On the role of energy convexity in the web function approximation

Transcription

1 NoDEA Nonlinear differ. equ. appl. 2 (25) /5/93 7 DOI.7/s c Birkhäuser Verlag, Basel, 25 On the role of energy convexity in the web function approximation Graziano CRASTA Università di Roma La Sapienza Dipartimento di Matematica P.le Aldo Moro 2 85 Roma (Italy) crasta@mat.uniroma.it Ilaria FRAGALÀ Dipartimento di Matematica del Politecnico piazza Leonardo da Vinci Milano (Italy) fragala@mate.polimi.it Filippo GAZZOLA Dipartimento di Matematica del Politecnico piazza Leonardo da Vinci Milano (Italy) gazzola@mate.polimi.it Abstract. For a given p> and an open bounded convex set R 2,we consider the minimization problem for the functional J p(u) = ( p u p u) over W,p (). Since the energy of the unique minimizer u p may not be computed explicitly, we restrict the minimization problem to the subspace of web functions, which depend only on the distance from the boundary. In this case, a representation formula for the unique minimizer v p is available. Hence the problem of estimating the error one makes when approximating J p(u p)byj p(v p) arises. When varies among convex bounded sets in the plane, we find an optimal estimate for such error, and we show that it is decreasing and infinitesimal with p. As p, we also prove that u p v p F.G. was supported by MURST project Metodi Variazionali ed Equazioni Differenziali non Lineari

2 94 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA converges to zero in W,m () for all m<. These results reveal that the approximation of minima by means of web functions gains more and more precision as convexity in J p increases. 2 Mathematics Subject Classification: 49K3, 52A, 49Q. Key words: Minimization, web functions, convexity, planar shapes. Introduction Let R 2 be an open bounded convex domain. Following [2], we call web functions on those functions u which have the same family of level lines as the distance function d (x) := dist(x, ) from the boundary of. The class of web functions turns out to have relevant properties regarding the minimization of integral functionals J on. In fact, if K p () denotes the subspace of web functions in W,p (), and J has the form J(u) = h( u ) u, u W,p (), suitable assumptions on the integrand h (not involving convexity), guarantee the existence of a minimizer for J on K p () for some p. Moreover, the energy of such minimizer admits a simple representation formula, see (4) below for the case h(s) =s p /p. It has therefore some interest to ask whether the minimum value of J over K p () can be used to approximate efficiently the infimum value of J over W,p (), when the explicit computation of the latter is not possible. To this end, as suggested in [8, 9], one is led to consider the quotient functional E() = min u K p() J(u) inf u W,p () J(u). Assuming that inf u W,p () J(u) < and J() =, the ratio E() is well-defined, and its value falls into the closed interval [, ]. Of course, the closer E() is to, the better is the approximation with web functions for the energy J. So, in order to determine a sharp lower bound for the error when the shape varies, one has to consider the optimization problem inf{e() : R 2, convex bounded}. () In our previous work [7], we studied the above problem for the quadratic energy ( ) J 2 (u) = 2 u 2 u and we proved that the corresponding value for the infimum in () is 3/4, and it is not attained. In this paper we consider the same problem for the energy ( ) J p (u) = p u p u, (2)

3 Vol. 2, 25 On the role of energy convexity in the web function approximation 95 where p belongs to the interval (, + ). The Euler equation for the functional J p on W,p () is { p u = in (3) u = on, where p u = div( u p 2 u) denotes the p-laplacian. It is worth noticing that, thanks to an integration by parts in (3), we get min J p (u) =J p (u p )= p u p p. u W,p () p Thus the integral u p p equals C /(p ), where C = C() is the best constant for the Sobolev inequality u p C u p p on W,p (), and the lower bound for E() can also be used to obtain an upper bound for C() in terms of the minimum of J p over K p (). From the web function point of view, the interest in studying the case of a general exponent p is twofold: to understand the role of convexity in the approximation error E and to investigate the limit of such error as p.it is known [2] that the solutions to (3) converge uniformly to the distance function d (x) asp. Therefore, a better (larger) lower bound for E is expected when the convexity exponent p increases. Our results confirm this expectation. In fact we show that, when J p is given by (2), the infimum in () has the value E p := 2p p 2. p/(p ) Again, such value is not attained, namely, there does not exist an optimal design. Note that E p tends to as p. Actually we prove a stronger result about the asymptotic behaviour as p : the difference between the solutions to (3) and the web minimizers of J p tends to zero in the W,m ()-norm for all m<. In order to obtain the sharp lower bound for E, we follow the same line of our proof in [7]. Nevertheless, the quasilinear case of general p is far from being a straightforward extension of the linear case p = 2. In particular, the turning point is a p-dependent geometric-integral inequality for convex sets, see Proposition 4. In [7], it was stated just for polygons when p = 2. The proof for arbitrary p is more delicate as it is based directly on the Brunn-Minkowski Theorem, without involving the isoperimetric inequality for convex polygons. As a consequence, the validity of the statement is enlarged to the whole class of convex bodies. This makes the inequality under consideration more interesting from the point of view of possible applications. For instance, as a corollary of Proposition 4 applied with p = 2, it is easy to show that the disk supports the largest sandpile among all planar domains with a given area (see Remark below). The contents are organized as follows. In Section 2 we state the main results: we establish the optimal lower bound for E in Theorem, and the asymptotic behaviour of solutions as p in Theorem 2. In Section 3, we study the extremal domains for the functional E, by showing that E attains its maximum on a unique shape

4 96 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA (disks) and exhibiting a minimizing sequence (made of isosceles triangles) which shows that the lower bound E p is sharp. In Section 4, we give bounds for both the denominator and the numerator of E (the proofs are postponed respectively to Sections 6 and 7) and we show that, combined together, they entail Theorem. The proof of Theorem 2 is given in Section 5. 2 Main results Before stating our results, some preliminaries are in order. Let C denote the class of all nonempty open bounded convex subsets of R 2. For every C, we denote by R its inradius, i.e. the supremum of the radii of the disks contained in. Then, the level lines of web functions are none other than the boundary of the so-called parallel sets t := {x : d (x) t} for t [,R ]. In terms of parallel sets, when J p is given by (2), the representation formula [8, (25)] for the minimum of J p over K p () reads min J p(u) = p R ( t p ) /(p ) dt, (4) u K p() p t where t and t indicate respectively the area and the perimeter of t. Formula (4) will be heavily exploited throughout the paper. It is also convenient to set N () = p p min J p(u), D() = p u K p() p min J p (u), u W,p () so that both the numerator N and the denominator D of E are nonnegative. Note also that N and D are homogeneous, while the functional E() = N () D() is invariant under dilations. We may now give the sharp lower bound for E: Theorem For all C we have E() > inf C E = 2p p 2 p/(p ) =: E p. Note that the map p E p is strictly increasing on (, + ), and that lim p E p = lim E p =. (5) p This tells us that more convexity in the functional gives more precision in the approximation with web functions. Clearly, the limiting behaviour (5) as p for the uniform lower bound implies the pointwise asymptotic lim E()= C; (6) p

5 Vol. 2, 25 On the role of energy convexity in the web function approximation 97 note however that the monotonicity of the map p E p does not imply the same property for each map p E() with fixed. Investigating this property could be a further point of interest. On the other hand, we can make more precise how (6) occurs. By using a result in [2] (see also [3]), we prove Theorem 2 Let C, and for all p> denote by u p and v p the minimizers of J p over W,p () and K p () respectively. Then, for all m [, + ), u p d and v p d in W,m (); (7) in particular, lim p u p v p,m =. Moreover, lim J p(u p ) = lim J p(v p )= d (x). p p Note that the convergence in (7) may not be improved to the W, () norm topology. To see this, it suffices to consider the case when is a disk: for all p>, we have u p = v p C (), whereas d C (). The convergence in (7) may also be seen in a different, more geometric, fashion. For all p denote by p,t the superlevels of u p : p,t := {x : u p (x) t}. By [4, Theorem 2], u (p )/p p is a concave function over, hence the sets p,t are convex. Their limit as p can be identified to the parallel set t. Indeed, as a consequence of the W,m convergence of u p to d, we obtain the following corollary of Theorem 2. It enlightens the role of p in the web function approximation for the p-laplace problems (3): larger p yield level sets of the solutions more similar to parallel sets. Corollary Let C. For all t, we have lim p p,t = t in the Hausdorff topology. 3 The maximal shape and a minimizing sequence In this section, we investigate the existence of extremal domains for the functional E. Wefixp> and for convenience we introduce the conjugate exponent to p, namely q :=. Then, we may rewrite N () in terms of q as p p R t q N () = dt, (8) t q and the statement of Theorem reads E() > inf E = q + C 2 q C. We start proving that E attains its maximum on if and only if is a disk. In other words, the solution u p to (3) is a web function if and only if is a disk.

6 98 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA Proposition E()=if and only if is a disk. Proof. If is a disk then the solution to (3) is radially symmetric, hence E()=. Assume now that E() =, namely that the solution u p to (3) coincides with the minimizing web function v p.if is smooth, the statement follows from the extension of a classical result by Serrin to the quasilinear case (see [, Theorem.3] for <p<2 and [3, Theorem 4] for arbitrary p). For general, to prove the result we exploit the representation formula for the web minimizer, which gives d(x) ( ) q t v p (x) = dt =: φ(d (x)), (9) t see [6, Theorem 3.]. On the other hand, since v p = u p, it is well-known [, 6] that v p C,α (). Since φ (t) > for every t [,R ), this is possible only if d is of class C in the open set {x ; <d (x) <R }, or equivalently if the sets t for t [,R ) have C boundary. Following the arguments of [, Section I..4], we deduce that this regularity property of parallel sets is satisfied uniquely when the equality sign holds in the Bonnesen s inequality + πr. R On the other hand, the extremal domains for this inequality consist of a rectangle ended by two half disks, that is, are of the form = S+B R, where S is a compact (possibly degenerate) line segment, and B R is a disk of radius R. Finally, on sets of this shape a direct computation shows that the solution u p to (3) can be a web function if and only if S is degenerate, that is, if and only if is a disk. Wesaythatisatangential body to a disk D if through each point of there exists a tangent line to which is also tangent to D. For such domains, the following simple characterization of N () in terms of R holds. Proposition 2 Let C be a tangential body to a disk. Then we have N () = (q + 2)2 q R q. Proof. By the density of polygons and the continuity of N() with respect to the Hausdorff topology [6, Section 6], it suffices to prove the statement when is a circumscribed polygon. In that case, by the arguments in the proof of [7, Proposition 3], we know that =2 /R, and for all t (,R ) t = R 2 (R t) 2, t = 2 R 2 (R t). Therefore, t q t q = 2 q 2 R (R t) q+ so that the statement follows from (8) after integration over (,R ).

7 Vol. 2, 25 On the role of energy convexity in the web function approximation 99 We now determine an upper bound for the infimum of E: Proposition 3 There exists a sequence of isosceles triangles {T h } h Csuch that lim sup h + E(T h ) q + 2 q. () In particular, inf C E() (q + )2 q. Proof. For all integer h consider the isosceles triangle T h Cdefined by T h = { (x, y) R 2 ;<y< π h, h 2 } y π h<x<h h2 y. π Taking into account that R T h letting h, we obtain = πh(h 2 + h 4 + π 2 ), using Proposition 2 and N (T h ) πq+ 2 2q q +2 h q as h. () In order to find an asymptotic lower bound for D(T h ), we choose a particular function v h W,p (T h ). We take v h (x, y) = [( ) q βh (x) β h (x) q] q 2 y 2, where β h is the function defining the two equal sides of T h, namely { π β h (x) = min h 2 (h + x), π } h 2 (h x) x [ h, h]. We have D(T h )= q min u W,p (T h ) J p (u) qj p (v h ) h. By the symmetry of T h and v h we may restrict our attention to the (half) triangle T h = {(x, y) T h ; x>} and double all the integrals involved. In T h we have v h (x, y) = [( ) q π(h x) q 2h 2 π(h x) 2h 2 q] y. With the change of variables x = hs, y = t/h, T h transforms into the fixed triangle { T = (s, t) R 2 ;<t<π, <s< t } ; π

8 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA moreover, we have: { [( π ) q v h (s, t) =h q ( s) q π q] } q 2 2 ( s) t =: h q α q (s, t) ( π ) q x v h (s, t) =h { q ( s) q + π π ( s) t q 2 ( π ) } 2 ( s) t =: h q β q (s, t) { y v h (s, t) =h q+ π 2 ( s) t q 2 ( π )} ( s) t =: h q+ γ q (s, t). 2 Hence, with the above change of variables, we obtain { q J p (v h )=2 [h 2( q ) βq 2 (s, t)+h 2( q+) γq 2 (s, t)] q/(2(q )) T q } h q α q (s, t) ds dt { } q 2h q γ q (s, t) q/(q ) α q (s, t) ds dt as h. q Some calculations give α q (s, t) ds dt = T Then we get T T ( π ) q+ 2 γ q (s, t) q/(q ) ds dt = 2 (q + )(q +2). D(T h ) qj p (v h ) πq+ 2 q (q + )(q +2) h q as h. Combining this asymptotic inequality with () proves () by letting h. 4 Proof of Theorem (optimal lower bound) In view of Proposition 3, the proof of Theorem is complete if we show that E() > q + 2 q C. (2) We first recall the definition of the piercing function λ, see [7] and previous work by Cellina [5]. For a.e. y the outer unit normal is well-defined and it will be denoted by n(y). For a.e. x the point Π(x) such that x Π(x) = d (x) is uniquely determined. Then we set: λ(y) = sup{k ; Π(y kn(y)) = y} for a.e. y. (3)

9 Vol. 2, 25 On the role of energy convexity in the web function approximation We clearly have λ(y) R on. We also extend the definition of λ to points x : λ(x) =λ(π(x)) x Π(x) for a.e. x. (4) When R 2 is a convex polygon, (4) enables us to write the measure of the parallel set t as t = λ(y) dy. (5) t Moreover, we have the following upper bound for D() in terms of the piercing function: Theorem 3 Let R 2 be a convex polygon and let u p be the minimizer of J p in W,p (). Then there exists δ = δ() such that D() = q + λ q+ (y) dy δ() (6) and δ() (q ) [ u p (x) p u p (x) n(π(x)) p ] dx. (7) We also have a dual statement to Theorem 3, namely a lower bound for N () in terms of the piercing function: Theorem 4 For all convex polygon R 2 we have N () 2 q λ q+ (y) dy. We postpone the proofs of Theorems 3 and 4 until Sections 6 and 7 respectively. We are now in a position to prove inequality (2). If is a disk, the inequality is obvious since E() =, see Proposition. If is a polygon, (2) is a direct consequence of Theorems 3 and 4. If is a bounded convex set other than a disk, we take an inner approximating sequence {P h } of polygons (that is, P h for all h and P h in the Hausdorff topology as h ). By Theorems 3 and 4 we have E(P h )= N (P h) D(P h ) E D(P h )+δ(p h ) p h N. D(P h ) Therefore, it is enough to show that (a subsequence of) {δ(p h )} converges to a strictly positive constant C. We argue by contradiction, assuming that δ(p h ). Consider the distance functions d h = d Ph and d, and let u h and u p be the minimizers of J p over W,p (P h ) and W,p (). Extending them by zero on \ P h,

10 2 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA we may think of d h and u h as defined on the whole. We have [ = lim inf δ(p h ) (q ) lim inf u h (x) p u h (x) d h (x) p] dx h h =(q ) [ u p (x) p u p (x) d (x) p ] dx. The first inequality in (8) follows from (7) and Fatou s lemma. The subsequent equality is deduced from the a.e. convergence d h d and u h u p, which hold up to subsequences. For the former, see (28) in [7]. The latter can be deduced using the weak W,p -convergence of u h to u p (see for instance [4]) combined with the convergence u h L p () u p L p () (deriving from the Euler equations (3) in and P h ). Now we conclude from (8) that u p is parallel to d a.e. in, that is u p is a web function. By Proposition, this contradicts the assumption that is not a disk. (8) 5 Proof of Theorem 2 (asymptotic behaviour) The first part of (7) (the convergence u p d ) is just Proposition 2. in [2]. In order to prove the convergence v p d, recall the explicit form (9) of v p. Then, for all m, we have [ ( ) /(p ) ] m d(x) d v p m = d (x) m as p, and (7) follows. Finally, the limit d(x) lim J p(u p ) = lim J p(v p )= p p follows from (7) by direct computation. d (x) 6 Proof of Theorem 3 (upper estimate of D) We follow the same line of proof adopted for the linear case in [7]. We first show (6). Assume that has N sides and denote them by F,...,F N. For simplicity, for all j =,...,N we denote by F j the open segment, namely the j-th side of without its endpoints. Note that the function λ introduced in (3) is defined in every point of except for the N vertices. Moreover, n(y) n j is a constant vector on F j. We take a partition of into N open subpolygons P,...,P N defined as follows: P j = {y tn j ; y F j, <t<λ(y)}.

11 Vol. 2, 25 On the role of energy convexity in the web function approximation 3 Each polygon P j may also be seen as the (open) epigraph Z j of the function λ on F j, namely Z j = {(y, t); y F j, <t<λ(y)}. For all j {,...,N} let W,p (P j ) := {v W,p (P j ); v =onf j }, W,p (Z j ) := {v W,p (Z j ); v(y, )= y F j } and consider the functional H j (v) = P j ( ) p v p v v W,p (P j ). Note that H j (v) = F j λ(y) [ ] p v(y tn j) p v(y tn j ) dt dy v W,p (P j ). (9) Let u p be the minimizer of J p. Let also u j denote the restrictions of u p to P j (j =,...,N) and set w j (y, t) =u j (y tn j ) (y, t) Z j. (2) Since u j C W,p (P j ), we have w j W,p (Z j ) and wj = u j n j so that t w j (y, t) t uj (y tn j ) (y, t) Z j. (2) Therefore we get H j (u j ) I j (w j ) (j =,...,N) (22) where λ(y) [ ( ) p v I j (v) := v] dt dy v W,p (Z j ). F j p t On the other hand, at each fixed y F j,wehave { λ(y) [ ] } min p g (t) p g(t) dt ; g W,p (,λ(y)),g()= = q(q +) λq+ (y). Therefore the minimum of I j on W,p (Z j ), which is attained by the function w(y, t) = q (λq (y) [λ(y) t] q ),

12 4 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA may be evaluated as min{i j (v); v W,p Then, by (22) and (23) we have j= (Z j )} = λ q+ (y) dy. (23) q(q +) F j N N J p (u p )= H j (u j ) I j (w j ) j= N λ q+ (y) dy q(q +) j= F j = λ q+ (y) dy. (24) q(q +) This yields D() = qj p (u p ) λ q+ (y) dy. (25) q + To complete the proof of (6), it remains to show that the inequality in (25) is strict. We may have equality in (25) only if we have equalities in (2) for all j =,...,N. But this is equivalent to u p K p () (i.e. u p web function), and in turn, to E() =, a contradiction in view of Proposition. We now prove (7). By using (24), (9) and (2) we obtain δ() = λ q+ D() q + [ ] = q J p (u p )+ λ q+ q [H j (u j ) I j (w j )] q(q +) j = q λ(y) [ j F j p uj (y tn j ) p u j (y tn j ) w j p ] p (y, t) t + w j (y, t) dt dy =(q ) λ(y) [ u j (y tn j ) p u j (y tn j ) n j p ]dt dy j F j =(q ) [ u p (x) p u p (x) n(π(x)) p ] dx and (7) follows.

13 Vol. 2, 25 On the role of energy convexity in the web function approximation 5 7 Proof of Theorem 4 (lower estimate of N ) The main tool for the proof of Theorem 4 is the following geometric inequality for convex domains, involving the family of its parallel sets. We believe it might have an independent interest in the framework of convex bodies (see Remark below). Proposition 4 For every Cand every q [, ) the following inequality holds: ( ) q q(q +) R t 2 q tq dt. (26) If q>, the equality sign holds if and only if is a tangential body to a disk. Proof. Let us define the function α(s) := sr, s [, ]. With a linear change of variables, the inequality (26) can be rewritten as ( q 2 α(s) ds) q(q +) α(s)s q ds. (27) The function α is non-negative, strictly monotone decreasing and, by the Brunn- Minkowski Theorem (see [5, Thm ]), it is concave down on [, ]. Hence the function β(s) :=α(s) ( s), s [, ], is non-negative, concave down, and satisfies β() =. Ifβ(s), then (27) is trivially satisfied; so assume that β(s). Then, from the concavity of β, we infer that β (s) β(s) for a.e. s (, ). (28) s In terms of the function β, the inequality (27) now becomes ( q 2 β(s) ds +) q(q +) β(s)s q ds. (29) Let us define the following transform of the function β: F (γ) := β(s)s γ ds, γ.

14 6 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA As β(s), we know that F (γ) >. Since β AC[, ], by an integration by parts and (28) we obtain the following estimate on the derivative of F : F (γ) = (log s)β(s)s γ ds = β(s)s γ ds γ γ γ F (γ) γ F (γ), (log s)β (s)s γ ds hence F (γ) F (γ), γ. γ + Since F () >, such differential inequality implies 2F () F (γ), γ. (3) γ + Finally, from (3) we obtain ( q 2 β(s) ds +) =(2F ()+) q 2qF() q(q +)F (q) = q(q +) β(s)s q ds, (3) so that (29) follows and the first part of the proposition is proved. In order to study the equality case in (26), it is enough to observe that, from (3), it holds if and only if F () =, that is if and only if β. In turn, this is equivalent to α(s) = s, which holds if and only if is a tangential body to a disk (see [5, Lemma 3..]). Remark From the classical isoperimetric inequality we have that ( ) q (q )/2 2. (32) π From the identity dq (x) dx = R t t q dt, combining (26) and (32) we obtain the following inequality: d q (x) dx 2 q(q +)π (q )/2 (q+)/2, C,q [, + ), (33) where the equality sign holds if and only if is a disk. For q = 2 the last inequality reads d (x) dx 3 π 3/2.

15 Vol. 2, 25 On the role of energy convexity in the web function approximation 7 As an application, this special case of inequality (33) can be used in order to prove that among all plane regions of a given area, the circle can support the largest sandpile (see for example [, p. 8]). Corollary 2 Let R 2 be a convex polygon. Then, for every q [, ) the function ψ q (t) := t q q + t q 2 q λ q (z) dz t t [,R ] (34) is nonnegative. Proof. Since is a polygon, we may apply Fubini s Theorem to obtain R λ q (z) dz = q (s t) q s ds. t t Then, with a linear change of variables, (34) follows from Proposition 4 applied to the convex domains t. Remark 2 If q N, the proof of Corollary 2 does not require Proposition 4. Indeed, by (5) we have ψ (t). Then, making use of the isoperimetric inequality for convex polygons in a similar way as in the proof of [7, Lemma 2], one sees that ψ q implies ψ q+. Therefore, Corollary 2 follows arguing by induction on q N. We are now ready to prove Theorem 4. Let v p K p () be the (unique) minimizing web function, i.e. By [6, Theorem 3.] we have J p (v p ) = min J p(u). u K p() v p (x) = d(x) ν q (t) dt, ν(t) := t t. Then, since N () = qj p (v p ), using the coarea formula and an integration by parts we infer N () = q = q = 2 q λ(y) λ(y) [ ν q (t) t ] ν q (s)ds dt dy p [ ν q ] (t) (λ(y) t)ν q (t) dt dy p λ q+ (y) dy + () (35)

16 8 Graziano Crasta, Ilaria Fragalà and Filippo Gazzola NoDEA where () := λ(y) [ q + ] 2 q [λ(y) t] q (q )ν q (t)+q[λ(y) t]ν q (t) dt dy. By Fubini s Theorem and recalling that (4) defines λ in the whole, we may rewrite () as () = R t [ q + ] 2 q λ q (z) (q )ν q (t)+qλ(z)ν q (t) dz dt. Finally, by (5), we have that ν(t) is the integral mean value of λ in t, and the above equation becomes () = R [ t q q + t q 2 q t λ q ] dt = R This, combined with (35) and Corollary 2, proves Theorem 4. ψ q (t) dt. References [] C. BANDLE, Isoperimetric inequalities and applications, Pitman 98 [2] T. BHATTACHARYA, E. DIBENEDETTO, J. MANFREDI, Limits as p + of p u p = f and related extremal problems, Rend. Sem. Mat. Univ. Pol. Torino, Some topics in nonlinear PDE s, Special issue (989), 5 6. [3] F. BROCK, J. PRAJAPAT, Some new symmetry results for elliptic problems on the sphere and in Euclidean space, Rend. Circ. Mat. Palermo 49 (2), [4] D. BUCUR, G. BUTTAZZO, Variational Methods in some Shape Optimization Problems, Lecture Notes of courses at Dipartimento di Matematica Università di Pisa and Scuola Normale Superiore di Pisa, Series Appunti di Corsi della Scuola Normale Superiore (22). [5] A. CELLINA, Minimizing a functional depending on u and on u, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (997), [6] G. CRASTA, Variational problems for a class of functionals on convex domains, J. Differential Equations 78 (22), [7] G. CRASTA, I. FRAGALÀ, F. GAZZOLA, A sharp upper bound for the torsional rigidity of rods by means of web functions, Arch. Rational Mech. Anal. 64 (22), 89 2.

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