Int. J. Anal. Appl. (2022), 20:20 Payne-Sperb-Stakgold Type Inequality for a Wedge-Like Membrane Abir Sboui1,3,5,∗, Abdelhalim Hasnaoui2,4 1Department of Mathematics, Faculty of Arts and Science (TURAIF), Northern Border University, KSA 2Faculty of Arts and Science (RAFHA), Northern Border University, KSA 3Department of Mathematics, ISSATM, University of Carthage, Tunisia 4Department of Mathematics, FST, University of Tunis El Manar, Tunisia 5Laboratory of partial differential equations and applications (LR03ES04), Faculty of sciences of Tunis, University of Tunis El Manar, 1068 Tunis, Tunisia ∗Corresponding author: abir.sboui@nbu.edu.sa, abirsboui@yahoo.fr Abstract. For a bounded domain contained in a wedge, we give a new Payne-Sperb-Stakgold type inequality for the solution of a semi-linear equation. The result is isoperimetric in the sense that the sector is the unique extremal domain. 1. Introduction For a two-dimensional bounded domain D, Payne and Rayner proved [9,10] that the eigenfunction u of the Dirichlet Laplacian corresponding to the fondamental eigenvalue λ(D) satisfies the following inequality ∫ D u2da ≤ λ(D) 4π (∫ D u da )2 , (1.1) where da denotes the Lebesgue measure. Equality is achieved if, and only if, D is a disk. The impor- tance of this inequality is that it is a reverse Cauchy-Schwarz type inequality for the first eigenfunction . This inequality was extended to higher dimension by kohler Kohler-Jobin [5,6]. Her inequality states that Received: Feb. 23, 2022. 2010 Mathematics Subject Classification. 35P15, 45A12, 58E30. Key words and phrases. Payne-Sperb-Stakgold inequality; semi-linear equation; isoperimetric inequality. https://doi.org/10.28924/2291-8639-20-2022-20 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-20 2 Int. J. Anal. Appl. (2022), 20:20 ∫ D u2 da ≤ λd/2 2d Cd j d−2 d/2−1,1 (∫ D u da )2 (1.2) where D is a bounded domain in Rd, Cd denotes the volume of the unit ball in Rd, and jd/2−1,1 is the first positive zero of the Bessel function Jd/2−1. Using the comparison method due to Giorgio Talenti, Chiti [1] proved that(∫ D uq da )1 q ≤ K(p,q,d) λ d 2 ( 1 p −1 q ) (∫ D up da )1 p for q ≥ p > 0. (1.3) Here K(p,q,d) = (d Cd) 1 q −1 p j d ( 1 q −1 p ) d 2 −1,1 (∫ 1 0 rd−1+q(1− d 2 )J q d 2 −1 (jd 2 −1,1 r)dr )1 q (∫ 1 0 rd−1+p(1− d 2 )J p d 2 −1 (jd 2 −1,1 r)dr )1 p . Equality holds if and only if D is a ball. A more interesting inequality in the spirit of the above has been proved by Payne, Sperb and Stakgold [11] for the following nonlinear problem ∆ u + f (u) = 0 in Ω ⊂R2, (1.4) u > 0 in Ω ⊂R2, (1.5) u = 0 on ∂Ω, for a given continuous function f (t), with f (0) = 0. This includes Dirichlet eigenvalue problem for the Laplace operator when f (t) = λt. For this problem, the Payne-Rayner inequality takes the form(∫ Ω f (u) dx )2 ≥ 8π ∫ Ω F (u) dx (1.6) where F (u) = ∫u 0 f (t)dt. Finally, Mossino [7] prove a generalization of the latest inequality for the p-Laplacian and the case of equality was discussed by Kesavan and Pacella [4]. Our aims is to give a version of Payne-Sperb - Stakgold inequality for the case of wedge like domains. 2. Preliminary Tools and main result Before stating our result, we give some notation . Let α ≥ 1 and W be the wedge defined in polar coordinates (r,θ) by W = { (r,θ) ∣∣ r > 0, 0 < θ < π α } . (2.1) Whenever pertinent, the arc length element will be denoted by ds2 = dr2 + r2dθ2 while the element of area is denoted by da = rdrdθ, and we let v(r,θ) = rα sin αθ. (2.2) Then, v is a positive harmonic function in W which is zero on the boundary ∂W. Int. J. Anal. Appl. (2022), 20:20 3 We are interested in the solution u of the following quasi-linear problem: P1 :   ∆u + f (u v )v = 0 in D u > 0 in D u = 0 on ∂D, where D is a sufficiently smooth bounded domain completely contained in the wedge W and the g((r,θ),t) = f ( t v(r,θ) )v(r,θ) is locally Hölder continuous and satisfies the following hypotheses. (H1) There exists A ∈ L1(D) and C > 0 such that |g((r,θ),t)| ≤ A(r,θ) + C|t|p,∀((r,θ),t) ∈ D×R, where p > 0. (H2) For t > 0, we have g((r,θ),t) > 0. The role of hypothesis (H1)is to ensure that every weak solution of the problem (P1) is a C2-solution of (P1). Notice that, The problem(P1) includes the eigenvalue problem for the Laplace operator with Dirichlet boundary condition, when we take f (u v ) = λu v . Now, if we write the solution of (P1) as u = vw, then the problem above transforms to P2 :   −div(v2∇w) = f (w)v2 in D v > 0 in D v = 0 on ∂D∩W. The solution w may be interpreted as a solution of the nonlinear classical problem (P1) for the 4-dimensional domain symmetric about the x2-axis when α = 1 and for the 6-dimensional domain bi-axially symmetric about the x1-axis and the x2-axis when α = 2, see [8] and [2]. Now, we need to introduce some notations and definitions. Let µ denoted measure defined by dµ = v2da. Then, the weighted unidimensional decreasing rearrangement of the function w with respect to measure µ is the function w∗ : [0,µ(D)] → [0, +∞) defined by w∗(0) = sup w, w∗(ξ) = inf { t ≥ 0; mw (t) < ξ } , ∀ξ ∈ (0,µ(D)], where mw (t) = µ ({ (r,θ) ∈ D; w(r,θ) > t }) , ∀t ∈ [0, sup w]. (2.3) The main result is given in the following theorem. Theorem 2.1. Let D be a smooth bounded domain completely contained in the wedge. Assume that (H1) and (H2) are satisfied. Let F be the primitive of f such that F (0) = 0. Then the solution u of the problem (P1) satisfies the inequality 4(2α + 2)(2α + 1) ( π 2α(2α + 2) ) 1 α+1 ∫ µ(D) 0 ξ α α+1 F ( ( u v )∗(ξ) ) dξ ≤ ∫ D F ( u v )v2 da. 4 Int. J. Anal. Appl. (2022), 20:20 Equality holds if and only if D is a perfect sector SR. The proof of this inequality and the equality case will be discussed in the next section . 3. The weighted version of Payne-sperb-stackgold inequality To beginning, we introduce the space W (D,dµ) of measurable functions ϕ that possess weak gradient denoted by |∇ϕ| and satisfy the following conditions (i) ∫ D |∇ϕ|2dµ + ∫ D |ϕ|2dµ < +∞ (ii) There exists a sequence of functions ϕn ∈ C1(D) such that ϕn(r,θ) = 0 on ∂D∩W and lim n→+∞ ∫ D |∇(ϕ−ϕn)|2dµ + ∫ D |ϕ−ϕn|2dµ = 0. (3.1) Using the fact that v is harmonic and the divergence theorem , we see ∫ D |∇u|2da = ∫ D |∇(wv)|2da = ∫ D |∇w|2v2da = ∫ D |∇w|2dµ. Thus w satisfies the first condition (i). Since u is a smooth solution of the problem P1, then w is also smooth and by the boundary condition in P2,we conclude that w satisfies the second condition (ii). Then w is in the space W (D,dµ). We introduce now the function Φ(t) = ∫ Dt f (w)dµ. (3.2) Since w and w∗ are equimeaserable then we have Φ(t) = ∫ Dt f (w)dµ = ∫ m(t) 0 f (w∗)dξ. (3.3) To proceed further, we need to show that m(t) is absolutely continuous on (0,M). Indeed, assume that µ({w = t}) is positive. Recall that w ∈ W (D,dµ) and proceeding as in the proof of Stampacchia’s theorem [3] to conclude that ∇w = 0 almost everywhere on the set {w = t}. Substitute this into P2, we obtain f (w) = 0 on and so g((r,θ),u) = f (u v )v = 0 on this set, which contradicts the hypothesis H2. Thus, w is continuous on (0,M) and By the fact that w∗ is the left inverse of m(t),we get Φ′(t) = f (w∗(m(t))m′(t) = f (t)m′(t). (3.4) By a weak solution to the problem P2 we mean a function w belong to W (D,dµ) and satisfies the equality ∫ D ∇w ·∇ϕdµ = ∫ D f (w)ϕdµ, (3.5) for every ϕ in C1(D), such that ϕ = 0 on ∂D∩W. Choose the test function ϕ defined by ϕ(r,θ) = { ( w(r,θ) − t ) , if w(r,θ) > t 0, otherwise , (3.6) Int. J. Anal. Appl. (2022), 20:20 5 where 0 ≤ t < M. Plugging (3.6) into (3.5) we get∫ w>t |∇w|2dµ = ∫ w>t f (w)(w − t)dµ. (3.7) Then, for � > 0, we have 1 � (∫ w>t |∇w|2dµ− ∫ w>t+� |∇w|2dµ ) = ∫ w>t f (w)dµ+ ∫ tt |∇w|2dµ− ∫ w>t+� |∇w|2dµ ) = ∫ w>t f (w)dµ. (3.9) The same computation for −� gives the same value of the limit. Thus − d dt ∫ w>t |∇w|2dµ = ∫ w>t f (w)dµ. (3.10) Now, applying the Cauchy Schwarz inequality( 1 � ∫ tt |∇w|dµ )2 ≤−m′(t)Φ(t). (3.12) From the coarea formula, we have − d dt ∫ w>t |∇w|dµ = ∫ ∂{w>t} v2ds. (3.13) Then, an application of the Payne-Weinberger isoperimetric inequality for the wedge-like membrane [12] leads to ( π 2α )2 ( 4α(α + 1) π m(t) )2α+1 α+1 ≤ (∫ ∂{w>t} v2ds )2 ≤−m′(t)Φ(t). (3.14) By appealing to (3.13), we obtain ( π 2α ) 1 α+1 (2α + 2) 2α+1 α+1 (m(t)) 2α+1 α+1 f (t) ≤−Φ′(t)Φ(t). (3.15) Integrating both sides of the last relation from 0 to M, then we have ( π 2α ) 1 α+1 (2α + 2) 2α+1 α+1 ∫ M 0 (m(t)) 2α+1 α+1 f (t) ≤ 1 2 Φ2(0), (3.16) 6 Int. J. Anal. Appl. (2022), 20:20 since Φ(M) = 0. But on the left hand side we have ∫ M 0 (m(t)) 2α+1 α+1 f (t)dt = ∫ M 0 2α + 1 α + 1 f (t) ∫ m(t) 0 ξ α α+1 dξdt (3.17) = 2α + 1 α + 1 ∫ M 0 f (t) ∫ µ(D) 0 ξ α α+1 χ{w∗>t}(ξ)dξdt = 2α + 1 α + 1 ∫ µ(D) 0 ∫ w∗(ξ) 0 f (t)ξ α α+1 dtdξ = 2α + 1 α + 1 ∫ µ(D) 0 F (w∗(ξ))ξ α α+1 dtdξ. Substituting the last result into (3.16), the desired inequality in Theorem 2.1 follows. Moreover, if equality is achieved in Theorem 2.1, then obviously inequality (3.15) reduces to equality. Since Φ′(t) = f (t)m′(t) and f (t) > 0, then equality in (3.15) implies equality in (3.14) and so Payne- Weinberger Lemma [12] implies that almost all level sets Dt are concentric sectors with fixed angle π α . Since D = {w > 0} is the increasing union of such sectors then D is a sector as well. Acknowledgement: The authors gratefully acknowledge the approval and the support of this research study by the grant number 7912-SAT-2018-3-9-F from the Deanship of Scientific Research at Northern Border University, Arar, KSA. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] G. Chiti, An Isoperimetric Inequality for the Eigenfunctions of Linear Second Order Elliptic Operators, Boll. Unione Mat. Ital., VI. Ser., A. 1 (1982), 145-151. [2] A. Hasnaoui, L. Hermi, Isoperimetric Inequalities for a Wedge-Like Membrane, Ann. Henri Poincaré. 15 (2014), 369–406. https://doi.org/10.1007/s00023-013-0243-y. [3] S. Kesavan, Topics in Functional Analysis and Applications, New Age International, 1989. [4] S. Kesavan, P. Filomena, Symmetry of Positive Solutions of a Quasilinear Elliptic Equation via Isoperimetric In- equalities, Appl. Anal. 54 (1994), 27–37. https://doi.org/10.1080/00036819408840266. [5] M.-T. Kohler-Jobin, Sur la première fonction propre d’une membrane: une extension àN dimensions de l’inégalité isopérimétrique de Payne-Rayner, J. Appl. Math. Phys. (ZAMP). 28 (1977), 1137–1140. https://doi.org/10. 1007/BF01601680. [6] M.-T. Kohler-Jobin, Isoperimetric Monotonicity and Isoperimetric Inequalities of Payne-Rayner Type for the First Eigenfunction of the Helmholtz Problem, Z. Angew. Math. Phys. 32 (1981), 625–646. https://doi.org/10.1007/ BF00946975. [7] J. Mossino, A Generalization of the Payne-Rayner Isoperimetric Inequality, Boll. Unione Mat. Ital., VI. Ser., A. 2 (1983), 335–342. https://doi.org/10.1007/s00023-013-0243-y https://doi.org/10.1080/00036819408840266 https://doi.org/10.1007/BF01601680 https://doi.org/10.1007/BF01601680 https://doi.org/10.1007/BF00946975 https://doi.org/10.1007/BF00946975 Int. J. Anal. Appl. (2022), 20:20 7 [8] L.E. Payne, Isoperimetric Inequalities for Eigenvalue and Their Applications, Autovalori e autosoluzioni: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Chieti, Italy, 1962, G. Fichera (ed.), C.I.M.E. Summer Schools, 27 (1962), 1-58. [9] L.E. Payne, M.E. Rayner, An Isoperimetric Inequality for the First Eigenfunction in the Fixed Membrane Problem, J. Appl. Math. Phys. (ZAMP). 23 (1972), 13–15. https://doi.org/10.1007/BF01593198. [10] L.E. Payne, M.E. Rayner, Some Isoperimetric Norm Bounds for Solutions of the Helmholtz Equation, J. Appl. Math. Phys. (ZAMP). 24 (1973), 105–110. https://doi.org/10.1007/BF01594001. [11] L.E. Payne, R. Sperb, I. Stakgold, On Hopf Type Maximum Principles for Convex Domains, Nonlinear Anal., Theory Methods Appl. 1 (1977), 547–559. https://doi.org/10.1016/0362-546X(77)90016-5. [12] L.E. Payne, H.F. Weinberger, A Faber-Krahn Inequality for Wedge-Like Membranes, J. Math. Phys. 39 (1960), 182–188. https://doi.org/10.1002/sapm1960391182. https://doi.org/10.1007/BF01593198 https://doi.org/10.1007/BF01594001 https://doi.org/10.1016/0362-546X(77)90016-5 https://doi.org/10.1002/sapm1960391182 1. Introduction 2. Preliminary Tools and main result 3. The weighted version of Payne-sperb-stackgold inequality References