Research related information

 email:       kroegerpawel@gmail.com

Publications: scholar.google   MathSciNet   Scopus   zbMATH   orcid.org

The best constant in the Poincaré inequality on a compact Riemannian manifold and related topics

An estimate under a lower bound on the Ricci curvature and an upper bound for the diameter was obtained in P. Kröger "On the spectral gap for compact manifolds", 

J. Differ. Geom. 36, 315--330 (1992), Spectral gap compact manifolds.pdf . The final publication is available at proyecteuclid.org .

That estimate is sharp for all manifolds of a given dimension that satisfy the above bounds. The proof is based on the maximum principle technique. 

The main problem in earlier works by S. T. Yau, P. Li and S. T. Yau, J. Q. Zhong and H. C. Yang was that the ranges of the eigenfunctions are not symmetric with respect 

to the origin. That motivates our paper "On the ranges of eigenfunctions on compact manifolds", Bull. London Math. Soc. 30, 651--656 (1998)

Ranges of Eigenfunctions.pdf   Ranges of Eigenfunctions2.pdf where sharp bounds for the quotients max u/(-min u) for eigenfunctions u are obtained.

Estimates for sums of eigenvalues of the Laplacian

Our estimates are based on the variational characterization of eigenvalues as applied by P. Li and S. T. Yau to the task of

estimating Dirichlet eigenvalues. We applied similar techniques to Neumann eigenvalues in P. Kröger "Upper bounds for

the Neumann eigenvalues on a bounded domain in Euclidean space", J. Funct. Anal. 106, 353--357 (1992)

https://doi.org/10.1016/0022-1236(92)90052-K   Upper bounds Neumann eigenvalues.pdf .

Our aim in P. Kröger "Estimates for sums of eigenvalues of the Laplacian", J. Funct. Anal. 126, 217--227 (1994)

https://doi.org/10.1006/jfan.1994.1146   Sums of eigenvalues.pdf  was to complement the previous bounds by upper bounds

for Dirichlet eigenvalues and lower bounds for Neumann eigenvalues.

Upper bounds for Neumann eigenvalues of convex domains in Euclideans space

We improve the bounds obtained earlier by S. Y. Cheng to sharp bounds for convex domains with given diameter in P. Kröger

"On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space", Proc. Amer. Math. Soc. 127,

1665--1669 (1999) Upper bounds Neumann eigenvalues convex.pdf .

A second-order differential inequality on area growth

We extend Günther's volume comparison theorem in order to obtain a second-order differential inequality.

Günther's theorem can be obtained from our theorem by integration.

P. Kröger "An extension of Günther's  volume comparison theorem",  Math. Ann. 329, 593--596 (2004)

The final publication is available at Springer via: https://doi.org/10.1007/s00208-004-0520-7  Günther's Volume Comparison.pdf.

Gradient estimates for solutions of the Schrödinger equation and the heat equation

Both articles are joint work with R. Bañuelos (Purdue) and are based on the maximum principle techique: R. Bañuelos, P. Kröger

"Gradient estimates for the ground state Schrödinger eigenfunction and applications", Comm. Math. Phys. 224, 545--550 (2001) with link to final

publication Springer https://doi.org/10.1007/s002200100551 Gradient estimates.pdf and "Isometric-type bounds for solutions of the heat equation",

Indiana Univ. Math. J. 46, 83--91 (1997) with link to Indiana Univ. Math. J.: Isoperimetric Bounds.

Potential theory, regularity of solutions of elliptic and parabolic equations

The papers P. Kröger "Regular Boundary Points and Exit Distributions for Parabolic Differential Operators", Potential Analysis 49, 203--207 (2018) with

Springer link to .pdf: Regular Boundary Points.pdf , P. Kröger "A counterexample in parabolic potential theory", Mathematika 42, 392--396 (1995)

Counterexample parabolic potential.pdf and P. Kröger "Harmonic spaces associated with parabolic and elliptic differential operators", Math. Ann. (1989) 

available at Springer via: https://doi.org/10.1007/BF01455064 Springer link: Harmonic Spaces Parabolic Elliptic Operators deal with problems from potential theory. 

The article with K.-Th. Sturm (Bonn) is concerned with Hölder continuity of solutions of the Schrödinger equation. It turns out that

quotients of solutions are more regular that the solutions themselves: P. Kröger, K.-Th. Sturm "Hölder continuity of normalized solutions of Schrödinger equations", 

Math. Ann. 297, 663--670 (1993) and final publication available at Springer via: https://doi.org/10.1007/BF01459522  Hölder continuity Schrödinger.pdf.

The articles P. Kröger, "Regularity conditions on parabolic measures", Ark. Mat. 32, 373--391 (1994) https://doi.org/10.1007/BF02559577  

Regularity parabolic measures.pdf  and P. Kröger, "A counterexample for L1-estimates for parabolic differential equations", Z. Anal. Anwend. 11, 401--406 (1992) 

Counterexample L^1 estimates.pdf  deal with singular solutions.

Main Papers (with links)

Regular Boundary Points and Exit Distributions for Parabolic Differential Operators. Potential Analysis 49, 203--207 (2018)

Springer link to .pdf: Regular Boundary Points

An extension of Günther's  volume comparison theorem. Math. Ann. 329, 593--596 (2004)

The final publication is available at Springer via:  https://doi.org/10.1007/s00208-004-0520-7   Günther's Volume Comparison.pdf  

(with R. Bañuelos) Gradient estimates for the ground state Schrödinger eigenfunction and applications. 

Comm. Math. Phys. 224, 545--550 (2001) The final publication is available at Springer via: https://doi.org/10.1007/s002200100551   Gradient estimates.pdf

(with E. Harrell and K. Kurata) On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue.

SIAM J. Math. Anal. 33, 240--259 (2001) https://doi.org/10.1137/S0036141099357574   Placement Obstacle Eigenvalue.pdf

On upper bounds for high order Neumann eigenvalues of convex domains in Euclidean space.

Proc. Amer. Math. Soc. 127, 1665--1669 (1999) https://doi.org/10.1090/S0002-9939-99-04804-2   Upper bounds Neumann eigenvalues convex.pdf

On the ranges of eigenfunctions on compact manifolds. Bull. London Math. Soc. 30, 651--656 (1998) Ranges of Eigenfunctions.pdf  Ranges of Eigenfunctions2.pdf

On explicit bounds for the spectral gap on compact manifolds. Soochow J. Math. 23, 339--344 (1997)  Explicit bounds spectral gap.pdf

(with Bañuelos, R.) Isometric-type bounds for solutions of the heat equation. Indiana Univ. Math. J. 46, 83--91 (1997)

Link to Indiana Univ. Math. J.: Isoperimetric Bounds

On the ground state eigenfunction of a convex domain in Euclidean space. Potential Anal. 5, 103--108 (1996)

The final publication is available at Springer via: https://doi.org/10.1007/BF00276699   Ground state convex domain.pdf

Estimates for eigenvalues of the Laplacian. In: Proc. of the ICPT 94. Walther de Gruyter.

A counterexample in parabolic potential theory. Mathematika 42, 392--396 (1995) https://doi.org/10.1112/S0025579300014662 

Counterexample parabolic potential.pdf

Regularity conditions on parabolic measures. Ark. Mat. 32, 373--391 (1994) https://doi.org/10.1007/BF02559577   Regularity parabolic measures.pdf

Estimates for sums of eigenvalues of the Laplacian. J. Funct. Anal. 126, 217--227 (1994) https://doi.org/10.1006/jfan.1994.1146   Sums of eigenvalues.pdf

(with Sturm, K.-Th.) Hölder continuity of normalized solutions of Schrödinger equations. Math. Ann. 297, 663--670 (1993)

The final publication is available at Springer via: https://doi.org/10.1007/BF01459522   Hölder continuity Schrödinger.pdf

On the spectral gap for compact manifolds. J. Differ. Geom. 36, 315--330 (1992) Spectral gap compact manifolds.pdf, final publication available at proyecteuclid.org

Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space. J. Funct. Anal. 106, 353--357 (1992)

 https://doi.org/10.1016/0022-1236(92)90052-K  Upper bounds Neumann eigenvalues.pdf

A counterexample for L1-estimates for parabolic differential equations. Z. Anal. Anwend. 11, 401--406 (1992)  Counterexample L^1 estimates.pdf

Harmonic spaces associated with parabolic and elliptic differential operators, Math. Ann. 285 (3), 393-403 (1989)

The final publication is available at Springer via: https://doi.org/10.1007/BF01455064, Springer link: Harmonic Spaces Parabolic Elliptic Operators.

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