# Extremum Problems for Eigenvalues of Elliptic Operators

@inproceedings{Henrot2006ExtremumPF, title={Extremum Problems for Eigenvalues of Elliptic Operators}, author={Antoine Henrot}, year={2006} }

Eigenvalues of elliptic operators.- Tools.- The first eigenvalue of the Laplacian-Dirichlet.- The second eigenvalue of the Laplacian-Dirichlet.- The other Dirichlet eigenvalues.- Functions of Dirichlet eigenvalues.- Other boundary conditions for the Laplacian.- Eigenvalues of Schrodinger operators.- Non-homogeneous strings and membranes.- Optimal conductivity.- The bi-Laplacian operator.

## 787 Citations

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Using three different notions of the generalized principal eigenvalue of linear second‐order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of…

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In this paper, we analyze an eigenvalue problem for a quasi-linear elliptic operators involving Dirichlet boundary condition in an open smooth bounded set of R N . We investigate a bifurcation…

### Eigenvalues of elliptic operators with density

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We consider eigenvalue problems for elliptic operators of arbitrary order 2m subject to Neumann boundary conditions on bounded domains of the Euclidean N-dimensional space. We study the dependence of…

### Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

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A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the…

### On the principal frequency of non-homogeneous membranes

- Mathematics
- 2023

. We obtained estimates for ﬁrst eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in…

### Domain dependence of eigenvalues of elliptic type operators

- Mathematics
- 2012

The dependence on the domain for the Dirichlet eigenvalues of elliptic operators considered in bounded domains is studied. The proximity of domains is measured by a norm of the difference of two…

### Domain dependence of eigenvalues of elliptic type operators

- MathematicsMathematische Annalen
- 2013

The dependence on the domain for the Dirichlet eigenvalues of elliptic operators considered in bounded domains is studied. The proximity of domains is measured by a norm of the difference of two…

### Eigenvalues of poly-harmonic operators on variable domains

- Mathematics
- 2012

We consider a class of eigenvalue problems for poly-harmonic oper- ators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric…

### Estimates for Variation of the First Dirichlet Eigenvalue of the Laplace Operator

- MathematicsJournal of Mathematical Sciences
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We obtain an estimate for variation of the first eigenvalue of the Laplace operator with the Dirichlet boundary condition in a bounded domain. The method is based on the use of superposition…

### Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator

- Mathematics
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We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and…

## References

SHOWING 1-3 OF 3 REFERENCES

### Generalized isoperimetric inequalities

- Mathematics
- 1973

Continuing the development of a previous paper on generalized isoperimetric inequalities (i.e., rearrangement inequalities for Green's functions), we extend the theory to the case of Green's…

### The theory of sound

- Engineering
- 1894

Volume 1: Preface 1. Sound due to vibrations 2. Composition of harmonic motions of like period 3. Systems with one degree of freedom 4. Generalized co-ordinates 5. Cases in which the three functions,…

### Isoperimetric inequalities in mathematical physics

- Physics, Mathematics
- 1951

The description for this book, Isoperimetric Inequalities in Mathematical Physics. (AM-27), will be forthcoming.