Vol. 29, No. 3, pp. 1{8, May WHERE BEST TO HOLD A DRUM FAST *

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1 SIAM J. OPTIMIATION c 998 Society for Industrial and Applied Mathematics Vol. 29, No. 3, pp. {8, May 998 WHERE BEST TO HOLD A DRUM FAST * STEVEN J. COX y AND PAUL X. UHLIG z Abstract. Allowed to fasten say one half of a drum's boundary which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk while, for general shapes, we establish existence, necessary conditions, and build and test a pair of numerical methods. Key words. Membrane, eigenvalue, Robin, mixed, optimal design AMS subject classications. 35P5, 65K5, 73D3. Introduction. We consider the fundamental mode of vibration of a drumhead that is fastened along part of its boundary and free on the remainder. More precisely, we study the least eigenvalue of?u = u in ; u = on?; u = on n?; n where is a smooth, open, bounded, connected, planar set and? is a measurable subset of its boundary. We denote this least eigenvalue by (?) and seek its extremes as? varies over subsets of of prescribed measure. Closely related questions for one{ dimensional continua have been raised in the engineering literature, see, e.g., Mroz and Rozvany [4] and Chuang and Hou [5]. We begin the analysis of our model problem by expressing the two boundary conditions in the single equation (:)? u + (?? )u=n = on ; where? denotes the characteristic function of?. With an eye toward a convenient variational characterization of (?) we note that (.) is not a boundary condition of the third (or Robin) type. To achieve this the coecient of u=n must be constant. Before blindly dividing through by?? we introduce a simple regularization. In particular, we arrive at (.) in the limit as "! in or equivalently,? u + ( + "?? )u=n = ; on ; (:2) "?? u + u=n = ; on : Physically, the drumhead remains free on n? while on? it is elastically supported by a fastener of stiness =". We denote by " (?) the least eigenvalue of? subject *Received by the editors January, 997; accepted by the editors Month, x, xxxx. This work was supported by NSF Grant DMS and a fellowship from the Humboldt Foundation. y Dept. Computational & Appl. Math., Rice University, 6 Main St., Houston, TX 775. z Math. Dept., St. Mary's University, San Antonio, TX.

2 2 STEVEN J. COX AND PAUL X. UHLIG to (.2). This boundary condition is indeed of the third type and so we may record the weak formulation (:3) ru rv dx + "?? uv ds = uv dx; 8 v 2 H (); and the associated variational characterization (:4) " (?) = inf jruj u2h 2 dx + "?? u 2 ds; () where H () is the class of H () functions with L 2 () norm one. The advantage of the chosen regularization lies in the fact that, in both (.3) and (.4), the underlying function space, H (), does not vary with?. We now x a number 2 (; ) (the Dirichlet fraction) and formulate the optimal design problems whose solutions will determine the range of " (?) as? varies over those subsets of of size jj. In particular, we study where inf " (?) and sup "(?)?2ad ()?2ad () ad () f? :? ; j?j = jjg; and j?j denotes the one{dimensional Hausdor measure of?. Generally speaking, we shall see that minimal designs favor a connected? while maximal designs tend to fragment?. Accordingly, in x2, we establish existence of minimizers and (relaxed) maximizers by showing that " is weak* continuous on the weak* closure of ad (). In x3 we characterize minimizers via rst order necessary conditions and provide an explicit minimal design for the disk. In x4 analogous rst order conditions lead to the uniqueness of the maximizer and its characterization in terms of the normal derivative of the rst eigenfunction of the pure Dirichlet problem. In x5 we construct distinct approaches to the numerical minimization and maximization of ". We test these methods on elliptical and L{shaped drums in x6. Though stated in the context of the planar Laplacian our arguments apply, without change, to second order selfadjoint elliptic equations on smooth bounded domains in an arbitrary number of dimensions. Though isoperimetric inequalities for mixed and Robin problems have received considerable attention, see, e.g., Bandle [], the paper of Buttazzo [4] appears to be the rst and only to consider an extremal Robin problem on a xed domain. Upon completion of this work we learned that Denzler [] had been simultaneously pursuing the same set of questions. Via methods quite distinct from those invoked here he showed that attains its minimum on ad () and that the supremum of is (), the least Dirichlet eigenvalue. 2. Existence. We shall denote by L(; [; ]) those measurable functions on that take values in the interval [; ]. With respect to the weak* topology on L () Friedland [2] has shown that Proposition 2.. The weak* closure of ad () is ad () f 2 L(; [; ]) : (x) ds = jjg:

3 WHERE BEST TO HOLD A DRUM FAST 3 In addition, ad () is the set of extreme points of ad (). For 2 ad () we denote by " () the rst eigenvalue of? subject to (2:) "? u + u=n = on : The analogous variational characterization (2:2) " () = inf R " (u; ) where R " (u; ) u2h () leads immediately to jruj 2 dx + "? (2:3) < " () () 8 2 ad () and 8 " > ; u 2 ds where () is the rst eigenvalue of? subject to Dirichlet conditions over the entire boundary. As R " (u; ) = R " (juj; ) it follows from (2.2) that " () is simple and may be associated with a nonnegative eigenfunction. Proposition 2.2. The mapping 7! " () is continuous with respect to the weak* topology on L(; [; ]). Proof. Suppose n * and that un is the positive rst eigenfunction, associated with n, normalized such that (2:4) u 2 n dx = and jru n j 2 dx + "? n u 2 n ds = "( n): From (2.3) and (2.4) it follows that fu n g n is bounded in H () and hence that u n * u in H () and u n! u in L 2 () and the traces u n j! uj in L 2 (). In addition "( n)!. These observations permit us to pass to the limit in the weak form ru n rv dx + "? n u n v ds = "( n) u n v dx; and so conclude that and u constitute an eigenpair for. As u is positive it follows that = " (). As ad () is weak* compact it now follows that Corollary 2.3. inf "(?) = min "?2ad () 2ad () () and sup "(?) = max "():?2ad () 2ad () Our interest is in characterizing those at which " attains its extremes. A number of previous studies have produced lower and upper bounds for "(). Regarding the latter, such bounds are typically achieved by replacing with a constant and with a disk. Polya and Szego accomplish this for starlike via the method of similar level lines, see Bandle [, Thm. III.3.2]. Hersch uses conformal transplantation and so requires that merely be simply connected. More precisely, he demonstrates, see [, Thm. III.3.7], that (2:5) " (; ) " (jj=jd j; D ) 8 2 ad ()

4 4 STEVEN J. COX AND PAUL X. UHLIG where D is the disk with radius equal to the conformal radius of. Of course when is itself a disk this results states that is maximal. The construction of useful lower bounds is considerably more dicult. All attempts to bound " () from below apply only to the case of constant. We cite Philippin [5], Bossel [3], and Sperb [7]. 3. Minimizing. " We show that 7! "() possesses a classical, i.e., ad (), minimizer. We compute it in the case of the disk while in the general case we produce pointwise optimality conditions. Returning to (2.2) we recognize that 7! " () is an inmum of ane functions of. As a result, Proposition 3.. 7! " () is concave on ad (). If we now recall, see, e.g., Bauer [2], that a bounded concave function on a compact convex set attains its minimum at an extreme point, we arrive at Corollary ! " () attains its minimum on ad (). We now produce an explicit minimizer in the case that is a disk, D. This is accomplished through circular symmetrization, dened as follows. Given v 2 H (D) we take u(r; t) = v(x) where x = r(cos t; sin t) and? < t. Now, at each r we replace t 7! u(r; t) with its symmetrically increasing rearrangement u _ (r; t) = inf fc : t 2 fs : u(r; s) cg g where A is simply the interval (?jaj=2; jaj=2). One then takes v _ (x) u _ (r; t) to be the circular (increasing about t = ) rearrangement of v. The corresponding symmetrically decreasing rearrangement is u^(r; t) = u _ (r;? jtj): As a simple example we note that if? 2 ad (D) then ^? (t) =? = n if jtj, otherwise. We now recall, see, e.g., Cox and Kawohl [9], that circular rearrangement can not increase the Dirichlet integral and that u _ and ^? are oppositely ordered. As a result, R " (v;? ) R " (v _ ;? ) 8 (v;? ) 2 H (D) ad (D); and so Proposition 3.3.? 7! " (?) attains its minimum at?. As? is clearly independent of " we proceed to let " approach. Our preliminary result does not require the domain to be a disk. Lemma 3.4. If? then "(?)! (? ) as "!. Proof. Let u " 2 H () denote the eigenfunction associated with " (?). Now, recalling (2.3), we nd (3:) jru " j 2 dx + "? u 2 " dx = "(?) ():?

5 WHERE BEST TO HOLD A DRUM FAST 5 As a result, fu " g "> is clearly bounded in H () and moreover? u 2 " dx = O("): Hence (a subsequence of) u " converges weakly in H () to some u 2 H (;?), those functions in H () with vanishing trace on?. We now show that u is the eigenfunction associated with (? ). Taking the limit inferior throughout (3.) gives jru j 2 dx lim inf "(?): "! Now if there exists a u 2 H (;?) and a > for which jruj 2 dx jru j 2 dx? then (3.) implies R " (u;? ) < " (?) for some ", contrary to Rayleigh's principle. Hence, (?) = jru j 2 dx lim inf "(?): "! The simple observation, " (?) (?), completes the argument. Corollary 3.5.? 7! (? ) attains its minimum at? Figure. Minimal fastening of the disk. In gure we have plotted?, for = =2 on the disk of unit diameter, along with the contours of the associated rst eigenfunction, computed by the pdeeig routine in Matlab [3] via a piecewise linear approximation on triangles. The computed value of (? ) is As the eigenvalue problem for such a design does not yield to separation of variables we return to the question posed at the close of the last section, namely, can one bound (? ) from below? Even in this simplest of all possible geometries our best analytical bound requires the majority of the boundary to be Dirichlet. More precisely, if is the disk of radius R and > =2 then (? ) 2? 2R 2 j2 ;.2.4.6

6 6 STEVEN J. COX AND PAUL X. UHLIG where j is the rst zero of the Bessel function J. This follows from Bandle's generalization of a result of Nehari, see [, Thm. III.3.9]. We now return to a general domain and denote by " the minimizer of " over ad (). We take u " 2 H () to be the positive eigenfunction associated with " and record " ( " ) = R " (u " ; " ) = In other words, R " (u " ; " ) = min 2ad () min R " (u; ) = u2h () min u2h () min R " (u; ): 2ad () min R " (u; " ) and R " (u " ; " ) = min R " (u " ; ): u2h () 2ad () The former simply states that u " is an eigenfunction corresponding to ". The latter however informs us that (3:2) " ju " j 2 ds = min ju 2ad () " j 2 ds: We remove the integral constraint on " at the cost of a Lagrange multiplier. More precisely, from the Lagrange Multiplier Rule, [6, Theorem 6..], we deduce that (3.2) implies the existence of and j j + j 2 j > such that (3:3) " ( ju " j ) ds = min 2L(;[;]) ( ju " j ) ds: From ju " j 2 we deduce from (3.3) that 2. If 2 = then (3.3) implies that " u " must vanish on the full boundary. Now the boundary condition, (2.), implies that u " is a Neumann eigenfunction. As u " does not change sign it can only be the constant eigenfunction. Now "u " = implies that " is identically zero, contrary to its integral constraint. Therefore, 2 <. Now, if = then, as 2 <, (3.3) implies that " is identically one, contrary to its integral constraint. Therefore, >. With 2? 2 = we deduce from (3.3) the following pointwise necessary conditions, (3:4) (3:5) (3:6) " (x) = ) u " (x) ; < " (x) < ) u " (x) = ; " (x) = ) u " (x) : Recalling that " may be assumed a member of ad () it follows that " jumps across a level set of the trace of its corresponding eigenfunction, u ". 4. Maximizing. Recalling (2.5) we begin with a simple proof of the fact that constant is maximal for the disk. Noting only that u, the eigenfunction corresponding to on the disk, is radial we nd (4:) " () R "(u ; ) = R " (u ; ) = " () 8 2 ad (D): With regard to general we shall see that where the maximizing is neither zero or one the trace of its corresponding eigenfunction is, like u, constant. In addition, we

7 WHERE BEST TO HOLD A DRUM FAST 7 establish uniqueness of the maximizer and show that when it lies everywhere between zero and one it is (to lowest order in ") proportional to the normal derivative of the rst Dirichlet eigenfunction on. The rst step is the derivation of pointwise conditions analogous to (3.4){(3.6). These shall stem from knowledge of the gradient of 7! " (). Proposition 4.. 7! " () is smooth and h "(); i = "? u 2 ds where u 2 H () is the nonnegative eigenfunction associated with. Proof. The gradient of a simple eigenvalue of a self{adjoint operator is the gradient of the Rayleigh quotient evaluated at the corresponding eigenfunction. See Cox [8] for details. If ^ " maximizes " over ad () then " (^ " ) 2 N ad () (^ " ), the cone of normals to ad () at ^ ". As ad () is convex this means that h " (^ " ); ^ " i = max 2ad () h " (^ " ); i; that is, (4:2) ^ " j^u " j 2 ds = max j^u 2ad () " j 2 ds; where ^u " is the positive eigenfunction corresponding to ^ ". As above, in order to shed the integral constraint, we invoke the Lagrange Multiplier Rule of Clark. This gives a and 2 for which j j + j 2 j > and (4:3) ^ " ( j^u " j ) ds = max 2L(;[;]) ( j^u " j ) ds From j^u " j 2 we deduce from (4.3) that 2 >. Similarly, <. With 2? 2 = we arrive at the pointwise necessary conditions (4:4) (4:5) (4:6) ^ " (x) = ) ^u " (x) ; < ^ " (x) < ) ^u " (x) = ; ^ " (x) = ) ^u " (x) : From Proposition 2.4 we note that these conditions are also sucient. A further consequence of (4.2) is that (^u " ; ^") is a saddle point of R ", i.e. R " (^u " ; ) R " (^u " ; ^ " ) R " (u; ^ " ) 8 (u; ) 2 H () ad (): From this observation comes Proposition 4.2. ^ " is unique. Proof. Suppose that and 2 are both maximizers of 7! " () and that u and u 2 are the respective rst eigenfunctions. We nd R " (u ; 2 ) R " (u ; ) R " (u 2 ; ) R " (u 2 ; ) R " (u 2 ; 2 ) R " (u ; 2 )

8 8 STEVEN J. COX AND PAUL X. UHLIG However, as R " (u ; ) = R " (u 2 ; 2 ) we nd that u and u 2 are both eigenfunctions for and hence u = u 2. Recalling the respective weak forms we nd (? 2 )u v ds = 8 v 2 H (); and hence = 2 on the support of u j, the trace of u. O of the support of u j it follows from (4.4) that = 2 =. From uniqueness we are able to ascertain symmetry. In particular, if is symmetric with respect to a line ` we may reect ^ " across ` to ^ "`. By simply reecting the associated u " it follows that "(^ " ) = "(^ "`) and hence, by uniqueness, that ^ " = ^ "`. We have proven Proposition 4.3. ^ " is symmetric about every line of symmetry of. This leads to a third proof of (4.). Proposition 4.4. If is a disk then ^ ". Disks are the only (smooth) sets with a constant maximizer. Proof. Full symmetry implies that ^ " must be constant. The only admissible constant is. Given a constant maximizer, it follows from (4.5) that u " is identically on. From the boundary condition, (2.), we then nd that u " =n =?=" on. Serrin [6, Theorem 2] has shown that a disk is the only C 2 domain on which one may solve ( + )u = subject to constant Dirichlet and Neumann data. If = D a is a disk of radius a then u(r) = J ( p r) is a radial solution of?u = u. The best eigenvalue, " (), is therefore the least positive for which u(a) + "u (a) = : It follows immediately then that " ()! (D a ) as "!, where (D a ) is the least positive root of 7! J ( p a), i.e., the rst Dirichlet eigenvalue of D a. This approach to the Dirichlet eigenvalue holds in fact for every domain. Proposition 4.5. If ^ " maximizes 7! "() over ad () then "(^ " )! () as "!. Proof. As "() " (^ " ) () it suces to show that (4:7) () lim inf "(): "! Let us denote by u " 2 H() the positive eigenfunction corresponding to " (). As ku " k 2 = and kru " k2 2 () it follows that there exists a u 2 H () for which u " * u in H () as "!. Given the normalization of u " we nd that ju " j2 ds = " jru " j2 dx + " " ()! as "!, i.e., u " j! in L 2 (). As u " j! u j in L 2 () it follows that u 2 H (). Now, given the weak lower semicontinuity of u 7! kruk2 2 and the nonnegativity of the boundary term, we nd jru j 2 dx lim inf "! jru " j2 dx + " ju " j2 ds = lim inf "(): "!

9 WHERE BEST TO HOLD A DRUM FAST 9 As u 2 H () and ku" k 2 = it follows from Rayleigh's principle that the left hand side is larger than (). This establishes (4.7). This proposition addresses the limiting behavior of the eigenvalue but says nothing about the limiting optimal design. We shall now show that if the limiting design takes values strictly between and then it is proportional to the normal derivative of the rst Dirichlet eigenfunction. We begin at the necessary condition (4.5) and note that for constant and < () one may solve?u = u in ; u = on ; in terms of the Dirichlet eigenfunctions, f j g, and Dirichlet eigenvalues, f j g, of. In particular, X h j ; i u = + j? j: The Robin condition, (2.), now suggests (4:8) =? " Integrating this expression over we nd (4:9) jj = ds =?" j= u X n =?" h j ; i j j= j? n : X j= h j ; i j? j X n ds = " h j ; i 2 j= j? j: We view this as an equation for. As the right side is continuous and strictly increasing from (at = ) to (at = ()) there exists a unique solution, ", depending smoothly on ". Expressing " as a power series, identication of like powers in (4.9) brings (4:) " = ()? 2()h ; i 2 " + O(" jj 2 ): Substituting this into (4.8) we arrive at (4:) " = n n + O("); where n = jj n ds: Hence, if ^ " takes values strictly between and it must necessarily be of this form. Moreover, as the necessary conditions are also sucient, whenever the above derivation produces an admissible design this design is maximal. Regarding the admissibility of " we note that, by construction, it is nonnegative and has the correct average. It remains only to check whether it is bounded above by. One scenario in which this bound is assured is when is smooth (in which case 2 C ()) and " and are suciently small. Finally, we remark that (4.) provides a nice renement of Proposition 4.5 in that it expresses, in terms of the Dirichlet fraction,, the rate at which "(^ " ) approaches ().

10 STEVEN J. COX AND PAUL X. UHLIG 5. Algorithms. We conne the design,, and the eigenfunction, u, to nite{dimensional spaces and so arrive at optimization problems amenable to a computer. We write as the closure of the disjoint union of m open edges, f? j g m j=, and then restrict to (s) = mx j= where 2 R m satises the box constraints j?j (s); (5:) j ; j = ; : : :; m and the integral constraint (5:2) mx j= j j? j j = jj: In order to compute " at such a we restrict our search to eigenvectors of the form u(x) = px i= U i T i (x) where p < and the T i comprise a so{called Galerkin basis for a p{dimensional subspace of H (). On substituting this expansion into the weak form (.3) with v running through the T i we arrive at the p p eigensystem (5:3) (K + "? Q())U = MU; where K and M are independent of while (5:4) Q ij () = T i T j ds = mx k= k? k T i T j ds: Let us denote the least eigenvalue of (5.3) by " (). As this approximation procedure respects the symmetry of the original problem we retain a variational characterization, (5:5) " () = min R " (U; ); hmu;ui= R " (U; ) h(k + "? Q())U; Ui: As 7! Q() is linear it follows from (5.5) that 7! " () is concave. Now, denoting by AD those 2 R m satisfying (5.) and (5.2), we may pose the nite{dimensional optimization problems min " 2AD () and max "(): 2AD As AD is compact and convex and " is bounded and concave it follows that 7! "() attains its minimum at an extreme point of AD, i.e., on AD, those 2 AD each component of which is either zero or one. Let us now turn to the gradient of 7! " (). For well chosen basis functions, e.g., piecewise linear hats, it can be shown that "()! " () as m and p approach

11 WHERE BEST TO HOLD A DRUM FAST. In particular, " () is simple for suciently large m and p. As a result we may apply the nite{dimensional analog of Proposition 4., (5:6) " () k = " Q() k U " ; U " where the associated eigenvector, U ", is normalized according to hmu" ; U " i =. The implementation of (5.6), in particular the application of Q()= k requires a careful accounting of the assembly of Q. Recalling (5.4) we nd Q ij () k =? k T i T j ds: To begin, let us evaluate these integrals under the assumption that? k is the interval [a; b] and that this interval is partitioned by the rst components of the grid points x i = (s i ; ), i.e., a = s < s 2 < < s n? < s n = b: We also suppose T i (x j ) = ij and that T i is piecewise linear. As a result?k T i T j ds = 3 8 >< >: Substituting the above into (5.6) we nd " () k = 3" n? X i= ; js? s 2 j if i = j = js i?? s i j + js i? s i+ j if < i = j < n js n?? s n j if i = j = n js i? s j j=2 if ji? jj = otherwise. (U " )2 i + (U " ) i(u " ) i+ + (U " )2 i+ js i+? s i j: In the general case, i.e., where the T i remain piecewise linear though? k may be a planar segment whose edges and grid points are ordered by a black{box grid generator (as in Matlab's PDE toolbox), the gradient takes the form (5:7) " () k = 3" X i2ik hu " i ij! i j; hu " i i (U " )2! + i + (U ")! + (U ")!? + (U " )2 ; i i!? i where I k is the set of indices of mesh edges! i contained in? k and! i are the indices of the grid points constituting the endpoints of! i. From here it is a simple matter to derive the nite{dimensional analogs of our pointwise optimality conditions. In particular, if each? k corresponds to a single mesh edge and " 2 AD is a classical minimizer of " and U " its associated eigenvector, then there exists a such that (5:8) " k = ) h U " i k > ; " k = ) h U " i k < : These conditions are reminiscent of those that arise in Krein's problem of the optimal distribution of mass, see, e.g., Cox [7]. As such we apply the simple alternating search strategy of [7] to our minimum problem. More precisely, given (j) 2 AD,

12 2 STEVEN J. COX AND PAUL X. UHLIG (I) Compute U (j), the minimizer of U 7! R " (U; (j) ) subject to hmu; Ui =. (II) Compute (j+), the minimizer of 7! R " (U (j) ; ) subject to 2 AD. (III) If (j+) 6= (j) then set j = j + and go to (I). The implementation of (I) simply requires the solution of (5.3) with = (j). The optimality conditions (5.8) animate the implementation of (II). More precisely, we compute J fk : hu (j) i k < g, where is chosen in such a way that and then dene X k2j (j+) k = j? k j = jj; n if k 2 J otherwise. This completes our description of the minimization algorithm. With respect to the maximization problem, recalling that we have a smooth, concave function subject only to box and linear constraints, we may invoke any of a number of standard optimization packages. 6. Numerical Results. For the maximization of " we used the constr function found in Matlab's Optimization Toolbox. The assembly of (5.3) and the computation of " and U " was carried out by the pdeeig function found in Matlab's PDE toolbox. Given U " we coded the gradient computation, (5.7), ourselves. The details of our implementation are spelled out in [8]. We present here only the results of our computations for two representative domains. In the rst case we consider the drumhead whose boundary is the ellipse x y2 9 = 6 : Recalling the discussion at the close of x4 we expect the maximizer, ^ ", as "!, to coincide with n n ; the product of and the normalized normal derivative of the rst Dirichlet eigenfunction of the ellipse. For the purpose of illustration we have, in g., plotted the underlying ellipse, the contours of the associated, and the graph of its corresponding, with = =2. The eigenfunction was computed at the p = vertices of 9488 triangles. The boundary was partitioned into m = edges and the associated Dirichlet eigenvalue was 2:45. Next, we set " = r, let r range from to -6, and denote by ^ the maximizer returned by constr, on the grid quoted above using r the default stopping criteria. We measured the pointwise distance from ^ to via r R(r) k? ^ r k max k j k? ^ r k j and have recorded its graph in g. 3. That no improvement is seen for " <?3 is most likely due to the fact that our computed is itself only accurate to?2.

13 WHERE BEST TO HOLD A DRUM FAST R r Figure 2. The limiting maximal fastener,. Figure 3. k? ^ " k as "!. As a nonconvex example, we pursue the maximizer over the L{shaped region familiar to users of Matlab Figure 4. Maximal fastening of the L. It is well known, see, e.g., Fox, Henrici, and Moler [], that the gradient of the rst Dirichlet eigenfunction is not bounded in a neighborhood of the reentrant corner. As a result, we may not expect (4.) to hold along the entire boundary. In g. 4 we have plotted ^ ", the maximizer returned by constr along with the level sets of its corresponding eigenfunction. Working over a grid of p = vertices, triangles and m = 92 boundary segments with " =?3 and = =2 we found "( ^ " ) 9:59. Note that the level sets indeed resemble those of the rst Dirichlet eigenfunction and that ^ " behaves like a clipped version of its normal derivative. Finally, we wish to present numerical results for the minimization problem. As above, we concentrate on the ellipse and the L. With respect to the former we oer in gures 5 and 6, respectively, the initial iterate supplied to, and nal iterate delivered by, the alternating search minimization algorithm presented at the close of the previous section. The domain was approximated by 3374 triangles with p = 7288 vertices. Its boundary was partitioned into m = 2 edges. With = =2 and " = : the algorithm came to rest in 69 iterations. The eigenvalue, 6:68, of the initial iterate was diminished to 3:7. In both cases we have also plotted the contours of the associated eigenfunction..5.5

14 4 STEVEN J. COX AND PAUL X. UHLIG Figure 5. Initial iterate. Figure 6. Final iterate. The initial and nal iterates, along with the contours of their associated eigenfunctions, for the L{shaped drum are depicted in gures 7 and Figure 7. Initial iterate. Figure 8. Final iterate. In this case the domain was approximated by 8238 triangles with p = 9936 edges. Its boundary was partitioned into m = 632 edges. With = =2 and " = : the algorithm came to rest in 3 iterations and reduced the eigenvalue of the initial iterate, 4.8, to.88. We note that the nal iterate pulled the Dirichlet data away from the reentrant corner and wrapped it around the outer corner. The resulting eigenvalue is indeed less than.9, the eigenvalue of the L with Dirichlet data on the three legs above the diagonal x = y. REFERENCES [] C. Bandle, Isoperimetric Inequalities and Applications, Pitman, Boston, 98. [2] H. Bauer, Sur le prolongement des formes lineair positives dans un espace vectorial ordonne, C.R. Acad. Sci. Paris, 244 (957), pp. 289{292. [3] M.{H. Bossel, Membranes elastiquement liees inhomogenes ou sur une surface: Une nouvelle extension de theoreme isoperimetrique de Rayleigh-Faber-Krahn,. Angew. Math. Phys. 39 (988), pp. 733{742. [4] G. Buttazzo, Thin insulating layers: The optimization point of view, in Material Instabilities in Continuum Mechanics and Related Mathematical Problems, J.M. Ball, ed., Oxford University Press, Oxford, 988, pp. {9. [5] C.H. Chuang and G.J-W., Eigenvalue sensitivity analysis of planar frames with variable joint and support locations, AIAA J. 3 (992), pp. 238{247. [6] F. Clarke, Nonsmooth Analysis and Optimization, SIAM, Philadelphia, 99.

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