Eigencurves for linear elliptic equations

Rivas, Mauricio A.; Robinson, Stephen B.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 45.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

This paper provides results for variational eigencurves associated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a, b, m) of continuous symmetric bilinear forms on a real separable Hilbert space V . Geometric characterizations of eigencurves associated with (a, b, m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.

[1] G.A. Afrouzi and K.J. Brown, On principle eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions. Proc. Amer. Math. Soc. (1999) 125–130. | DOI | MR | Zbl

[2] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. MPS-SIAM, Philadelphia (2005). | MR | Zbl

[3] G. Auchmuty, Bases and comparison results for linear elliptic eigenproblems. J. Math. Anal. Appl. 390 (2012) 394–406. | DOI | MR | Zbl

[4] G. Auchmuty and Q. Han, Representation of solutions of Laplacian boundary value problems on exterior regions. Appl. Math. Optim. 69 (2014) 21–45. | DOI | MR | Zbl

[5] G. Auchmuty and M.A. Rivas, Laplacian eigenproblems on product regions and tensor products of Sobolev spaces. J. Math. Anal. Appl. 435 (2016) 842–859. | DOI | MR | Zbl

[6] C. Bandle and A. Wagner, Isoperimetric inequalities for the principle eigenvalue of a membrane and the energy of problems with Robin boundary conditions. J. Convex Anal. 22 (2014) 627–640. | MR | Zbl

[7] P. Binding and H. Volkmer, Eigencurves for two-parameter Sturm–Liouville equations. SIAM Rev. 38 (1996) 27–48. | DOI | MR | Zbl

[8] P. Blanchard and E. Brüning, Variational Methods in Mathematical Physics. Springer-Verlag, Berlin (1992). | DOI | MR | Zbl

[9] D. Daners and J.B. Kennedy, On the asymptotic behaviour of the eigenvalues of a Robin problem. Differ. Integ. Equ. 23 (2010) 659–669. | MR | Zbl

[10] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). | MR | Zbl

[11] M.A. Filinovskiy, On the eigenvalues of a Robin problem with a large parameter. Math. Bohem. 139 (2014) 341–352. | DOI | MR | Zbl

[12] T. Giorgi and R. Smits, Eigenvalue estimates and critical temperatures in zero fields for enhanced surface superconductivity. Z. Angew. Math. Phys. 58 (2007) 224–245. | DOI | MR | Zbl

[13] T. Kato, Perturbation Theory for Linear Operators, 2nd Edn. Springer-Verlag, New York (1976). | MR | Zbl

[14] B. Ko and K.J. Brown, The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems. Nonlin. Anal.: Theory Methods. Appl. 39 (2000) 587–597. | DOI | MR | Zbl

[15] A.A. Lacey, J.R. Ockendon and J. Sabina, Multidimensional reaction diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58 (1998) 1622–1647. | DOI | MR | Zbl

[16] N. Mavinga, Generalized eigenproblem and nonlinear elliptic equations with nonlinear boundary conditions. Proc. R. Soc. Edinburgh 142A (2012) 137–153. | DOI | MR | Zbl

[17] N. Mavinga and M.N. Nkashama, Steklov-Neumann eigenproblems and nolinear elliptic equations with nonlinear boundary conditions. J. Differ. Equ. 248 (2010) 1212–1229. | DOI | MR | Zbl