The application of isoperimetric inequalities for nonlinear eigenvalue problems

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1 The alication of isoerimetric inequalities for nonlinear eigenvalue roblems GABRIELLA BOGNAR Institute of Mathematics University of Miskolc 355 Miskolc-Egyetemvaros HUNGARY Abstract: - Our aim is to show the interlay between geometry analysis and alications of the theory of isoerimetric inequalities for some nonlinear roblems. Reviewing the isoerimetric inequalities valid on Minkowskian lane we show that we can get estimations of hysical quantities, namely, estimation on the first eigenvalue of nonlinear eigenvalue roblems, on the basis of easily accessible geometrical data. Key-Words: - nonlinear eigenvalue roblems, isoerimetric inequalities, Minkowskian geometry The classical isoerimetric inequality The classical isoerimetric inequality after which all such inequalities are named states that of all lane curves of given erimeter the circle encloses the largest area. This extremal roerty is exressed in the inequality: L 4 A, () where A denotes the area of the domain and L the length of its boundary curve, and where equality holds only for circles. This inequality was known already to the Greeks. Paus, in whose writings these results are reserved, attributes their discovery to Zenodorus. In their famous book Isoerimetric Inequalities in Mathematical Physics, Pólya and Szegő extended this notion to include inequalities for domain functionals, rovided that the equality sign is attained for some domain or in the limit as the domain degenerates [5]. Isoerimetric inequalities in broader sense There are several interesting and imortant geometrical and hysical quantities deending on the shae and size of a curve: -the length of its erimeter, the area included, -the moment of inertia, with resect to the centroid, of a homogeneous late bounded by the curve, -the torsional rigidity of an elastic beam the cross section of which is bounded by the given curve, -the rincial frequency of a membrane of which the given curve is the rim, -the electrostatic caacity of a late of the same shae and size, -and several other quantities. By the hel of the isoerimetric inequalities we estimate hysical quantities on the basis of easily accessible geometrical data. The study of isoerimetric inequalities in a broader sense began with the conjecture of St Venant in 856, that of all cylindrical beams of given cross-sectional area the circular beam has the highest torsional rigidity. In 877 Lord Rayleigh conjectured that of all vibrating elastic membranes of constant density and fixed area the circular membrane has the minimum rincial frequency. He gave some evidence to suort the conjecture. In 93 H. Poincaré made the conjecture that of all solids of given volume the shere has the minimum exterior electrostatic caacity. The roofs of these conjectures were given later. Around 93 G. Faber [7] and E. Krahn [] obtained indeendently the statement of Rayleigh. The roof were based on the introduction of a secial system of curvilinear coordinates. G. Szegő and G. Pólya gave another roof by using the Steiner symmetrization [5]. In recent literature the statement of the Rayleigh conjecture is usually referred to as the Faber-Krahn inequality. This roerty is exressed by the inequality j λ () A with equality only for the circle, and where j is the first ositive zero of the Bessel function of the first kind J ( x ), moreover A is the area of the domain. In 93 G. Szegő gave a rigorous roof of H. Poincaré s conjecture [6]. In 948 G. Pólya verified the conjecture of B. de St. Venant [4].

2 3 Geometrical inequalities The theory of isoerimetric inequalities is a subject of great diversity and comlexity. Our aim is to show the interlay between geometry analysis and alications for some nonlinear roblems. 3. The Bonnesen inequality Let curve c be given as follows y = x (3) R, < <. This is a central symmetric convex curve, which lays the same role as the circle. If =, then curve c is a circle with radius. The Minkowskian length of curve c defined by (3) is L x c ( ) = 4 = x= x P, (4) and the area of the domain bounded by this curve is where A( c ) = 4 x = P, (5) x= P = B,. (6) If =, then P =. In aer [5] G. D. Chakerian roved and alied the Bonnesen inequality in the Minkowskian lane for any convex n-gon (and consequently for any convex curve) L A P. (7) This inequality was roved by L. Fejes Tóth [8] for nonconvex curves in the Euclidean lane. The roof given in [8] can be generalized without difficulty to such Minkowskian geometry where the circle is any centrally symmetric convex curve. The Bonnesen inequality (7) is valid for non-convex curves in Minkowskian geometry [4]. If =, inequality (7) is reduced to the Bonnesen inequality valid on the Euclidean lane. 3. The isoerimetric inequality in the Minkowskian lane From the Bonnesen inequality (7) for a simly connected convex domain G. D. Chakerian [5] showed that the isoerimetric inequality in the Minkowskian metric for a simly connected convex domain has the form L 4PA. (8) Inequality (8) can be considered as the generalization of the classical isoerimetric inequality (). In (8) equality holds if and only if domain is bounded by curve c y = x. 4 Eigenvalue roblems We consider the following eigenvalue roblems: 4. The linear roblem (the roblem of a vibrating membrane) We consider a homogeneous membrane covering a region R. The deformations u ( x, y) normal to the lane has to satisfy the differential equation u λ u = in. (9) On the boundary of we have the Dirichlet boundary condition: u =, if the mebrane is fixed. The solutions u of roblem (9) under Dirichlet boundary condition are called eigenfunctions and the corresonding values of λ are eigenvalues. In [6] it is showed that there exist countably many number of distinct normalized eigenfunctions with associated eigenvalues to the eigenvalue roblem (9). For the eigenvalues λ j ( ) of the Dirichlet eigenvalue roblem of (9) the relation λ j ( ) holds when k. Every eigenvalue is ositive. Only in certain cases, the solutions of (9) can be calculated exlicitly. For examle, in the case of fixed membranes when is bounded by - rectangle, - circle, - circular segments, - triangles with angles: 3 3 3

3 In the case of the linear roblem (9) many aers were ublished on the estimation of the first eigenvalue. Such bounds are based on geometrical data of the domain. The Faber-Krahn inequality () gives a lower bound for the smallest (the first) eigenvalue. For the case of convex domains J. Hersch [9] roved the following bound on the first eigenvalue λ, 4 where is the radius of the greatest inscribed circle in. For a simly connected domain, E. Makai [] showed that there exists a constant C such that C < and 4 4 R. Osserman [] gave the bound, if λ k, if 4 For k-fold connected domains. C λ. k, k =, 4. The nonlinear roblems We seek eigenfunctions u j and corresonding eigenvalues λ j ( j =,,...) of the following nonlinear eigenvalue roblem Q = λ u u in where the nonlinear oerator Q u = R () Q is defined by u u y y for < <. u y If =, roblem () is equivalent to the linear roblem (9). The boundary condition corresonding to the Dirichlet roblem of () is u =. In [3] it is showed that there exist countably many number of distinct normalized eigenfunctions in, W ( ) with associated eigenvalues to the eigenvalue roblem (). For the eigenvalues λ j ( ) of the Dirichlet eigenvalue roblem of () the relation λ j ( ) holds when k. Every eigenvalue is ositive. Here the first eigenfunction has also many secial roerties. The first eigenfunction does not change sign and the corresonding eigenvalue, the first eigenvalue is simle [3]. We have showed (see []) that the Dirichlet roblem of () has solutions belonging to C ( ) C ( ) when is bounded by rectangle {( x y) : x a, y b} =,. For the nonlinear roblem with Dirichlet boundary condition the eigenvalues and corresonding eigenfunctions can be given as follows where ~ k l λ = k l a b k ~ l ~ uk, l = Ak, l S xs y, a b k, l =,,..., ~ =, sin A k, l = const. determined from the normalization of the eigenfunction, and function S is the solution of the differential equation under condition S S S S = S ( ) =, S ~ ( ) =. The function S is the generalized sine function, which lays the same role in case of nonlinear roblem () as the sine function in case of the linear roblem (9).

4 Isoerimetric inequalities are also useful in the derivation of exlicit a riori inequalities emloyed in the determination of a riori bounds in various tyes of initial or boundary value roblems. As an examle, we know that for domain with sufficiently smooth boundary the first eigenvalue in the fixed membrane roblem (with boundary condition u = ) admits the following characterization: du du dy dy λ = min. u W, ( ) u dy This characterization gives us a bound for λ, i.e., for, any v W ( ) dv λ. v dv dy dy dy The equality sign will always hold for some choice of v In [4] we gave a lower bound for the first eigenvalue of the nonlinear eigenvalue roblem (). By using geometrical data we get ( ) Ph λ () A where A is the area of R, P is defined in (6), and h is the first ositive zero of the generalized nonlinear Bessel function H ( x) satisfying the nonlinear ordinary differential equation (see[]) d with conditions and λ H x H H () = ( ) =. = In () equality holds if and only if domain is bounded by curve c. Inequality () is a generalization of the Faber-Krahn inequality for nonlinear eigenvalue roblems. Another lower bound can be given for the first eigenvalue of the nonlinear roblem () by using the method of Steiner symmetrization. We know that the eigenfunction associated to λ has the same sign in. If domain is a simly connected convex domain in R, then A σ λ ( ) () A where A is the area of R, is the radius of the greatest inscribed curve c of, and σ = P is the area of the region bounded by the greatest inscribed curve c of. In () equality holds if and only if domain R is bounded by curve c. Inequality () is the generalization of the estimation given by E. Makai for the linear eigenvalue roblem (9) in []. References: [] G. Bognár, On the solution of some nonlinear boundary value roblem, Proc. WCNA, August 9-6, 99. Tama, Walter de Gruyter, Berlin-New York, 956. [] G. Bognár, Numerical aroximations of some nonlinear roblems, Problems in Modern Mathematics (Edited by Nikos E. Mastorakis) World Scientific Engineering Society Press [3] G. Bognár, Existence theorem for eigenvalues of a nonlinear eigenvalue roblem, Communications on Alied Nonlinear Analysis, 4, 997, No., [4] G. Bognár, A lower bound for the smallest eigenvalue of some nonlinear ellitic eigenvalue roblem on convex domain in two dimensions, Alicable Analysis, 5, 993, No.-4, [5] G. D. Chakerian, The isoerimetric roblem in the Minkowskian lane, Amer. Math. Monthly, 67, 96,. -4. [6] R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol.I., New York, Interscience, 953. [7] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Flache und gleicher Sannung die kreisförmige den tiessten Grundton gibt, Sitz. ber. bayer. Akad. Wiss., 93,

5 [8] L. Fejes Tóth, Elementarer Beweis einer Isoerimetrischen Ungleichung, Acta Math. Acad. Scie. Hung. 95, [9] J. Hersch, Physical interretation and strengthening of M. H. Protter's method for vibrating nonhomogeneous membranes; its analogue for Schrödinger's equation, Pacific J. Math. 96, [] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94, 94, [] E. Makai, A lower estimation of the rincial frequencies of simly connected membranes, Acta Math. Acad. Sci. Hung., 6, 965, [] R. Osserman, Bonnesen-style isoerimetric inequalities, The Amer. Math. Monthly, 86, 979,. -9. [3] W. Pielichowski, A nonlinear eigenvalue roblem related to Gabriella Bognar s conjecture, Studia Scie. Math. Hung., 33, 997, [4] G. Pólya, Torsional rigidity, rincial frequency, electrostatic caacity and symmetrization, Quart. Al. Math., 6, 948, [5] G. Pólya, G. Szegő, Isoerimetric Inequalities in Mathematical Physics, Princeton Univ. Press, 95. [6] G. Szegő, Über einige neue Extremalaufgaben der Potential-theorie, Math. Z., 3, 93,