad. interested reader can consult [3]. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 3 (1957), 25-43. [3] E. DiBenedetto, Degenerate Parabolic Equations, These ideas have been explored to obtain regularity re- Springer-Verlag, 1993. sults for other partial differential equations, like the [4] E. Fabes and D. Stroock, A new proof of Moser’s porous medium equation or doubly nonlinear parabolic parabolic Harnack inequality using the old ideas of equations. I’ll comment briefly on two extensions for Nash, Arch. Rational Mech. Anal. 96 (1986), no. 4, which I am partly responsible. 327-338.

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