The Mathematics Page
of
Thomas Bieske

University of South Florida

I am currently an Asst. Prof. at the University of South Florida. Before that, I was a NSF VIGRE Postdoctoral Fellow at the University of Michigan working with Prof. Juha Heinonen and a Visiting Asst. Prof. at the University of Arkansas. I did my graduate work at the University of Pittsburgh under Prof. Juan Manfredi. Here is an updated vita .

Personal information can be found here.

I can be reached via tbieske at math dot usf dot edu .

The following is a list of papers with abstracts. The links are to pdf versions of the paper.

1. On Infinite Harmonic Functions on the Heisenberg Group
   Comm. PDE. (2002) 27 (3&4), 727-761.
   Abstract:  This paper examines infinite harmonic functions in the viscosity sense on the Heisenberg group by extending Aronsson's concept of Absolute Minimizing Lipschitz Extensions (1967) to the Heisenberg group. Existence of infinite harmonic functions in the viscosity sense is proved following the scheme of Bhatthacharya, DiBenedetto, and Manfredi (1989). Uniqueness of infinite harmonic functions is proved using an extension of Jensen's proof (1993). Both the existence and uniqueness proofs utilize the concept of subelliptic jets. By establishing a natural relationship between Euclidean and subelliptic jets, the technology of viscosity solutions found in Crandall, Ishii, and Lions (1992) can be used.
   
   
2. Viscosity solutions on Grushin-type planes
   Illinois J. Math. 46 (2002), 893-911. 
    Abstract:  This paper examines viscosity solutions to a class of fully nonlinear equations on Grushin-type planes. First, viscosity solutions are defined, using subelliptic second order superjets and subjets. Then, a Grushin maximum principle is proved, and as an application, comparison principles for certain types of nonlinear functions follow. This is accomplished by establishing a natural relationship between Euclidean and subelliptic jets, in order to use the viscosity solution technology of Crandall, Ishii, and Lions (1992). The particular example of infinite harmonic functions on certain Grushin-type planes is examined in further detail.
   
   
3. Lipschitz Extensions on generalized Grushin spaces
   Michigan Math J. (2005), 53 (1), 3--31.
   Abstract: In Viscosity solutions on Grushin-type planes , viscosity solutions to a class of non-linear differential equations are defined and Euclidean results are extended to Grushin-type planes, a sub-Riemannian environment without a group structure. In this paper, we examine the same class of equations but now consider generalized Grushin-type spaces of higher dimension. In addition, we show that C^1_{sub} absolute minimizers are viscosity infinite harmonic.
   
   
4. The Aronsson-Euler equation for Absolutely minimizing Lipschitz extensions with respect to Carnot-Caratheodory metrics
   Joint with Luca Capogna.
   Trans AMS (2005), 357 (2), 795--823.
   Abstract: We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the L^{\infty} variational problem ``inf ||\nabla_0 u||_{L^{\infty}(Omega)''
   subject to the condition that u=g is Lipschitz on the boundary of Omega, where Omega is an open subset of a Carnot group, \nabla_0 u denotes the horizontal gradient of a real-valued function u on Omega, and the Lipschitz class is defined in relation to the Carnot-Caratheodory metric. In particular we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen (1993) and Crandall, Evans and Gariepy (2001). We also derive the Aronsson-Euler equation for more ``regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carath\'eodory metrics which are associated to ``free" systems of vector fields.
   
   
5. Absolute Minimizers on Carnot Groups. 
   Future Trends in Geometric Function Theory. RNC Workshop. Jyvaskyla 2003.
   University of Jyvaskyla, Dept. of Mathematics and Statistics, Report 92, 15--21.
   Abstract: In this note, we extend the concepts of viscosity solutions and absolute minimizers to the setting of Carnot groups. In particular, the existence-uniqueness of infinite harmonic functions in the viscosity sense and the relationship between absolute minimizers and infinite harmonic functions are discussed. As a consequence, the uniqueness of absolute minimizers follows.
   
   
6. The Maximum Principle for Vector Fields
   Joint with Frank Beatrous and Juan Manfredi.
   Contemp. Math. 370, Amer. Math. Soc., Providence, RI, 2005. 1--9.
   Abstract: We present an extension of Jensen's uniqueness theorem for viscosity solutions of second order partial differential equations to the case of equations generated by vector fields.
   
   
7. Properties of Infinite Harmonic Functions in Riemannian Vector Fields
   Abstract: We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existence-uniqueness of infinite harmonic functions in Riemannian spaces. We then show such functions are equivalent to those that enjoy comparison with Riemannian cones. Using comparison with cones, we show that the Riemannian distance is a supersolution to the infinite Laplace equation, but is not necessarily a solution. We find some geometric conditions under which the Riemannian distance is a solution and under which it fails to be a solution.
   
   
8. Comparison principle for parabolic equations in the Heisenberg group
   Electron. J. Diff. Eqns. (2005), 2005 (95), 1-11.
   Abstract: We prove a comparison principle for a class of parabolic equations in the Heisenberg group and show the sufficiency of considering test functions that concern only the past.
   
   
9. Equivalence of Weak and Viscosity Solutions to the P-Laplace Equation in the Heisenberg Group
   Ann. Acad. Sci. Fenn. Math. (2006), 31, 363--379.
   Abstract: We prove weak and viscosity solutions to the P-Laplace equation in the Heisenberg group coincide. In particular, the viscosity sub(super-)solutions coincide with the potential theoretic P-sub(super-)harmonic functions. We are then able to obtain a comparison principle for the P-Laplacian.
   
   
10. The P-Laplace Equation on a class of Grushin-type Spaces
    Joint with Jasun Gong.
    Proceedings Amer. Math. Soc. (2006), 134 (12), 3585-3594.
    Abstract: We find the fundamental solution to the P-Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity. A solution to the 2-Laplace equation in a wider class of spaces is presented.
    
    
11. A Comparison principle for a class of subparabolic equations in Grushin-type spaces
    Electron. J. Diff. Eqns. (2007), 2007 (30), 1--9.
    Abstract: We define two notions of viscosity solutions to subparabolic equations in Grushin-type spaces, depending on whether the test functions concern only the past or both the past and the future. We then prove a comparison principle for a class of subparabolic equations and show the sufficiency of considering the test functions that concern only the past.
12. Properties of Infinite Harmonic Functions on Grushin-type Spaces
    To appear in Rocky Mountain Journal.
    Abstract: In this paper, we examine potential-theoretic and geometric properties of viscosity infinite harmonic functions in Grushin-type spaces, which are sub-Riemannian spaces lacking a group structure. In particular, we prove such functions enjoy comparison with Grushin cones. As a consequence, the distance function is viscosity infinite superharmonic, but we show that it is not necessarily viscosity infinite subharmonic.
13. Fundamental solutions to P-Laplace equations in Grushin vector fields
    Submitted for publication.
    Abstract: We find the fundamental solution to the P-Laplace equation in Grushin-type spaces. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized Grushin operator in Euclidean space. We then use this solution to find an infinite harmonic function with specific boundary data and to compute the capacity of annuli centered at the singularity.