STRICT CONVEXITY OF LEVEL SETS OF SOLUTIONS OF SOME NONLINEAR ELLIPTIC EQUATIONS

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1 STRICT CONVEXITY OF LEVEL SETS OF SOLUTIONS OF SOME NONLINEAR ELLIPTIC EQUATIONS FRANCESCA GLADIALI AND MASSIMO GROSSI Abstract. I this paper we study the covexity of the level sets of solutios of the problem u = fu i Ω 0. u > 0 i Ω u = 0 o Ω, where f is a suitable fuctio with subcritical or critical growth. Uder some assumptios o the Gauss curvature of Ω we prove that the level sets of the solutio of 0. are strictly covex.. Itroductio I this paper we study the shape of the level sets of the problem u = fu i Ω. u > 0 i Ω u = 0 o Ω where Ω is a strictly covex domai of R N, N ad f is a smooth oliearity. I particular we are iterested to establish the covexity of the level set. This problem has bee ivestigate by may author sice 950. A first importat results cocers the liear case ft = λ t where λ is the first eigevalue of the laplacia i Ω ad u = ϕ is the first eigefuctio. I [], Brascamp ad Lieb showed that ϕ is log cocave if Ω is covex, ad so it has covex level sets. The proof of this result was rather ivolved ad uses the fact that the parabolic operator uder homogeeous Dirichlet boudary coditio preserves the t log cocavity of the positive iitial data. Lios i [5] exteded this method to other oliearities ad he showed that if f is of the type ft = λt µt p with p >, λ, µ > 0, ad λ > λ, the u is log cocave. The proof eeds that each solutio of. ca be obtaied as the limit as t of the solutio of the parabolic associated operator 99 Mathematics Subject Classificatio. 35B05,35B50. Supported by MIUR, atioal project Variatioal Methods ad Noliear Differetial Equatios.

2 GLADIALI AND GROSSI with some log cocave iitial coditio. To study the covexity of a solutio of some elliptic ad parabolic equatios Korevaar i [] itroduced a cocavity fuctio Cu, x, y, µ = uµx + µy µux µuy o Ω Ω where 0 µ. The fuctio C measures how much the fuctio u fails to be cocave. Of course u is cocave if ad oly if C 0 o Ω Ω. Korevaar used a maximum priciple ad some boudary poit lemmas to show that if u is a solutio to the geeralied eigefuctio equatio the v = log u is cocave. Subsequetly Caffarelli ad Spruck i [] proved the log cocavity of solutios of a eigevalue problem with weight, as well as further applicatio, simplifyig the Brascamp ad Lieb result, ad those of Korevaar also. Kawohl i [9], [0], exploited the cocavity fuctio ad showed that if u is a solutio of. with f 0, f strictly decreasig ad f harmoic-cocave i.e. is covex the the solutio u is cocave. f I [0] he used this result to show that if u is a solutio of., with ft = t p, 0 < p <, the the fuctio v = u p is cocave if Ω is covex ad sufficietly smooth. I particular the level sets of u are covex. But if the expoet p is greater tha this method caot be applied. Other importat results are due to Gabriel who first show the covexity of the level sets for the Gree s fuctio of i Ω. I [4] ad [5] he showed that the level sets are strictly covex if N = 3, ad the result exted to all dimesios. His method was geeralied by Lewis i [] where he applied it to the p-capacitary fuctios ad by may authors see for example[], [3], [], to show the covexity of level sets i rig shaped domais. For other results i this topic see [9] ad refereces therei. We poit out that oe of this works if ft = t p with p >. To our kowledge the oly result o the strict covexity of the level set for the oliearity ft = t p is due to C. S. Li [4]. I this paper the author proves the result for strictly covex domai of R ad he also assume that u is the least eergy solutio for.. I this paper we give two examples of solutios which have covex level sets for some superliear oliearities where the dimesio of the space is greater tha. We poit out that our method ca be used to hadle other sigularly perturbed elliptic problems.

3 CONVEXITY 3 The first example deals with the followig perturbed critical problem. u = NN u pε u > 0 u = 0 i Ω i Ω o Ω where Ω R N is a bouded smooth domai, N 3, ad p ε = N+ N ε for ε > 0. I [6] Grossi ad Molle have show that for ε small eough the level sets of a solutio u are strictly starshaped. I this paper we exted this result provig the strict covexity of the level set provided that Ω has positive curvature More precisely we have the followig result: THEOREM.. Let u ε be a solutio of. such that.3 lim u Ω ε ε 0 = S Ω u ε where S is the best costat i the Sobolev embeddig ad = N N. If Ω has strictly positive Gauss curvature at ay poit p Ω, the there exists ε such that, for every 0 ε ε the level sets of u ε have strictly positive Gauss curvature at ay poit which is ot the maximum oe. I particular the level sets are strictly covex. The secod example is the followig subcritical perturbed problem.4 ε u + u = u p u > 0 u = 0 i Ω i Ω o Ω where Ω R N is a covex domai, N, ε > 0 ad < p < N+ N if N > 3. Agai we fid that the level sets are covex THEOREM.. Let u ε a family of sigle peak solutio of.4. If Ω is covex the there exists ε > 0 such that for each 0 < ε < ε, u ε has covex level sets. The proof requires the uderstadig of problem. ad.4 as ε 0. I the case of problem.4 oly the behavior ear the maximum poit is eeded to hadle the covexity result. I case of problem. the exact behavior of the solutio iside Ω is ivestigate ad our proof relies o the deep works [3] ad [8]. We thik that this techic ca be applied to other perturbed problems whe the asymptotic behavior is kow.

4 4 GLADIALI AND GROSSI The paper is orgaied as follows. I Sectio we state some kow result ad give some otatio. The proof of Theorem. is give i Sectio 3. I Sectio 4 we discuss the subcritical case.. Some prelimiaries I this sectio we state some kow facts about problem. that are ecessary i the proof of Theorem.. We also recall the otio of Gauss curvature ad ormal curvature ad their relatioship with the covexity of a surface. Let us itroduce some otatios. Let S f t the level set of a fuctio f, so S f t = {x Ω : fx = t}. We suppose f at least C ad f 0. The the level set is the surface S f t = f t ad the ormal vector field to S f t i the poit x is give by x = fx. We idicate by T fx x the taget space to S f t attached at the poit x S f t. The τ T x if ad oly if τ x = 0. If τ T x the value of the Secod fudametal form of S f t at x o τ is give by S x τ = fx N i,j= f xτ i τ j = x i x j fx τ t H f xτ where H f x deotes the Hessia matrix of f at the poit x. If τ T x is a uit vector τ =, the ormal curvature of S f t at x i the directio τ is k f,x τ = S x τ = fx τ t H f xτ. The Gauss-Kroecker curvature of S f t at x, deoted by K f x is the product of the pricipal curvatures of S f t at x, which are statioary values of the ormal curvature o the taget space T x. While the ormal curvature depeds o the taget vector τ the Gauss curvature is itrisic. We also metio the followig well kow facts THEOREM.. Let S be a compact coected orieted N - surface i R N whose Gauss-Kroecker curvature is owhere ero. The i The Gauss map N : S S N is oe to oe ad oto, ii S is strictly covex. If f is a radial fuctio, the its level sets are spheres ad k f,x τ = has the same value for all taget directio τ at ay poit of the x sphere x 0. The the Gauss curvature K f x = x 0 x N

5 CONVEXITY 5 for all radial fuctio. For refereces about curvatures ad Secod fudametal form look at [8]. Now we state some kow result about the covexity of level sets i covex rig, see [] THEOREM.. Let u be the uique solutio of u = fu i Ω \ Ω u = 0 o Ω u = o Ω with fu cotiuous ad oicreasig i u, f0 = 0, Ω ad Ω bouded covex sets i R N. The the level surface of u are covex C +α hypersurfaces. Here we quote some results o the covexity of the level set for the Gree s fuctio. I [] Lewis geeralies the work of Gabriel i [4], [5] about the covexity of the level sets of G. The result is THEOREM.3. Give a covex rig Ω \ Ω, let u be a harmoic fuctio such that u = 0 o Ω ad u = o Ω. The i the set {x : ux > t} is covex for 0 t <, ii if u 0 ad x Ω \ Ω, the all the ormal curvature at x of the level surface {y : uy = ux} are positive. Theorem.3 proves the covexity of the level sets S G t of the Gree s fuctio i a aular covex domai, ad shows that all the ormal curvature of S G t are positive. This imply also that the Gauss curvature of S G t is positive. COROLLARY.. Let Ω be a covex bouded domai i R N, N 3 ad let x 0 Ω. The the level sets of Gree s fuctio Gx, x 0 for the Laplacia i Ω with Dirichlet boudary coditio have strictly positive Gauss curvature at ay poit. Proof. The result is proved by Gabriel i [4] for N = 3, ad geeralie to all dimesio. The idea is that i a eighborhood of x 0 the Gree s fuctio behave essetially as ω N x x 0, where ω N N is the area of the uit sphere i R N, ad so the ormal curvature of its level sets is essetially the ormal curvature of a sphere which is strictly positive, ad icreasig as x x 0. Hece i a small eighborhood of x 0 the Gauss curvature of S G t is strictly positive ad the level sets of G are strictly covex. Now we ca apply the Theorem of Lewis at G i Ω\Ω where Ω is a strictly covex level set ear x 0 to have the positiveess of the Gauss curvature of S G t i all Ω \ {x 0 }. Let u ε be a solutio of problem. that satisfies.3. Let x ε Ω a poit such that ux ε = u ε. The asymptotic behavior of the solutio u ε was studied by Ha i [8] ad by Rey i [7]. They proved the followig covergece result

6 6 GLADIALI AND GROSSI THEOREM.4. Let Ω be a smooth bouded domai i R N, N 3. Deotig by x 0 = lim ε 0 x ε, we have x 0 Ω ad [ ] N NN 4 N Gx, x0 u ε u ε x S g x Ω \ {x 0 }, where Gx, x 0 is the Gree s fuctio for the laplacia i Ω with ero Dirichlet boudary coditio ad g = gx 0, x 0 where gx, y is the regular part of the Gree s fuctio. Moreover there exists δ > 0 idepedet of such that lim ε 0 u ε =. Grossi ad Molle i [6] have show the followig THEOREM.5. If Ω is covex the there exists ε such that, for every 0 < ε < ε it occurs:. x x ε u ε x < 0 x Ω \ {x ε }. I particular, the maximum poit x ε is the oly critical poit ad the superlevels are strictly starshaped. 3. The mai result Proof of Theorem.. We argue by cotradictio. Let us suppose that there exists a sequece ε > 0, ε 0 ad poits Ω \ {x } such that, if u is a solutio of problem. correspodig to the value p = p ε, that satisfies.3 the 3. K u 0 where K u is the Gauss curvature of the surface S u u = {x Ω suchthat u x = u } at the poit. We use u u ε ad x x ε deotes the uique critical poit of u. Notice that from. the ormal at the surface S u u is always defied by = u. Up to a subsequece u Ω. Step Firstly we suppose x 0, so Ω \ B δ x 0 for some δ > 0. I this [ ] N domai we have from Theorem.4 that g x NN 4 N Gx,x 0 S g i C Ω \ B δ x 0, where g = u u. From 3. we have K g = K u 0 ad passig to the limit lim K g = K G 0. If Ω is a iterior poit this is impossible from Theorem.3 ad Corollary.. If otherwise Ω, this is impossible sice we suppose the Gauss curvature of Ω is strictly positive.

7 CONVEXITY 7 Step Now x 0. We first cosider the case where B = Bx, for some R > 0. Cosider the fuctio ũ x = u u x + R u x u By stadard blow-up results, see [8] for example, ũ coverges to Ux = i C B0, R, for all R R. If B N + x the = u x B0, R ad up to a subsequece B0, R. Suppose 0. From 3. we have Kũ = u N K u 0 ad passig to the limit K U 0. But this is ot possible sice K U = > 0. If = 0, U0 = 0 ad we caot speak of N Gauss curvature of U i 0 sice S U = {0}. But 0 ad Kũ 0 implies there exists at least a taget directio τ T such that kũ, τ = τ t ũ Hũ τ 0. Now τ τ with τ =, ad τ t Hũ τ τ t H U 0τ 0. But 0 is a o degeerate maximum poit for the fuctio Ux ad the τ t H U 0τ > 0 ad so a cotradictio follows. Step 3 Fially we cosider the case where x 0 but / B. From equatio. we have 3. u x = NN Gx, yu p ydy where Gx, y = N ω gx, y ad gx, y is the regular part x y N of the Gree s fuctio of the laplacia i Ω. Here we use some ideas by [3]. Let r = x ; we have r. Let v x = r 3. Lettig y = x + v x = NN r v x = NN u r Ω R u, ad r 0 as u x + r x for x Ω = Ω x r. The by we get u N Ω Ω G Gx + r x, yu p ydy. x + r x, x + u p x + u u d.,

8 8 GLADIALI AND GROSSI where Ω = u Ω x. Multiplyig by v = r ad recallig the defiitio of ũ we get 3.3 v v x = N ω 4N NN r 4 N ε r u N ε I [8] it is proved that 4N 4 N ε u N ε Ω g 0 ũ CU = C Ω r N x r u x + r x, x + N + u From this estimate we derive Ω ũ p d R N Sice gx, is a bouded fuctio we get g x + r x, x + ũ p Ω u d gx 0, x 0 Now we cosider the first itegral i 3.3 i.e. N ε 4 N ε N r ω u N ε Ω x r u By the defiitio of r we have r u each R > 0, ad so r u ad x 0 the x r u N ũp R > u N ũp i R N. N+ + R N d. u u d ũ p d. d = ω N. N+. + = R for as. If is large eough < C ad we ca pass to the limit as x. Moreover we kow from Theorem.4 that lim u ε =. The also lim r ε =. We ca pass to the limit i 3.3 gettig 3.4 v v x x N i CΩ \ {0} It is easily see that the covergece i 3.4 is C K, for ay compact set K Ω \ {0}. Now let = x r. The, where =. From 3. ad the C covergece of v v to V x = we get x N K v v = r N K u 0

9 CONVEXITY 9 ad passig to the limit K V 0 ad this is ot possible sice K V = N > 0. This fiishes the proof of Theorem.. 4. The subcritical case We start this sectio by recallig the defiitio of sigle-peak solutio to the problem.4. Deote by W the uique solutio of: W + W = W p i R N 4. W > 0 i R N lim y W y = 0, W 0 = W. ad defie the eergy of the solutio W as E = W + W W p+. R N R p + N R N Let us recall that W is a radial fuctio ad strictly radially decreasig DEFINITION 4.. Let u ε C Ω C Ω a family of solutios of.4, for small ε, ad let x ε Ω be such that u ε = u ε x ε. The u ε is said a family of sigle-peak solutios ear a poit x 0 Ω if i x ε [ x 0 as ε 0, ii ε N Ω ε u ε + u ε p+ Ω up+ ε ] E as ε 0. The followig lemma is well kow see [6] for example LEMMA 4.. If u ε is a family of sigle-peak solutios of.4, the, for ε sufficietly small, x ε is the oly local maximum poit ad u ε x 0, as ε 0, for ay x Ω \ x 0. There is a rich literature o the existece of sigle peak solutios to the problem.4 see for example [6], [7] ad the refereces therei. We are i positio to give the proof of Theorem.. Proof of Theorem.. Let v ε y = u ε x ε + εy, where x ε are the maximum poits of u ε. The v ε solves v ε + v ε = vε p i Ω ε v ε > 0 i Ω ε v ε = 0 o Ω ε where Ω ε = Ω x ε /ε ad Ω ε R N as ε 0. It is easy to see that v ε W i C loc RN, ad W satisfies 4.. Furthermore 0 is a odegeerate maximum poit for W. We wat to show that i each ball B ε = Bx ε, εr for R R the fuctio u ε has strictly covex level sets. This follows from the covergece of v ε W as i Theorem.. Let us suppose by cotradictio that there exists a sequece ε 0, ad poits B B ε, x, with Gauss curvature less or equal tha ero. The the same is true for the fuctio v, so that

10 0 GLADIALI AND GROSSI K v = ε N K u 0 where = x ε ad B0, R. Up to a subsequece B0, R, ad if 0 the lim K v = K W 0. But this is ot possible sice W is radial ad its Gauss curvature is strictly positive if 0. Now cosider the case = 0; the coditio K v 0 implies that there exists at least a uit vector τ T ad k v, τ = τh t v τ 0. Up to a subsequece τ τ, τ = ad τh t v τ τ t H W 0τ 0 ad this is ot possible sice 0 is a o degeerate maximum poit for W, ad its Hessia matrix is egative defied i 0. Reasoig as i [6], δ > 0, δ R we ca fid a ball B ε Ω such that u ε x δ i Ω \ B ε. Now cosider the level set S uε δ; it is covex sice S uε δ B ε. The fuctio u ε satisfies u ε = up ε u ε i Ω \ S ε uε δ u ε = 0 o Ω u ε = δ o S uε δ p ad Ω \ S uε δ is a covex rig. Takig δ < p is decreasig i 0 s δ ad so Theorem. applies. the fs = sp s ε Refereces [] H.J. Brascamp, E.H. Lieb, O extesios of the Bru-Mikowski ad Prékoph-Leidler theorems, icludig iequalities for log cocave fuctios ad with a applicatio to the diffusio equatio, J. Fuct. Aal., Vol, 976, pp [] L.A. Caffarelli, J. Spruck, Covexity properties of solutios to some classical variatioal problems, Comm. PDE., Vol 7, 98, pp [3] J.I. Dia, B. Kawohl, O covexity ad starshapedess of level sets for some oliear elliptic ad parabolic problems o covex rigs, J. Math. Aal. Apll.,Vol 77, 993, pp [4] R.M. Gabriel, A exteded priciple of the maximum for harmoic fuctios i 3-dimesios, J. Lodo Math. Soc., Vol 30, 955, pp [5] R.M. Gabriel, A result cocerig covex level surfaces of 3-dimesioal harmoic fuctios, J. Lodo Math. Soc., Vol 3, 957, pp [6] M. Grossi, R. Molle, O the shape of the solutios of some semiliear elliptic problems, Comm. Cot. Math. to appear [7] M. Grossi, A. Pistoia, O the effect of critical poits of distace fuctio i superliear elliptic problems, Adv. Diff. Eqs Vol 5, pp [8] Z.C. Ha, Asymptotic approach to sigular solutios for oliear elliptic equatios ivolvig critical Sobolev expoet, A. Ist. H. Poicaré, Aal. No Liéare, Vol 8, 99, pp [9] B. Kawohl, Rearragemets ad covexity of level sets i PDE, Lectures Notes i Math., Vol 50, Spriger-Verlag, Heidelber, 985. [0] B. Kawohl, A remark o N. Korevaar s cocavity maximum priciple ad o the asymptotic uiqueess of solutios to the plasma problem, Math. Meth. i the Appl. Sci., Vol 8, 985, pp

11 CONVEXITY [] N. Korevaar, Covex solutios to oliear elliptic ad parabolic boudary value problems, Idiaa U. Math. J. Vol 3, 983, pp [] J.L. Lewis, Capacitary fuctio i covex rigs, Arch. Rat. Mech. Aal., Vol 66, 977, pp [3] Y.Y. Li, Prescribig scalar curvature o S ad related problems, part I J. Diff. Eq., Vol 0, 995, pp [4] C.S. Li, Uiqueess of least eergy solutios to a semiliear elliptic equatio i R. Mauscripta Math. Vol , 3-9. [5] P.L. Lios, Two geometrical properties of solutios of semiliear problems, Appl. Aal., Vol, 98, pp [6] W.M. Ni ad J. Wei, O the locatio ad profile of spike-layer solutio to sigular perturbed semiliear Dirichlet problems, Comm. Pure Appl. Math., Vol 48, 995, pp [7] O. Rey, Proof of two cojectures of H. Breis ad L.A. Peletier, Ma. Math., Vol 65, 989, pp [8] J.A. Thorpe Elemetary topics i differetial geometry, Spriger-Verlag, 978. Dipartimeto di Matematica, Uiversità di Roma La Sapiea - P.le A. Moro Roma - Italy. address: gladiali@mat.uiroma.it Dipartimeto di Matematica, Uiversità di Roma La Sapiea - P.le A. Moro Roma - Italy. address: grossi@mat.uiroma.it