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Bulk-boundary eigenvalues for Bilaplacian problems

• *Corresponding author: María del Mar González

Dedicated to Prof. Juan Luis V´azquez for his 75th birthday

• We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.

Mathematics Subject Classification: 35A09, 35B30, 35G05, 35P15.

 Citation:

• Figure 1.  Function $\Psi(\lambda)$

Figure 2.  Comparison between function $\frac{s J_{\ell+s}(r)}{J'_{\ell+s}(r)}$ and its limit, a cotangent function scaled and displaced. The black line is the identity function

Figure 3.  First four eigenvalues ($k = 1,2,3,4$) for $\ell = 0,1,2,3$ and different values of $\gamma$ and $a$ (in dimension $n = 2$)

Figure 4.  $\det W_\ell(\lambda)$ as a function of $\lambda$ for $n = 2, \gamma = 0.4,\ell = 0,1,2$ for the cases $a = 0.19$ and $a = 0.21$

Figure 5.  First four eigenvalues ($k = 1,2,3,4$) for $\ell = 0,1,2,3$ and different values of $\gamma$ and $a$ ($n = 4$)

•  [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, US Department of Commerce, (National Bureau of Standards Applied Mathematics series 55, 1965), (1965). [2] V. Adolfsson, $L^2$-integrability of second-order derivatives for Poisson's equation in nonsmooth domains, Math. Scand., 70 (1992), 146-160.  doi: 10.7146/math.scand.a-12391. [3] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math., 12 (1959), 623-727.  doi: 10.1002/cpa.3160120405. [4] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104. [5] E. Almansi, Sull'integrazione dell'equazione differenziale $\Delta^{2}\Delta^{2}u = 0$, Rendiconti, (Reale Accademia dei Lincei. Classe de scienze fisiche, matematiche e naturale), (1899) vol III, 104–107. [6] E. Almansi, Sull'integrazione dell'equazione differenziale $\Delta^{2n} = 0$, Annali di Matematica, Serie III, 2 (1899) 1–51. doi: 10.1007/BF02419286. [7] B. Bogosel, The Steklov spectrum on moving domains, Appl. Math. Optim., 75 (2017), 1-25.  doi: 10.1007/s00245-015-9321-5. [8] D. Bucur, A. Ferrero and F. Gazzola, On the first eigenvalue of a fourth order Steklov problem, Calc. Var. Partial Differential Equations, 35 (2009), 103-131.  doi: 10.1007/s00526-008-0199-9. [9] D. Bucur and F. Gazzola, The first biharmonic Steklov eigenvalue: Positivity preserving and shape optimization, Milan J. Math., 79 (2011), 247-258.  doi: 10.1007/s00032-011-0143-x. [10] D. Buoso, Analyticity and Criticality Results for The Eigenvalues of The Biharmonic Operator, Geometric Properties for Parabolic and Elliptic PDE's, 65–85, Springer Proc. Math. Stat., 176, Springer, [Cham], 2016. doi: 10.1007/978-3-319-41538-3_5. [11] D. Buoso, L. M. Chasman and L. Provenzano, On the stability of some isoperimetric inequalities for the fundamental tones of free plates, J. Spectr. Theory, 8 (2018), 843-869.  doi: 10.4171/JST/214. [12] D. Buoso and J. B. Kennedy, The Bilaplacian with Robin boundary conditions, SIAM J. Math. Anal., 54 (2022), 36-78.  doi: 10.1137/20M1363984. [13] D. Buoso and E. Parini, The buckling eigenvalue problem in the annulus, Commun. Contemp. Math., 23 (2021), Paper No. 2050044, 19 pp. doi: 10.1142/S0219199720500443. [14] D. Buoso and L. Provenzano, A few shape optimization results for a biharmonic Steklov problem, J. Differential Equations, 259 (2015), 1778-1818.  doi: 10.1016/j.jde.2015.03.013. [15] V. I. Burenkov, Sobolev Spaces on Domains, Teubner-Texte zur Mathematik, 137. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. doi: 10.1007/978-3-663-11374-4. [16] V. I. Burenkov, P. D. Lamberti and M. Lanza de Cristoforis, Spectral stability of nonnegative selfadjoint operators, (Russian) Sovrem. Mat. Fundam. Napravl., 15 (2006), 76–111; translation in J. Math. Sci. (N.Y.) 149 (2008), 1417–1452. doi: 10.1007/s10958-008-0074-4. [17] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [18] J. S. Case, Boundary operators associated with the Paneitz operator, Indiana Univ. Math. J., 67 (2018), 293-327.  doi: 10.1512/iumj.2018.67.6223. [19] S.-Y. A. Chang and J. Qing, The zeta functional determinants on manifolds with boundary. I. The formula, J. Funct. Anal., 147 (1997), 327-362.  doi: 10.1006/jfan.1996.3059. [20] S.-Y. A. Chang and J. Qing, The zeta functional determinants on manifolds with boundary. II. Extremal metrics and compactness of isospectral set, J. Funct. Anal., 147 (1997), 363-399.  doi: 10.1006/jfan.1996.3060. [21] S. Y. A. Chang and R. A. Yang, On a class of non-local operators in conformal geometry, Chin. Ann. Math. Ser. B, 38 (2017), 215-234.  doi: 10.1007/s11401-016-1068-z. [22] L. M. Chasman, An isoperimetric inequality for fundamental tones of free plates, Comm. Math. Phys., 303 (2011), 421-449.  doi: 10.1007/s00220-010-1171-z. [23] L. M. Chasman, Vibrational modes of circular free plates under tension, Appl. Anal., 90 (2011), 1877-1895.  doi: 10.1080/00036811.2010.534727. [24] C. V. Coffman and R. J. Duffin, On the fundamental eigenfunctions of a clamped punctured disk, Adv. in Appl. Math., 13 (1992), 142-151.  doi: 10.1016/0196-8858(92)90006-I. [25] C. V. Coffman, R. J. Duffin and D. H. Shaffer, The fundamental mode of vibration of a clamped annular plate is not of one sign, Constructive Approaches to Mathematical Models (Proc. Conf. in honor of R. J. Duffin, Pittsburgh, Pa., 1978), pp. 267-277, Academic Press, New York-London-Toronto, Ont., 1979. [26] R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. [27] A. Ferrero, F. Gazzola and T. Weth, On a fourth order Steklov eigenvalue problem, Analysis (Munich), 25 (2005), 315-332.  doi: 10.1524/anly.2005.25.4.315. [28] G. Fichera, Su un principio di dualità per talune formole di maggiorazione relative alle equazioni differenziali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 19 (1955), 411-418. [29] G. Françcois, Spectral asymptotics stemming from parabolic equations under dynamical boundary conditions, Asymptot. Anal., 46 (2006), 43-52. [30] A. Fraser and R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math., 226 (2011), 4011-4030.  doi: 10.1016/j.aim.2010.11.007. [31] H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach, SIAM J. Math. Anal., 52 (2020), 340-369.  doi: 10.1137/19M1258840. [32] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Springer Science & Business Media, 2010. doi: 10.1007/978-3-642-12245-3. [33] F. Gazzola and G. Sweers, On positivity for the biharmonic operator under Steklov boundary conditions, Arch. Ration. Mech. Anal., 188 (2008), 399-427.  doi: 10.1007/s00205-007-0090-4. [34] A. Girouard, A. Henrot and J. Lagacé, From Steklov to Neumann via homogenisation, Arch. Ration. Mech. Anal., 239 (2021), 981-1023.  doi: 10.1007/s00205-020-01588-2. [35] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn-Hilliard model in a domain with non permeable walls, Physica D: Nonlinear Phenomena, 240 (2011), 754-766.  doi: 10.1016/j.physd.2010.12.007. [36] M. González and M. Sáez, Eigenvalue Bounds for The Paneitz Operator and Its Associated Third-Order Boundary Operator on Locally Conformally Flat Manifolds, To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. [37] J. K. Hale, Eigenvalues and perturbed domains, Ten Mathematical Essays on Approximation in Analysis and Topology, 95–123, Elsevier B. V., Amsterdam, 2005. doi: 10.1016/B978-044451861-3/50003-3. [38] D. S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.), 4 (1981), 203-207.  doi: 10.1090/S0273-0979-1981-14884-9. [39] P. Knopf and C. Liu, On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems, Interfaces and Free Boundaries, 23 (2021), 507-533.  doi: 10.4171/IFB/463. [40] P. Knopf and A. Signori, On the nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and boundary penalization, J. Differential Equations, 280 (2021), 236-291.  doi: 10.1016/j.jde.2021.01.012. [41] P. Knopf, K. F. Lam, C. Liu and S. Metzger, Phase-field dynamics with transfer of materials: The Cahn-Hillard equation with reaction rate dependent dynamic boundary conditions, ESAIM Math. Model. Numer. Anal., 55 (2021), 229-282.  doi: 10.1051/m2an/2020090. [42] J. R. Kuttler and V. G. Sigillito, Inequalities for membrane and Stekloff eigenvalues, J. Math. Anal. Appl., 23 (1968), 148-160.  doi: 10.1016/0022-247X(68)90123-6. [43] P. D. Lamberti and L. Provenzano, On the explicit representation of the trace space $H^{\frac{3}{2}}$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems, Revista Matemática Complutense, 35 (2022), 53-88.  doi: 10.1007/s13163-021-00385-z. [44] G. Liu, On asymptotic properties of biharmonic Steklov eigenvalues, J. Differential Equations, 261 (2016), 4729-4757.  doi: 10.1016/j.jde.2016.07.004. [45] C. Liu and H. Wu, An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: Model derivation and mathematical analysis, Arch. Ration. Mech. Anal., 233 (2019), 167-247.  doi: 10.1007/s00205-019-01356-x. [46] V. G. Maz'ya and S. V. Poborchi, Differentiable Functions on Bad Domains, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. [47] I. Mitrea and M. Mitrea, Multi-Layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains, Lecture Notes in Mathematics, 2063, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32666-0. [48] F. Olver, D. Lozier, R. Boisvert and C. Clark, et al., NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. [49] J. von Below and G. Françcois, Spectral asymptotics for the Laplacian under an eigenvalue dependent boundary condition, Bull. Belg. Math. Soc. Simon Stevin, 12 (2005), 505-519.  doi: 10.36045/bbms/1133793338. [50] C. Xia and Q. Wang, Eigenvalues of the Wentzell-Laplace operator and of the fourth order Steklov problems, J. Differential Equations, 264 (2018), 6486-6506.  doi: 10.1016/j.jde.2018.01.041. [51] W. P. Ziemer, Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation, Graduate Texts in Mathematics, 120, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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