SYMMETRY AND REGULARITY OF AN OPTIMIZATION PROBLEM RELATED TO A NONLINEAR BVP

Transcription

1 lectronic ournal of Differential quations, Vol , No. 108, pp ISSN: URL: or ftp ejde.math.txstate.edu SYMMTRY AND RGULARITY OF AN OPTIMIZATION PROBLM RLATD TO A NONLINAR BVP CLAUDIA ANDDA, FABRIZIO CUCCU Abstract. We consider the functional Z ` q + 1 f 2 2 u f u f q f dx, where u f is the unique nontrivial weak solution of the boundary-value problem u = fu q in, u = 0, where R n is a bounded smooth domain. We prove a result of Steiner symmetry preservation and, if n = 2, we show the regularity of the level sets of minimizers. 1. Introduction Let be a bounded domain R n with smooth boundary. We consider the Dirichlet problem u = fu q in, u 1.1 = 0, where 0 q < min{1, 4/n} and f is a nonnegative bounded function non identically zero. We consider nontrivial solutions of 1.1 in H0 1. The equation 1.1 is the uler-lagrange equation of the integral functional v q Dv2 vv q f dx, v H 1 0. By using a standard compactness argument, it can be proved that there exists a nontrivial minimizer of the above functional. This minimizer is a nontrivial solution of 1.1. From the maximum principle, every nontrivial solution of 1.1 is positive. Then, by [5, Theorem 3.2] the uniqueness of problem 1.1 follows. To underscore the dependence on f of the solution of 1.1, we denote it by u f. Moreover, u f W 2,2 C 1,α for all α, 0 < α < 1 see [8, 10]. Let f 0 be a fixed bounded nonnegative function. We study the problem inf v H0 1, f Ff0 q Dv2 vv q f dx, Mathematics Subject Classification. 3520, 3560, 40K20. Key words and phrases. Laplacian; optimization problem; rearrangements; Steiner symmetry; regularity. c 2013 Texas State University - San Marcos. Submitted anuary 7, Published April 29,

2 2 C. ANDDA, F. CUCCU D-2013/108 where, denoting by A the Lebesgue measure of a set A, Ff 0 = {f L : {f c} = {f 0 c} c R}; 1.3 here Ff 0 is called class of rearrangements of f 0 see [9]. Problems of this kind are not new; see for example [1, 2, 3, 5]. From the results in [5] it follows that 1.2 has a minimum and a representation formula. Let f = inf v H 1 0 q Dv2 vv q f dx. 1.4 Renaming q the constant q and putting p = 2 and q = q + 1 in [5] we have where f = q 2 2 If, q 2 If = sup H0 1 2 q q fvq Dv 2 dx is defined in the same paper. By [5, Theorem 2.2] it follows that there exist minimizers of f and that, if f is a minimizer, there exists an increasing function φ such that f = φu f. 1.5 We denote by supp f the support of f, and we call a level set of f the set {x : fx > c}, for some constant c. In Section 2, we consider a Steiner symmetric domain and f 0 bounded and nonnegative, such that supp f 0 <. Under these assumptions, we prove that the level sets of the minimizer f are Steiner symmetric with respect to the same hyperplane of. As a consequence, we have exactly one optimizer when is a ball. Chanillo, Kenig and To [4] studied the regularity of the minimizers to the problem λα, A = inf Du 2 dx + α u 2 dx, u H0 1, u 2,D=A where R 2 is a bounded domain, 0 < A < and α > 0. In particular they prove that, if D is a minimizer, then D is analytic. In Section 3, following the ideas in [4], we give our main result. We restrict our attention to R 2. Let b 1,..., b m > 0 and 0 < a 1 < < a m <, m 2, be fixed. Consider f 0 = b 1 χ G1 + + b m χ Gm, where G i = a i for all i, G i G i+1, i = 1,..., m 1. We call η the minimum in 1.2; i.e., η = q u f u f q f dx, where f and u f are, respectively, the minimizing function and the corresponding solution of 1.1. In this case 1.5 becomes m f = b i χ Di, where i=1 D 1 = {u f > c 1 }, D 2 = {u f > c 2 },..., D m = {u f > c m }, D

3 D-2013/108 SYMMTRY AND RGULARITY 3 for suitable constants c 1 > c 2 > > c m > 0. We show regularity of D i for each i proving that > 0 in D i. Following the method used in [4], we consider q + 1 s, t = 2 + sdv2 u f + svu f + sv q f t dx η, 1.6 where v H 1 0, f t is a family of functions such that f t Ff 0 with f 0 = f, and s R. We have s, t 0, 0 = 0 s, t. Therefore, s, t = 0, 0 is a minimum point; it follows that 2 s 0, s t 0, t s 0, 0 2 t 0, 0 2 xpanding 1.7 in detail and using some lemmas from [4] we prove that the boundaries of level sets of f are regular. Theorem 1.1. Let R 2, f 0 = m i=1 b iχ Gi with m 2, and f = m i=1 b iχ Di a minimizer of 1.2. Then > 0 on D i, i = 1,..., m. 2. Symmetry In this section we consider Steiner symmetric domains. We prove that, under suitable conditions on f 0 in 1.3, minimizers inherit Steiner symmetry. Definition 2.1. Let P R n be a hyperplane. We say that a set A R n is Steiner symmetric relative to the hyperplane P if for every straight line L perpendicular to P, the set A P is either empty or a symmetric segment with respect to P. To prove the symmetry, we need [6, Theorem 3.6 and Corollary 3.9], that, for more convenience for the reader, we state here. These results are related to the classical paper [7]. Theorem 2.2. Let R n be bounded, connected and Steiner symmetric relative to the hyperplane P. Assume that u : R has the following properties: u C C 1, u > 0 in, u = 0; for all φ C0, Du Dφ dx = φf u dx, where F has a decomposition F = F 1 + F 2 such that F 1 : [0, R is locally Lipschitz continuous, while F 2 : [0, R is non-decreasing and identically 0 on [0, ɛ] for some ɛ > 0. Then u is symmetric with respect to P and u v x < 0, where v is a unit vector orthogonal to P and x belongs to the part of that lies in the halfspace with origin in P in which v points. Theorem 2.3. Let be Steiner symmetric and f 0 a bounded nonnegative function. If supp f 0 < and f Ff 0 is a minimizer of 1.2, then the level sets of f are Steiner symmetric with respect to the same hyperplane of.

4 4 C. ANDDA, F. CUCCU D-2013/108 Proof. Let u = u f be the solution of 1.2. Then u C 0 C 1 and satisfies Du Dψ dx = ψfu q dx ψ C0. Since u > 0 and since from 1.5 f = φu with φ increasing function, it follows that φu 0 on {x : ux < d} for some positive constant d. Then we have u q f = F 1 u + F 2 u with F 1 u 0 and F 2 u = φuu q. From Theorem 2.1 and f = φu we have the assertion. Remark 2.4. By this theorem, if is an open ball and supp f 0 <, then f is radially symmetric and decreasing. 3. Regularity of the free boundaries In this section we prove the following result. Theorem 3.1. Let R 2, f 0 = m i=1 b iχ Gi with m 2, and f = m i=1 b iχ Di a minimizer of 1.2. Then > 0 on D i, i = 1,..., m. We use the notation introduced in Section 1. Without loss of generality we can assume m = 3; the general case easily follows. Let f = b 1 χ D1 + b 2 χ D2 + b 3 χ D3. We will prove > 0 in D 2 ; we omit the proof for D 1 and D 3 because it is similar. We define the family f t by replacing only the set D 2 by a family of domains D 2 t. First of all, we explain how to define the family D 2 t. In the sequel we use the notation introduced in [4], reorganized according to our needs. We call a curve : [a, b] R, < a < b <, regular if: i it is simple, that is: if a x < y b and x a or y b, then x y; ii C2 a,b is finite; iii is uniformly bounded away from zero. If, in addiction, a = b, we say that the curve is closed and regular. If the domain of is a, b we say that is regular respectively, closed and regular if the continuous extension of to [a, b] is regular respectively, closed and regular. Now, we introduce the notation F := D 2 ; F := F { > 0}. Let = p k=1 k be a finite union of open bounded intervals k R, = 1, 2 : F a simple curve which is regular on each interval k and F. We suppose that dist k, h > 0 for 1 h k p. Assume also that θ on. For each ξ, we denote by Nξ = N 1 ξ, N 2 ξ the outward unit normal with respect to D 2 at ξ. We also define the tangent vector to N ξ = N 2 ξ, N 1 ξ, and N the first derivative of N. Reversing the direction of if necessary, we will assume, without loss of generality, that and N have the same direction; i.e.,, N =. We observe that, because is C 2 and simple on k, for each k there exists β k > 0 such that the function is injective. φ k : k [ β k, β k ] R 2, ξ, β x 1, x 2 = φ k ξ, β = ξ + βnξ

5 D-2013/108 SYMMTRY AND RGULARITY 5 Because dist k, h > 0 for all h k, we can find a number β 0 > 0 and we can paste together the functions φ k to obtain a function φ injective on [ β 0, β 0 ]. Choose β 0 such that dist φ [ β 0, β 0 ], D 1 > 0 and dist φ [ β 0, β 0 ], D 3 > 0. Now, we define K = D 2 \ φ β 0, 0] ; for t t 0, t 0 we define D 2 t = K {φξ, β : ξ, β < gξ, t}, 3.1 where g : t 0, t 0 R, t 0 > 0, is a function such that and gξ, t, g t ξ, t, g tt ξ, t C t t 0, t gξ, 0 0 ξ. 3.3 We observe that D 2 0 = D 2. Next we compute the measure of D 2 t. Put At = D 2 t and A = D 2 0 = D 2 ; we have gξ,t At = D 2 + ξ, βdβdξ, where ξ, β = x 1, x 2 ξ, β βn 1 N 1 = 2 + βn 2 N 2 =, N β N, N = + β N, N. We show that + β N, N 0. Indeed, from the fact that C 2 <, we have N, N L <. Substituting t 0 by a smaller positive number if necessary, we can assume that and g L t 0,t 0 < β 0 N, N L g L t 0,t 0 < θ. Note that the first of these assumptions guarantees that D 2 t has positive distance from D 1 and D 3. We have β N, N g L t 0,t 0 N, N L θ for all ξ and β g L t 0,t 0. Thus, ξ, β = + β N, N. Substituting into the formula for At we have gξ,t At = A + + β N, N dβdξ 0 = A + gξ, t gξ, t2 N, N dξ. To obtain D 2 t = D 2 for all t t 0, t 0, we find the further constraint on g: gξ, t N, gξ, t N dξ = 0 t t 0, t

6 6 C. ANDDA, F. CUCCU D-2013/108 Moreover, we calculate the derivatives of At, that we will use later. A t = gt ξ, t ξ + gξ, tg t ξ, t N, N dξ = 0; A t = gtt ξ, t ξ + gξ, tg tt ξ, t + gt 2 ξ, t N, N dξ = Once we have defined the family D 2 t, we can go back to the functional 1.6. The following lemma describes 1.7 with f t = b 1 χ D1 + b 2 χ D2t + b 3 χ D3. We find an inequality corresponding to [4, 2.3 of Lemma 2.1]. Lemma 3.2. Let f t = b 1 χ D1 + b 2 χ D2t + b 3 χ D3, where the variation of domain D 2 t is described by 3.1 and g : t 0, t 0 R, t 0 > 0, satisfies 3.2, 3.3 and 3.4. Then, for all v H0 1, the conditions 1.7 becomes Dv 2 qu f v 2 u f q 2 f dx gt 2 1, 0 dσ b 2 c q 2 g t 1, 0 vdσ Proof. We calculate the second derivative of the functional 1.6, with respect to s. We have = q + 1 Duf + sdv, Dv vu s f + sv q f t dx and 2 0, 0 = q + 1 s2 Dv 2 qu f v 2 u f q 2 f dx. 3.7 Before calculating the second derivative of with respect to t, we rewrite 1.6 in the form q + 1 s, t = Du 2 f + sdv 2 dx b 1 u f + svu f + sv D q dx 1 b 2 u f + svu f + sv q dx b 3 u f + svu f + sv D q dx η. 3 D 2t We observe that, if F : R 2 R is a continuous function, then gξ,t F F = F φξ, gξ, β ξ, βdβdξ; D 2t D 2 0 whence, from the Fundamental Theorem of Calculus, F = g t ξ, tf φξ, gξ, t ξ, gξ, tdξ. t D 2t Using the above relation with F = u + sv u + sv q, we have t = b 2 g t ξ, t u f + sv uf + sv q ξ, gξ, t dξ, where, for simplicity of notation, we set u f φξ, gξ, t = uf and v φξ, gξ, t = v. Moreover 2 t 2 = b 2 u f + sv q{ [ gtt ξ, tu f + sv + q + 1gt 2 ξ, t Du f + sdv, N ] } ξ, gξ, t + gt 2 ξ, tu f + sv N, N dξ,

7 D-2013/108 SYMMTRY AND RGULARITY 7 where we have used that t u f φξ, gξ, t = Duf φξ, gξ, t, N gt ξ, t, t ξ, gξ, t = g t N, N. We note that, when t = 0, u f φξ, gξ, t = uf ξ = c 2, Du f φξ, gξ, 0 = Duf ξ Nξ and ξ, gξ, 0 = ξ, 0 = ξ. valuating the above expression in 0, 0, we find 2 [ ] t 2 0, 0 = b 2c q+1 2 g tt ξ, 0 ξ + gt 2 ξ, 0 N, N dξ + b 2 c q 2 q + 1 gt 2 ξ, 0 ξ ξdξ. By using 3.5 with t = 0 we find 2 t 2 0, 0 = b 2c q 2 q + 1 = b 2 c q 2 q + 1 We also have that is, Using 1.7 in the form g 2 t ξ, 0 ξ ξdξ g 2 t 1, 0 dσ. 2 s t = b 2q + 1 g t ξ, tvu f + sv q ξ, gξ, t dξ; 2 s t 0, 0 = b 2c q 2 q + 1 g t ξ, 0 v ξ ξ dξ = b 2 c q 2 q + 1 g t 1, 0 v dσ. 2 s 2 0, t 2 0, 0 2, s t 0, 0 and using 3.7, 3.8 and 3.9 in this inequality, we obtain Note that in inequality 3.6 only g 1, 0 appears. Moreover, g 1, 0 has null integral on. Indeed, differentiating 3.4 with respect to t and putting t = 0, we obtain gξ, 0 dξ = 0. Now a natural question arises: does inequality 3.6 hold for any function h with null integral on? The answer is contained in the following result. Lemma 3.3. Let and be the same as described. Let h : R bounded, continuous and such that h dσ = 0. Then, for all v H1 0 and for all a R we have 2. Dv 2 qu f v 2 u f q 2 f dx h 2 dσ b 2 c q 2 hv a dσ 3.10

8 8 C. ANDDA, F. CUCCU D-2013/108 The proof of the above lemma is similar to that of [4, Lemma 2.2]; we omit it. The following lemma is an analogue to [4, Lemma 3.1]. Lemma 3.4. Let P be a point on F = {u f > c 2 }. Suppose that for all k Z + there exist a positive number r k, a bounded open interval k and a regular curve k : k F such that r 1 > r 2 > 0, k k F B rk P \ B rk+1 P. Then we must have 1 dσ <. k=1 k Proof. Without loss of generality, we assume that P is the origin. We suppose also that k h = for all k h, and denote all k with. We define { k k+1 m if m k, k,m = otherwise. We suppose by contradiction that k=1 k 1 dσ = Let V be a smooth radial function in R 2, decreasing in x, defined by V x = 2, x = 0 1 < V x < 2, 0 < x < 1/2 0 < V x < 1, 1/2 < x < 1 V x = 0, x 1. For all k Z + we define v k x = V x r k. Consider k large enough such that supp v k. Now we fix k; we have v k x 1 = 1, x = 0 0 < v k x 1 < 1, 0 < x < r k /2 1 < v k x 1 < 0, r k /2 < x < r k v k x 1 = 1, x r k. Since k and are bounded, k is of finite length. Moreover, is uniformly bounded away from 0 on k since k F. Together with the fact that 1,k 1 B rk C, we have < dσ = 1 dσ < 0. 1,k 1 1,k 1 Choose m such that r m < r k /2. From the facts that v k x 1 > 0 in B rm, l B rm for all l m and v k x 1 1 as x 0 and 3.11, we have dσ for l. m,l 1 m,l Consequently, there must be a number l m such that dσ dσ < 1,k 1 m,l dσ.

9 D-2013/108 SYMMTRY AND RGULARITY 9 Choose a subinterval l l such that m,l 1 dσ + l dσ = Then we have k dσ = 0, 1,k 1 dσ. where k = 1,k 1 m,l 1 l. Now we can apply Lemma 3.3 to k,, v k, a = 1 and h = v k 1 rearranging, obtain We find that Dv k 2 qu f v 2 ku f q 2 f dx b 2 c q 2 2 dσ. k Dv k 2 qu f vku 2 f q 2 f dx DV 2 dx. B 10 By the above estimate, for a suitable constant C, we have C DV 2 2 dx dσ B 10 k 2 dσ 1,k 1 k 1 2 = dσ. Then, when k, we have C DV 2 dx B 10 k=1 h=1 k h=1 h h 2 which is a contradiction. So we must have dσ <, as desired. dσ = +, and, after Lemma 3.5. Let P be a point on F = {u f > c 2 }. Suppose that there are numbers K Z and σ > 0 such that, for each k K, there exists a regular curve k : k F with the following two properties: Then P > 0. k F B 2 kp \ B 2 k+1p, H 1 k k = ξdξ > σ 2 k. k For a proof of the above lemma, see [4, Lemma 3.2]. From an intuitive point of view, this lemma says that, if the set {u f > c 2 } { > 0} is big enough around a point of {u f > c 2 }, then > 0 at this point. Now, we are able to prove our main theorem.

10 10 C. ANDDA, F. CUCCU D-2013/108 Proof of Theorem 3.1. By using the previous Lemmas and superharmonicity of u f the Theorem follows from the results of sections 5 and 6 in [4]. Open problems. The method used in this paper to prove regularity does not work when the number of level sets of f is infinite. Therefore it remains to study the boundaries of level sets of f in the case of the rearrangement class Ff 0 of a general function f 0. We can obtain an analogous result to Lemma 3.4 for the p-laplacian operator, but we cannot go further because we lack a suitable regularity theory for the p- Laplacian operator and its solutions. We think that it is reasonable to guess that a regularity result of the type that we have proven in this work will hold for the situation with the p-laplacian when p < 2. Acknowledgments. The authors want to thank the anonymous referees for their valuable comments and suggestions. References [1] A. Alvino, G. Trombetti and P. L. Lions; On optimization problems with prescribed rearrangements, Nonlinear Analysis, TMA , [2] S. Chanillo, D. Grieser, M. Imai, K. Kurata and I. Ohnishi; Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes, Commun. Math. Phys , [3] S. Chanillo, D. Grieser and K. Kurata; The free boundary problem in the optimization of composite membranes, Contemporary Math , [4] S. Chanillo, C.. Kenig and G. T. To; Regularity of the minimizers in the composite membrane problem in R 2, ournal of Functional Analysis , [5] F. Cuccu, G. Porru and S. Sakaguchi; Optimization problems on general classes of rearrangements, Nonlinear Analysis , [6] L.. Fraenkel; An Introduction to maximum principles and symmetry in elliptic problems, Cambridge Tracts in Mathematics, Cambridge Univ. Press [7] B. Gidas, W. M. Ni and L. Nirenberg; Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, N , [8] D. Gilbarg, N. S. Trudinger; lliptic partial differential equations of second order, Springer Verlag, Berlin, [9] B. Kawohl; Rearrangements and convexity of level sets in PD s, Lectures Notes in Mathematics, Springer [10] P. Tolksdorff; Regularity for a more general class of quasilinear elliptic equations, ournal of Differential quations , Claudia Anedda Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, Cagliari, Italy -mail address: canedda@unica.it Fabrizio Cuccu Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, Cagliari, Italy -mail address: fcuccu@unica.it