## Numerical approximation of the first eigenpair of the $$p$$-Laplacian using finite elements and the penalty method.(English)Zbl 0884.65103

For $$1<p <\infty$$ and for bounded convex domains $$\Omega$$, the authors are interested in the first eigenvalue $$\lambda_1$$ of $\text{div} \bigl(|\nabla |^{p-2} \nabla u\bigr) =\lambda |u|^{p-2} u\quad \text{in } \Omega, \quad u=0 \quad \text{on } \partial \Omega,$ where $$|\cdot |$$ denotes the norm in $$L^2(\Omega)$$. The eigenvalue $$\lambda_1$$ is simple, and the eigenfunction $$u_1$$ of one sign. $$\lambda_1$$ is the minimum of $$F(u)= \int_\Omega |\nabla u|^p dx$$ over $$W_0^{1,p} (\Omega)$$ under the constraint $$G(u) =1- |u|^p =0$$. To evaluate $$\lambda_1$$ and $$u_1$$, the sequence of penalty functionals $$L(u,\gamma_k) =F(u) +\gamma_k [G(u)]^2$$ is introduced, and $$u_1$$ is approximated by finite elements. Convergence is proved, and numerical results are presented for various values of $$p$$ and for $$\Omega= (0,1)$$, $$\Omega= (0,1) \times(0,1)$$, and $$\Omega =$$ unit ball in $$\mathbb{R}^2$$.

### MSC:

 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs 35J70 Degenerate elliptic equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 49R50 Variational methods for eigenvalues of operators (MSC2000)

Mathematica
Full Text:

### References:

  Adams R. A., Sobolev Spaces (1978)  Bhattacharya T., ” Ann. Acad. Sci Fenn. #I 14 pp 325– (1989)  Blanchard P., Variational Methods in Mathematical Physics (1992) · Zbl 0756.49023  Chavel I., Eigenvalues in Riemannian Geometry (1984)  Cheeger J., Problems in Analysis pp 195– (1970)  DOI: 10.1002/cpa.3160340404 · Zbl 0481.35003  DOI: 10.1007/BF02566216 · Zbl 0484.53034  Del Pino M., J. Differential Equations 80 pp 213– (1995)  Fletcher R., Practical Methods of Optimization:Unconstrained Optimization 1 (1980)  DOI: 10.1080/00036819108840034 · Zbl 0724.35015  DOI: 10.1016/0022-0396(86)90121-X · Zbl 0549.35102  DOI: 10.1080/00036817808839195 · Zbl 0383.35053  Lindqvist P., Fall School in Analysis Lectures 68 pp 33– (1995)  Manasevich R., Differential and Integral Equations 8 pp 1– (1989)  Pólya G., Isoperimetric Inequalities in Mathematical Physics (1951) · Zbl 0044.38301  Struwe M., Variational Methods (Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems) (1990) · Zbl 0746.49010  DOI: 10.1215/S0012-7094-79-04620-9 · Zbl 0418.35068  DOI: 10.1080/01630569108816455 · Zbl 0754.65082  Wolfram S., Mathematica. A System for Doing Mathematics by Computer (1993) · Zbl 0671.65002  Zeidler E., Nonlinear Functional Analysis and its Applications:Linear Monotone Operators 11 (1990)  Zeidler E., Nonlinear Functional Analysis and its Applications Variational Methods and Optimization (1985) · Zbl 0583.47051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.