# On the P\'olya conjecture for circular sectors and for balls

@inproceedings{Filonov2022OnTP, title={On the P\'olya conjecture for circular sectors and for balls}, author={Nikolai Filonov}, year={2022} }

In 1954, G. P´olya conjectured that the counting function N (Ω , Λ) of the eigenvalues of the Laplace operator of the Dirichlet (resp. Neumann) boundary value problem in a bounded set Ω ⊂ R d is lesser (resp. greater) than (2 π ) − d ω d | Ω | Λ d/ 2 . Here Λ is the spectral parameter, and ω d is the volume of the unit ball. We prove this conjecture for both Dirichlet and Neumann boundary problems for any circular sector, and for the Dirichlet problem for a ball of arbitrary dimension. We…

## One Citation

### A note on domain monotonicity for the Neumann eigenvalues of the Laplacian

- Mathematics
- 2022

. Given a convex domain and its convex sub-domain we prove a variant of domain monotonicity for the Neumann eigenvalues of the Laplacian. As an application of our method we also obtain an upper bound…

## References

SHOWING 1-10 OF 16 REFERENCES

### P\'olya's conjecture for the disk: a computer-assisted proof

- Mathematics
- 2022

The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from…

### Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces

- Mathematics
- 1997

Abstract We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L 2 ( Ω ) (or L 2 ( Ω , C m…

### Bounds for zeros of some special functions

- Mathematics
- 1970

For n> 1 let bn and c. be zeros (ordered by in- creasing values) of u(x) and v(x), respectively, which are non- trivial solutions of u"+p(x)u=O and v"+q(x)v=O with contin- uous p(x) and q(x). It is…

### On the remainder in the Weyl formula for the Euclidean disk

- Mathematics
- 2010

We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3).

### On the Schrödinger equation and the eigenvalue problem

- Mathematics
- 1983

AbstractIf λk is thekth eigenvalue for the Dirichlet boundary problem on a bounded domain in ℝn, H. Weyl's asymptotic formula asserts that
$$\lambda _k \sim C_n \left( {\frac{k}{{V(D)}}}…

### Bessel phase functions: calculation and application

- MathematicsNumerische Mathematik
- 2017

Some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory are given and a bound on the error of the first term of this asymPTotic expansion is found.