The Mathematics Page
of
Thomas Bieske

University of South Florida

I am currently a Professor of Mathematics at the University of South Florida. Before that, I was a NSF VIGRE Postdoctoral Fellow at the University of Michigan working with the late Prof. Juha Heinonen and a Visiting Assistant Professor at the University of Arkansas. I did my graduate work at the University of Pittsburgh under Prof. Juan Manfredi. Here is a vita (current as of 8/7/24).

Personal information can be found here.

I am also moderator of the Facebook community Avenue Carnot, a community dedicated to disseminating news and information about analysis in metric spaces and sub-Riemannian geometry.

Avenue Carnot

I can be reached via tbieske at usf dot edu.

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

  1. Viscosity solutions to the Infinite Laplace equation in Grushin-type spaces
    Joint with Zachary Forrest.
    Modern Mathematical Methods (2024) 2 (1), 41--54.
    Abstract: In this paper, we prove the existence and uniqueness of viscosity solutions to the infinite Laplace equation in Grushin-type spaces whose tangent spaces consist of arbitrary triangular vector fields.
  2. Generalizations of the Drift Laplace Equation over the Quaternions in a Class of Grushin-Type Spaces
    Joint with Keller Blackwell.
    Constr. Math. Anal. (2023), 6 (3), 164-175.
    Abstract: Beals, Gaveau, and Greiner in 1996 establish a formula for the fundamental solution to the Laplace equation with drift term in Grushin-type planes. The first author and Childers in 2013 expanded these results by invoking a p-Laplace-type generalization that encompasses these formulas while the authors explored a different natural generalization of the p-Laplace equation with drift term that also encompasses these formulas. In both, the drift term lies in the complex domain. We extend these results by considering a drift term in the quaternion realm and show our solutions are stable under limits as p tends to infinity.
  3. Existence and uniqueness of viscosity solutions to the infinity Laplacian relative to a class of Grushin-type vector fields
    Joint with Zachary Forrest.
    Constr. Math. Anal. (2023), 6 (2), 77-89.
    Abstract: In this paper, we pose the infinite-Laplace equation as a Dirichlet Problem in a class of Grushin-type spaces whose vector fields are of the form $X_k(p) := \sigma_k(p) \frac{\partial}{\partial x_k}$ and $\sigma_k$ is not a polynomial for indices $m + 1 \leq k \leq n$. Solutions to the infinite-Laplacian in the viscosity sense have been shown to exist and be unique by Bieske in 2005 when $\sigma_k$ is a polynomial; we extend these results by exploiting the relationship between Grushin-type and Euclidean second-order jets and utilizing estimates on the viscosity derivatives of sub- and supersolutions in order to produce a comparison principle for semicontinuous functions.
  4. Generalizations of the Drift Laplace Equation in the Heisenberg Group and a Class of Grushin-Type Spaces
    Joint with Keller Blackwell.
    Electron. J. Diff. Eqns. (2021), 2021 (99), 1-13.
    Abstract: We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation with drift term. Our results are independent of the results of Bieske and Childers, in that Bieske and Childers consider a generalization that focuses on a p-Laplace-type equation while we primarily concentrate on a generalization of the drift term.
  5. On the Lie Algebra of polarizable Carnot groups.
    Anal. and Math. Physics. (2020), 10, No. 80, 11pp.
    Abstract: Balogh and Tyson (2002) established the concept of polarizable Carnot groups, which are Carnot groups for which proper polar coordinates can be constructed. They also show that the groups of Heisenberg-type are polarizable and present an open question as to which Carnot groups are polarizable. Here, we explore the hidden nuance in this question as we demonstrate that polarization is not just a property of the algebraic group law, it is also dependent upon the Lie Algebra structure. We explore generalized Heisenberg-type groups and demonstrate the impact of the Lie Algebra.
  6. Correction to "A p(x) -Poincar\'e-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups"
    Joint with Robert Freeman.
    Anal. and Math. Physics. (2019), 9 (4), 1611-1612.
  7. On the p-Laplace equation in a class of H\"{o}rmander Vector Fields
    Joint with Robert Freeman.
    Electron. J. Diff. Eqns. (2019), 2019 (35), 1-13.
    Abstract: We find the fundamental solution to the $p$-Laplace equation in a class of H\"{o}rmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points, which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space. We then extend these solutions to a generalization of the $p$-Laplace equation and use these solutions to find infinite harmonic functions and their generalizations. We also compute the capacity of annuli centered at the singularity.
  8. Equivalence of Weak and Viscosity Solutions to the p(x)-Laplacian in Carnot Groups
    Joint with Robert Freeman.
    Anal. and Math. Physics. (2019), 9 (4), 1583-1610.
    Abstract: We show the equivalence of weak and viscosity solutions to the p(x)-Laplacian in Carnot groups, under certain natural restrictions on the function p(x). As a consequence, we obtain a comparison principle for viscosity solutions and thus uniqueness of viscosity solutions to the Dirichlet problem.
  9. A p(x)-Poincar\'e-type Inequality for Variable Exponent Sobolev Spaces with Zero Boundary Values in Carnot Groups
    Joint with Robert Freeman.
    Anal. and Math. Physics. (2018), 8 (2), 289-308.
    Abstract: We prove a p(x)-Poincar\'e-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups. We then establish the existence and uniqueness (up to a set of zero p(x)-capacity) of a minimizer to the Dirichlet energy integral for the variable exponent case.
  10. The parabolic infinite-Laplace equation in Carnot groups
    Joint with Erin Martin.
    Mich. Math. J. (2016), 65 (3), 489-509.
    Abstract: By employing a Carnot parabolic maximum principle, we show existence-uniqueness of viscosity solutions to a class of equations modeled on the parabolic infinite-Laplace equation in Carnot groups. We show stability of solutions within the class and examine the limit as t goes to infinity.
  11. The $\infty(x)$-equation in Grushin-type Spaces
    Electron. J. Diff. Eqns. (2016), 2016 (125), 1-13.
    Abstract: We employ Grushin jets which are adapted to the geometry of Grushin-type spaces to obtain the existence-uniqueness of viscosity solutions to the $\infty(x)$-Laplace equation in Grushin-type spaces. Due to the differences between Euclidean jets and Grushin jets, the Euclidean method of proof is not valid in this environment.
  12. The $\infty(x)$-equation in Riemannian Vector Fields
    Electron. J. Diff. Eqns. (2015), 2015 (164), 1-9.
    Abstract: We employ Riemannian jets which are adapted to the Riemannian geometry to obtain the existence-uniqueness of viscosity solutions to the $\infty(x)$-Laplace equation in Riemannian vector fields. Due to the differences between Euclidean jets and Riemannian jets, the Euclidean method of proof is not valid in this environment.
  13. The parabolic p-Laplace equation in Carnot groups
    Joint with Erin Martin.
    Ann. Acad. Sci. Fenn. (2014), 39, 605-623.
    Abstract: By establishing a parabolic maximum principle, we show uniqueness of viscosity solutions to the parabolic p-Laplace equation and then examine the limit as t goes to infinity. Additionally, we explore the limit as p goes to infinity.
  14. Generalizations of a Laplacian-Type Equation in the Heisenberg Group and a Class of Grushin-Type Spaces
    Joint with Kristen Childers.
    Proc. Amer. Math. Soc. (2014), 142 (3), 989-1003
    Abstract: Beals, Gaveau and Greiner (1996) find the fundamental solution to a 2-Laplace-type equation in a class of sub-Riemannian spaces. This solution is related to the well-known fundamental solution to the p-Laplace equation in Grushin-type spaces (Bieske-Gong, 2006) and the Heisenberg group (Capogna, Danielli, Garofalo, 1997). We extend the 2-Laplace-type equation to a p-Laplace-type equation. We show that the obvious generalization does not have desired properties, but rather, our generalization preserves some natural properties.
  15. A Sub-Riemannian Maximum Principle and its application to the p-Laplacian in Carnot Groups
    Ann. Acad. Sci. Fenn. (2012), 37, 119-134.
    Abstract: We prove a sub-Riemannian maximum principle for semicontinuous functions. We apply this principle to Carnot groups to provide a ``sub-Riemannian" proof of the uniqueness of viscosity infinite harmonic functions and to establish the equivalence of weak solutions and viscosity solutions to the p-Laplace equation. This result extends the author's previous work in the Heisenberg group.
  16. Fundamental solutions to P-Laplace equations in Grushin vector fields
    Electron. J. Diff. Eqns. (2011), 2011 (84), 1-10.
    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.
  17. The Infinite Dirac Operator
    Joint with John Ryan.
    2010 J. Phys.: Conf. Ser. 254 012003.
    Abstract: In this article, we define the infinite Dirac operator and explore some key properties, particularly its conformal invariance. En route, we also establish the conformal invariance of the p-Dirac equation. We also introduce the infinite Dirac operator on the sphere $S^{n}$ and establish the relationship between the two infinite Dirac operators via the Cayley transformation. Also we introduce an infinite Laplace operator on $S^{n}$.
  18. The Carnot-Carath\'eodory distance vis-a-vis the eikonal equation and the infinite Laplacian
    Bull. London Math. Soc. (2010), 42 (3), 395-404.
    Abstract: In R^n equipped with the Euclidean metric, the distance from the origin (smoothly) satisfies the eikonal equation and is (smoothly) infinite harmonic everywhere except the origin. Dragoni (2007) has shown that the Carnot-Carath\'eodory distance satisfies the eikonal equation in the viscosity sense outside of the origin but Bieske, Dragoni, Manfredi (2008) have shown that the distance is not viscosity infinite harmonic at all points outside the origin. We examine the behavior of the negative distance function and show that it is a viscosity solution to the eikonal equation exactly where it is viscosity infinite harmonic.
  19. The Carnot-Carath\'eodory distance and the infinite Laplacian
    Joint with Federica Dragoni and Juan Manfredi.
    J. of Geo. Anal. (2009), 19 (4), 737-754.
    Abstract: In R^n equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization for when the distance from the origin in an arbitrary Carnot-Carath\'eodory space is a viscosity infinite harmonic function at a point outside the origin. We show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity infinite harmonic subsolution. In addition, the distance function is not a viscosity infinite harmonic supersolution at the origin.
  20. Properties of Infinite Harmonic Functions on Grushin-type Spaces
    Rocky Mtn J. of Math. (2009), 39 (3), 729-756.
    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.
  21. Parabolic equations relative to vector fields
    Electron. J. Diff. Eqns. (2008), 2008 (124), 1-7.
    Abstract: We define two notions of viscosity solutions to parabolic equations defined using vector fields, depending on whether the test functions concern only the past or both the past and the future. Using the parabolic maximum principle, we then prove a comparison principle for a class of parabolic equations and show the sufficiency of considering the test functions that concern only the past.
  22. Properties of Infinite Harmonic Functions relative to Riemannian Vector Fields
    Le Matematiche (2008), LXIII (2), 19-37.
    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 infinite harmonic and under which it fails to be infinite harmonic.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. The Aronsson-Euler equation for Absolutely minimizing Lipschitz extensions with respect to Carnot-Carath\'eodory 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-Carath\'eodory 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.
  29. Lipschitz Extensions on generalized Grushin spaces
    Mich. 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.
  30. Absolute Minimizers on Carnot Groups.
    Future Trends in Geometric Function Theory. RNC Workshop. Jyv\"askyl\"a 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.
  31. Viscosity solutions on Grushin-type planes
    Illinois J. Math. (2002), 46, 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.
  32. 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.