Some properties of De Giorgi classes

Transcription

1 end. Istit. Mat. Univ. Trieste Volume , DOI: 0.337/ /359 Some roerties of De Giorgi classes Emmanuele DiBenedetto and Ugo Gianazza Dedicated to Giovanni Alessandrini for his 60th Birthday Abstract. The De Giorgi classes [DG] E; γ, defined in ± below encomass, solutions of quasilinear ellitic equations with measurable coefficients as well as minima and Q-minima of variational integrals. For these classes we resent some new results 2 and 3., and some known facts scattered in the literature 3 5, and formulate some oen issues 6. Keywords: De Giorgi classes, Hölder continuity, Harnack inequality, higher integrability, boundary behavior, decay estimates. MS Classification 200: 35J5, 49N60, 35J92.. Introduction Let E be oen subset of N and for y N, let y denote a cube of edge 2ρ centered at y. The De Giorgi classes [DG] ± E; γ in E are the collection of functions u W, loc E, for some >, satisfying Du k ± dx γ u k ± dx ± for all cubes y E, and all k, for a given ositive constant γ. We further define [DG] E; γ = [DG] + E; γ [DG] E; γ. 2 A celebrated theorem of De Giorgi [2] states that functions u [DG] E; γ are locally bounded and locally Hölder continuous in E. Moreover, non-negative functions u [DG] E; γ satisfy the Harnack inequality [7]. Local subsuer-solutions, in W, loc E, of quasi-linear ellitic equations in divergence form belong to [DG] + E; γ [2], with γ roortional to the ratio of uer and lower modulus of elliticity. Local minima and/or Q- minima of variational integrals with -growth with resect to Du belong to these classes [0]. Thus the [DG] -classes include local solutions of ellitic equations with merely bounded and measurable coefficients, only subject to

2 246 E. DIBENEDETTO AND U. GIANAZZA some uer and lower elliticity condition. They also include local minima or Q-minima of rather general functionals, even if not admitting a Euler equation. The interest in the De Giorgi classes stems from the large class of, seemingly unrelated functions they encomass, and from roerties, such as local Hölder continuity [2], and the Harnack inequality [7], tyically regarded as roerties of solutions of ellitic artial differential equations [2, 4]. The urose of this note is to resent some new results on De Giorgi classes 2 and 3., as well as collecting some known facts scattered in the literature 3 5, and formulate some oen issues 6 to serve as a basis for further investigations. 2. De Giorgi Classes and SubSuer-Harmonic Functions The generalized De Giorgi classes [GDG] ± E; γ, are the collection of functions u W, loc E, for some >, satisfying Du k ± dx N γ u k ± dx 3 ± for all cubes y E, and all k, for a given ositive constant γ. Convex, monotone, non-decreasing functions of sub-harmonic functions are sub-harmonic. Similarly, concave, non-decreasing, functions of suer-harmonic functions are suer-harmonic. Similar statements hold for weak, subsuer- solutions of linear ellitic equations with measurable coefficients [4]. The next lemma establishes analogous roerties for functions u [DG] ± E; γ. Given any such class, we refer to the set of arameters {, γ, N} as the data and say that a constant C = Cdata deends only on the data if it can be quantitatively determined a-riori only in terms of the indicated set of arameters. Lemma 2.. Let ϕ : be convex and non-decreasing, and let u [DG] + E; γ. There exists a ositive constant γ deending only on the data, and indeendent of u, such that ϕu [GDG] + E; γ. Likewise let ψ : be concave and non-decreasing, and let u [DG] E; γ. There exist a ositive constant γ deending only on the data, and indeendent of u, such that ψu [GDG] E; γ. Proof. By De Giorgi s theorem [2, 2], there exists a constant C = Cdata, such that for any u [DG] ± E; γ, there holds u k ±,Kρy C N u k ± dx 4

3 SOME POPETIES OF DE GIOGI CLASSES 247 for every air of cubes y E and all k. It suffices to rove the first statement for ϕ C 2, and verify that ϕu satisfies 3 + for cubes K centered at the origin of N. For any such ϕ and all h k ϕu ϕh + ϕ hu h + = u k + χ [k>h] ϕ kdk 5 + From this, a.e. in E D [ ϕu ϕh + ] ϕ hu h + Du k + χ [k>h] ϕ kdk. Integrate over, take the root of both sides, and majorize the resulting term on the right-hand first by the continuous version of Minkowski inequality, then by alying the definition + of the [DG] + E; γ-classes, and finally by using 4. This gives D [ ϕu ϕh + ] ϕ hu h +,Kρ Du k +,Kρ χ [k>h] ϕ kdk C C N C N N+ u k +,K +ρ χ [k>h] ϕ kdk 2 u k +,K +ρ χ [k>h] ϕ kdk 2 u k + dx χ [k>h] ϕ kdk K C N = N+ u k + χ [k>h] ϕ kdk dx K C N [ = ϕu ϕh N+ ] + ϕ hu h + dx K C N ϕu ϕh + ϕ hu h,k +. In these calculations, we have denoted by C = C, N, γ a generic constant deending only uon the data, and that might be different from line to line. In the last two stes we have interchanged the order of integration with the hel of Fubini s Theorem and have alied Hölder s inequality. By the convexity and monotonicity of ϕ, ϕu ϕh + ϕ hu h

4 248 E. DIBENEDETTO AND U. GIANAZZA Therefore, D ϕu ϕh +,Kρ C N ϕu ϕh + ϕ hdu h +,Kρ +,K Uon alying the definition of + of [DG] + E; γ, and then 6, the last term on the right-hand side is majorized by Combining these estimates yields D ϕu k + C ϕu ϕh+,k. N γ dx ϕu k dx 7 + for all k and all y E, for a constant γ = γdata. If u [DG] E; γ and ϕ is convex, there is no guarantee, in general, that ϕu [GDG] + E; γ for some γ = γ, N, γ. The next lemma rovides some sufficient conditions on ϕ for this to occur. Lemma 2.2. Let ϕ : a, +, for some a < be convex, non-increasing, and such that lim ϕt = lim t + t + tϕ t = 0, 8 and let u [DG] E; γ, with range in a, +. There exists a ositive constant γ deending only on the data, such that ϕu [GDG] + E; γ. Likewise let ψ :, a, for some a >, be concave, nonincreasing, and satisfying lim ψt = lim t t tψ t = 0, 9 and let u [DG] + E; γ, with range in, a. There exists a ositive constant γ deending only on the data, such that ψu [GDG] E; γ. Proof. It suffices to rove the first statement for ϕ C 2 over congruent cubes K centered at the origin. The starting oint is the analog of 5, i.e., ϕu = u k ϕ kdk. 0 Since u [DG] E; γ, by 4 the function u is locally bounded below in E, and without loss of generality we may assume u 0. Hence, the reresentation 0

5 SOME POPETIES OF DE GIOGI CLASSES 249 is well defined by virtue of the assumtion 8 on ϕ. From this, by taking the gradient of both sides, then taking the -ower, and finally integrating over gives Dϕu dx = Du k ϕ kdk dx. + The roof now arallels that of Lemma 2.. Secifically, aly sequentially the continuous version of Minkowski s inequality, the definition of the classes [DG] E; γ, the su-bound 4, interchange the order of integration, and use Hölder s inequality. This gives Dϕu,Kρ Du k,kρ ϕ kdk + C C N = = C N N+ C N N+ C u k,k +ρ + 2 u k,k +ρ ϕ kdk ϕ kdk K u k ϕ kdk ϕudx K N ϕu.,k Now if ϕ is convex, non-increasing and satisfying 8, the function ϕ l +, for all l in the range of ϕ, shares the same roerties. Hence, N D ϕu l C dx + ϕu l dx + for all cubes y E and all l. 2.. Some Consequences The su-bound in 4 can be given the following sharer form [7]. Lemma 2.3. Let u [DG] ± E; γ. Then for all σ > 0 there exists a constant C σ deending only uon the data and σ, such that N σ σ su u k ± C σ u k σ ±dx.

6 250 E. DIBENEDETTO AND U. GIANAZZA If u [DG] E; γ is non-negative, then Lemma 2.2 with ϕu = u and a = 0, imlies that u [GDG] + E; γ. Therefore Lemma 2.3, with k = 0, imlies that for all τ > 0, N inf u C τ τ τ u τ dx. 2 Proosition 2.4. Let u be a non-negative function in the De Giorgi classes [DG] E; γ. Then for any air of ositive numbers σ and τ su u N inf u C σ + τ σ τ σc τ u σ dx u τ dx. 3 Inequalities of the form are at the basis of Moser s aroach to the Harnack inequality for non-negative weak solutions to quasilinear ellitic equations with bounded and measurable coefficients [4]. The Harnack inequality will follow from 3 if ln u BMOE. This fact is established by Moser for non-negative weak solutions of ellitic equations. We will establish that for non-negative functions u [DG] E; γ, one has ln u BMOE by using the Harnack inequality established in [7]. 3. De Giorgi Classes, BM OE and Logarithmic Estimates The roof of the following lemma is in [7]. Lemma 3.. Let u [DG] E; γ be non-negative. There exist ositive constants C and σ, deending only uon the data, such that u σ dx C inf K uσ, 4 ρy for any cube y such that K 2ρ y E. Such an inequality, referred to as the weak Harnack inequality, was established by Moser for non-negative suer-solutions of ellitic equations with bounded and measurable coefficients [4]. It is noteworthy that it continues to hold for non-negative functions in [DG] E; γ, with no further reference to equations. Lemma 3.2. Let u [DG] E; γ be non-negative. Then ln u BMO.

7 SOME POPETIES OF DE GIOGI CLASSES 25 Proof. By Lemma 3. u σ dx u σ dx = u σ dx su u σ u σ dx inf C K uσ ρy 5 for any cube y such that K 2ρ y E. Set ln u σ ρ = ln u σ dx, and estimate e ln uσ ln u σ ρ dx e ln uσ ρ + e ln uσ ρ e ln uσ dx e ln uσ dx. The second term on the right-hand side is estimated by Jensen s inequality and 5 and yields e ln uσ ρ e ln uσ dx e ln uσ dx u σ dx u σ dx dx C. uσ The first term is estimated analogously. Hence, there exists a constant C, deending only uon the data, such that e ln uσ ln u σ ρ dx C for any cube y such that K 2ρ y E. Thus ln u BMOE. 3.. Logarithmic Estimates evisited Let u W, loc E be a non-negative weak suer-solution of an ellitic equation in divergence form, and with only bounded and measurable coefficients. Then there exists a constant C, deending only on, N, and the modulus of elliticity of the equation, such that D ln u C dx 6

8 252 E. DIBENEDETTO AND U. GIANAZZA for any air of cubes y E. Such an estimate, established by Moser, ermits one to rove that ln u BMOE, which in turn yields the Harnack inequality. Our aroach for functions in the [DG] E; γ classes is somewhat different. For non-negative functions in such classes we first establish the weak Harnack estimate 4, and then the latter is used to rove Lemma 3.2. It is not known, whether non-negative functions in [DG] E; γ satisfy 6. The next roosition is a artial result in this direction. Proosition 3.3. Let u [DG] E; γ be non-negative and bounded above by some ositive constant M. Then D ln u dx γ ln M dx 7 u for any air of cubes y E. Proof. The arguments being local may assume that y = {0}. By the definition of classes, for all 0 < t < M, Du t dx γ u t dx. K Multily both sides by t and integrate over 0, M. The left-hand side is transformed as M 0 dt M t + Du t dx = Du t dt dx 0 t+ M = Du 0 t + χ [u<t]dt dx M = Du dt dx t+ = = u Du M + Du u D ln u dx M dx Du dx.

9 SOME POPETIES OF DE GIOGI CLASSES 253 The integral on the right-hand side is transformed as M 0 t + u t dx dt K M t u = K u t + dt dx [ = t u ] M M t u dt K t + u u t dx t = M t u M M u dt dx + dx K K u t t M M u dx + ln M K K u dx. Combining the revious estimates gives D ln u dx M + Du dx γ K ln M u dx. γ M u dx K Since u [DG] E; γ, the term in round brackets on the right-hand side is non-ositive and can be discarded. emark 3.4. Alying Lemma 2.2 to ϕu = ln + M/u, gives the weaker estimate D ln u γ dx ln M dx. 8 u 4. Higher Integrability of the Gradient of Functions in the De Giorgi Classes Proosition 4.. Let u [DG] ± E. Then there exist constants C > and σ > 0, deendent only uon the data, such that, for any air of cubes y E, there holds Du +σ dx +σ C N ρ Du dx. 9

10 254 E. DIBENEDETTO AND U. GIANAZZA Proof. Let u be in the classes [DG] E; γ defined in 2. For any air of cubes y E, write down + and for the choice def k = u = udx. Adding the resulting inequalities gives Du γ dx By the Sobolev-Poincaré inequality u u dx C q Du q dx q for a constant C q = C q N, q. Hence, for all such q u u dx., for all q N q Du dx C q γ Du q dx ρ [ ] N N +, for all air of congruent cubes y E. The conclusion follows from this and the local version of Gehring s lemma [9], as aearing in []. emark 4.2. Hence, the higher integrability of the gradient of solutions of ellitic equations with measurable coefficients [5], and more generally of Q-minima [0], continues to hold for function in the De Giorgi classes. If u [DG] ± E; γ, the conclusion is in general false, as one can verify starting from subsuer-harmonic functions. However, essentially the same arguments give the inequality N Du k ± dx C q γ Du q q dx ρ K for all q [ N N+, ], and all k all k udx udx if u [DG] + E; γ, if u [DG] E; γ.

11 SOME POPETIES OF DE GIOGI CLASSES Measure Theoretical Decay Estimates of Functions in De Giorgi Classes For a non-negative function f L loc E one estimates the measure of the set [f > t] relative to a cube y E, as µ [f > t] y t f,kρy. Estimates of the measure of the set [f < t] relative to y are not, in general, a consequence of the mere integrability of f. One of De Giorgi s estimates of [2], is that if u is a non-negative function in [DG] E; γ, then [u < t] y CN,, γ ln t / asymtotically as t 0, 20 rovided [u > t] Kρ y 2. Here σ denotes the Lebesgue measure of a measurable set σ N. The next roosition imroves on this estimate. Proosition 5.. Let u [DG] E; γ be non-negative, and assume that for some t o > 0 and α 0,, there holds [u > to ] y] α. 2 There exist ositive constants C, t, σ = C, t, σn,, γ, t o, α, deending only on the indicated arameters and indeendent of u, such that [u < t] Kρ y C, for t < t ln t σ ln t Proof. In what follows we denote by C a generic ositive constant that can be determined a-riori only in terms of {N,, γ, t o, α} and that it may be different in the same context. The arguments being local to concentric cubes y K 2ρ y E, may assume y = {0} and write 0 =. Let n o be the smallest ositive integer such that 2 no t o, and for n n o set def A n,ρ = [ u < ] 2 n, for n n o. The discrete isoerimetric inequality [3, Chater I, Lemma 2.2], reads l h [u < h] Kρ ρ N+ CN [u > l] Du dx [h<u<l] for any two levels 0 < h < l. Alying it with l = 2 n, h = 2 n+, so that [h < u < l] = A n,ρ A n+,ρ,

12 256 E. DIBENEDETTO AND U. GIANAZZA and taking into account 2, yields 2 n+ A n+,ρ CN α ρn A n,ρ A n+,ρ Du dx. Majorize the right-hand side by the Hölder inequality, then raise both terms to the ower, and majorize the right-hand side by in the definition of the classes [DG] E; γ. These sequential estimates yield 2 n A n+,ρ Cρ C C 2 n D u 2 n dx A n,ρ A n+,ρ u 2 n dx A no,2ρ An,ρ A n+,ρ An,ρ A n+,ρ. This in turn yields the recursive inequalities A n+,ρ CN,, γ, α A no,2ρ A n,ρ A n+,ρ. Let n be a ositive integer to be chosen. Adding them from n o to n gives An,ρ CN,, γ, α n n o Ano,2ρ eturn now to the assumtion 2 and estimate [u > to ] K 2ρ y] K 2ρ [u > to ] y] 2 N Ano,ρ. 23 α 2 N. Therefore, the same arguments leading to 23 can be reeated over the cube K 2ρ and give An,2ρ CN,, γ, α n n o Ano,4ρ Ano,2ρ. 24 While the constant C in 24 differs from the one in 23, we may take them to be equal by taking the largest. The assumtion 2 continue to hold with t o relaced by 2 n. Hence, the revious arguments can be reeated and yield the analogues of 23 24, i.e., A2n,ρ A2n,2ρ CN,, γ, α n n o CN,, γ, α n n o An,2ρ An,4ρ An,ρ An,2ρ

13 SOME POPETIES OF DE GIOGI CLASSES 257 for the same constant C. Combining them gives A2n,ρ Iteration of this rocedure yields A jn,ρ C j 4 jn n n o j C 2 4 2N n n o 2. for all j N. Choose n so large that n n o > 2 n, and then take j = n. By ossibly modifying the various constants, the revious inequality yields A j 2,ρ Cj 4 jn j j for all j N. The constant C being fixed, for each 0 < ε < Aj 2,ρ j jε for all j j. there exists j so large that Fix now t 2 j 2 and let j be the largest integer such that 2 j+2 t 2 j2. For such choices [u < t] A j 2,ρ C. ln t ε 2 ln t 2 The arabolic version of this result has been used in [6]. 6. Boundary Behavior of Functions in the De Giorgi Classes Let h W, loc N C N. The De Giorgi classes [DG] + Ē; γ, h, in the closure of E are the collection of functions u W, loc Ē, such that u h Wo, E, for all cubes centered at some y E, and satisfying Du k + γ dx u k + dx 25 E E for all airs of congruent cubes y, centered at some y E and all levels k su h, k inf h. 26 E E

14 258 E. DIBENEDETTO AND U. GIANAZZA We let further [DG] Ē; γ, h = [DG]+ Ē; γ, h [DG] Ē; γ, h. Functions in [DG] Ē; γ, h are continuous u to oints y E, rovided E satisfies a ositive geometric density at y, i.e., there exist ρ o and η 0,, such that see [2] E c y η y, for all ρ ρ o. For < < N, the -caacity of the comact set E c y is defined by c [E c y] = inf Dψ dx. 27 N ψ W, o N C N E c Kρy [ψ ] For < < N, the relative -caacity of E c y with resect to y is δ y ρ = c [E c y] ρ N, < < N. 28 If = N, and for 0 < ρ <, the N-caacity of the comact set E c y, with resect to the cube K 2ρ y, is defined by c N [E c y] = inf Dψ N dx. 29 ψ Wo,N K 2ρ y CoK 2ρ y E c Kρy [ψ ] K 2ρy The relative caacity δ y ρ can be formally defined by 28, for all < N. For = N, we let δ y ρ c N [E c y], as defined by 29. For a ositive arameter ɛ denote by I,ɛ y, ρ the Wiener integral of E at y E, i.e., I,ɛ y, ρ = ρ [δ y t] ɛ dt t. 30 The celebrated Wiener criterion states that a harmonic function in E is continuous u to y E if and only if the Wiener integral I 2, y, ρ diverges as ρ 0 [6]. It is known that weak solutions of quasilinear equations in divergence form, and with rincial art exhibiting a -growth with resect to Du, when given continuous boundary data h on E, are continuous u to y E if I, y, ρ diverges as ρ 0 [8]. Since such solutions belong to the boundary [DG] Ē; γ, h classes [0], it is natural to ask whether the divergence of the Wiener integral I, y, ρ, is sufficient to insure the boundary continuity for functions u [DG] Ē; γ, h.

15 SOME POPETIES OF DE GIOGI CLASSES 259 The only result we are aware of in this direction is due to Ziemer [7]. It states that a function u [DG] Ē; γ, h is continuous u to y E if ρ ex δ y t dt t as ρ 0. 3 Ziemer s roof follows from a standard De Giorgi iteration technique. It has been recently established that local minima of variational integrals when given continuous boundary data h are continuous u to y E rovided [5] I,ε y, ρ diverges as ρ 0. Here ε is a number that can be determined a- riori only in terms of the growth roerties of the functional. While such minima are in the classes [DG] Ē; γ, h, the result is not known to hold for functions merely in such classes. Also the otimal arameter e = remains elusive. A similar result has been recently obtained with a different aroach in []. The significance of a Wiener condition for Q-minima, is that the structure of E near a boundary oint y E, for u to be continuous u to y, hinges on minimizing a functional, rather than solving an ellitic.d.e. Acknowledgements Emmanuele DiBenedetto was suorted by NSF grant DMS eferences [] J. Björn, Shar exonents and a Wiener tye condition for boundary regularity of quasiminimizers, rerint 205, 3. [2] E. De Giorgi, Sulla differenziabilità e l analiticità delle estremali degli integrali multili regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat , [3] E. DiBenedetto, Degenerate arabolic equations, Universitext, Sringer- Verlag, New York, 993. [4] E. DiBenedetto, eal analysis, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, [5] E. DiBenedetto and U. Gianazza, A Wiener-tye condition for boundary continuity of quasi-minima of variational integrals, Manuscrita Math , no. 3-4, [6] E. DiBenedetto, U. Gianazza, and V. Vesri, Continuity of the saturation in the flow of two immiscible fluids in a orous medium, Indiana Univ. Math. J , no. 6, [7] E. DiBenedetto and N. S. Trudinger, Harnack inequalities for quasiminima of variational integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 984, no. 4,

16 260 E. DIBENEDETTO AND U. GIANAZZA [8]. Gariey and W. P. Ziemer, A regularity condition at the boundary for solutions of quasilinear ellitic equations, Arch. ational Mech. Anal , no., [9] F. W. Gehring, The L -integrability of the artial derivatives of a quasiconformal maing, Acta Math , [0] M. Giaquinta and E. Giusti, Quasiminima, Ann. Inst. H. Poincaré Anal. Non Linéaire 984, no. 2, [] M. Giaquinta and G. Modica, egularity results for some classes of higher order nonlinear ellitic systems, J. eine Angew. Math. 3/32 979, [2] O. A. Ladyzhenskaya and N. N. Ural tseva, Linear and quasilinear ellitic equations, Translated from the ussian by Scrita Technica, Inc. Translation editor: Leon Ehrenreis, Academic Press, New York-London, 968. [3] V. G. Maz ja, The continuity at a boundary oint of the solutions of quasi-linear ellitic equations, Vestnik Leningrad. Univ , no. 3, ussian, with English summary, Vestnik Leningrad Univ. Math , English translation. [4] J. Moser, A new roof of De Giorgi s theorem concerning the regularity roblem for ellitic differential equations, Comm. Pure Al. Math , [5] E. W. Stredulinsky, Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J , no. 3, [6] N. Wiener, Une condition necéssaire et suffisante de ossibilité our le roblème de Dirichlet, C.. Acad. Sci. Paris , [7] W. P. Ziemer, Boundary regularity for quasiminima, Arch. ational Mech. Anal , no. 4, Authors addresses: Emmanuele DiBenedetto Deartment of Mathematics Vanderbilt University 326 Stevenson Center Nashville TN 37240, USA em.diben@vanderbilt.edu Ugo Gianazza Diartimento di Matematica F. Casorati Università di Pavia via Ferrata, 2700 Pavia, Italy gianazza@imati.cnr.it eceived Aril 25, 206 Acceted May 9, 206