Research Overview

My research is in theoretical mathematics, where our broad goal is to refine, develop, improve, and discover tools to analyze and understand the world we live in. The basis of this investigation is the study of measurable quantities and the relationships between them, while in particular we are interested in the now classical paradigms developed in the calculus of variations and partial differential equations, and especially in their connections with harmonic analysis and geometric measure theory.

 

Research Interests

Calculus of Variations

Harmonic Analysis

Partial Differential Equations

Geometric Measure Theory

 

Research Synopsis

A more precise description of my research is that it concerns semi-continuity problems in the Calculus of Variations and functional inequalities related to Partial Differential Equations. The underlying paradigm here is that one can model a given physical phenomenon through an energy – the action – where the reality we expect to happen is a local minimizer of the posited energy – the so-called least action principle. When the energy is finite valued, a necessary condition of being an energy minimizer is to satisfy a corresponding Euler-Lagrange equation, which in more than one variable is a Partial Differential Equation. For these equations, one is interested in well-posedness in the sense of Hadamard – the question of existence, uniqueness, and continuity with respect to data of the solution – a basic framework which is a minimal test of whether the equations are a reasonable approximation of reality.

 

Semicontinuity Problems

Concerning semi-continuity problems in the Calculus of Variations, the foundation is the now classical Direct Method of Tonelli. We recall that Tonelli's direct method approaches the question of minimization of a functional

\[ F: X \rightarrow (-\infty, \infty], \]

where \( X \) is a given infinite dimensional space \( X \) by separating the question of compactness of sequences

\[ F(u_n) \le C \]

and the question of lower-semicontinuity with respect to the topology of compactness

\[ F(u) \le \lim_{n \to \infty} F(u_n) \]

when \( u_n \rightarrow u \) in a suitable topology. A related question of interest is whether under additional hypothesis on \( F \) one has

\[ F(u) = \lim_{n \to \infty} F(u_n) \]

when \( u_n \rightarrow u \), usually in a stronger topology than in the case of lower semi-continuity. My study of these topics began at Carnegie Mellon under the direction of Giovanni Leoni and following the courses of Irene Fonseca. This training led to my first result [32], where I gave simplified proofs of the Reshetnyak continuity and lower semi-continuity theorems, while the perspective it developed led to my later introduction (with Tien-Tsan Hsieh) of the fractional gradient and the development of Calculus of Variations tools for fractional gradient energies [30, 31]. The former result is now a common reference for Reshetnyak's theorems, and has been used in the establishment of similar theorems in related context for which they are not immediately applicable. The latter has initiated a broad study on fractional gradient problems in the Calculus of Variations.

The Direct Method has been developed extensively since Tonelli's introduction, where an example of significant interest is when one has a sequence of energies

\[ F_n : X \rightarrow (-\infty, \infty] \]

which are posited as approximations of reality. This framework gives rise to several relevant questions. Firstly, one wonders whether there is a single energy \( F \) which is in some sense the limiting energy, i.e. can one find \( F \) such that

\[ F(u) = \lim_{n \to \infty} F_n(u) \]

for all \( u \in X \). Second, one studies the problem of Gamma convergence, which is a lower semi-continuity double limit:

\[ F(u) \le \lim_{n \to \infty} F_n(u_n) \]

for all sequences \( u_n \rightarrow u \) and the existence of a "recovery" sequence \( u_n \rightarrow u \) for which

\[ F(u) = \lim_{n \to \infty} F_n(u_n). \]

Here there is also a choice of topology related to the compactness one can derive from the double sequence functional bound

\[ F_n(u_n) \le C. \]

On these subjects I have made a number of contributions in the papers [9, 13-15, 17, 18, 33, 34] related to open questions posed by various authors and/or natural questions to ask given the results presented in the literature.

 

Functional Inequalities and PDE Regularity

A necessary condition of minimizers of finite valued energies \( F \) is that they satisfy a corresponding PDE, the Euler-Lagrange equation: For a local minimizer \( u \in X \), the functional derivative \( \delta F \) satisfies

\[ \delta F(u)[v] = 0 \]

for all \( v \) in a suitable space. When \( F \) is assumed to be a Lebesgue integral of a density function which depends on \( u \) through its derivatives, the preceding functional derivative is a weak formulation of a PDE. Along with well-posedness in the sense of Hadamard, a relevant question that next arises is that of regularity or partial regularity, which (in addition to questions of compactness in the establishment of the existence of minimizers) motivates the development of functional inequalities. For example, in the classical setting one often has \( X = W^{1,p}(\Omega) \), the Sobolev space of weakly differentiable functions whose weak derivative is \( p \)-integrable for some open, bounded, and suitably smooth \( \Omega \subset \mathbb{R}^n \), one is interested in inequalities of the form

\[ \|u\|_Y \le C \|\nabla u\|_{L^p(\Omega)}, \]

so-called Sobolev inequalities. The equation itself can function as a reverse Sobolev inequality, for example through the use of suitable choices of \( v \) to establish Caccioppoli inequalities, which by iteration can lead to regularity. My work on fractional gradient Calculus of Variations problems led naturally to the establishment of regularity of fractional PDE in the papers [26, 27], while the majority of my work has on functional inequalities [3, 6-8, 10-12, 16, 20, 21, 23-25, 28, 29, 35-43].

In the study of functional inequalities, the most interesting regimes are the case of critical scaling and the case which is dual to this. In the context of the Sobolev spaces \( X = W^{1,p}(\Omega) \) mentioned above, these correspond to \( p = n \), the spatial dimension, and \( p = 1 \). In the former, a now classical result asserts that \( \nabla u \in L^n(\Omega) \) implies \( u \) is of bounded mean oscillation, i.e.

 \[ \frac{1}{|Q|} \int_Q |u - \frac{1}{|Q|} \int_Q u| \, dx \le ||\nabla u||_{L^n(\Omega)} \]

for all cubes \( Q \subset \Omega \). Based on classical work of Yudovich and Adams, as well as recent contributions of Cianchi, Fontana, and Morpurgo, with (now former) my PhD student You-Wei Chen we showed that one can improve these embeddings to capture the fine properties of such functions through the introduction of a capacitary BMO space in [1]. The main point of this paper is the introduction of such a space, the demonstration of its role in Sobolev embeddings, and most importantly, to establish that these spaces enjoy an analogue of the John-Nirenberg inequality, a result of fundamental importance in the study of functions of bounded mean oscillation. These questions are a part of a general interest I have in results for capacities [3, 16, 20-22, 25, 35], and an area that I believe will see further development with our fundamental result, as with fractional gradient problems in the Calculus of Variations.

The case dual to the critical case is equally interesting, where from the study of embeddings for the fractional gradient I was led to connections with the work of Bourgain and Brezis, as well as Van Schaftingen. We recall that in their seminal 2004 Comptes Rendus announcement and 2007 JEMS paper, Bourgain and Brezis showed that when \( F \in L^1(\mathbb{R}^3; \mathbb{R}^3) \) is a given divergence free vector field, one can can show that the Hodge system

\[ \text{curl } Z = F \text{ in } \mathbb{R}^3, \]

\[ \text{div } Z = 0 \text{ in } \mathbb{R}^3, \]

admits a solution \( Z : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) with the estimate

\[ \|Z\|_{L^{3/2}(\mathbb{R}^3; \mathbb{R}^3)} \le C \|F\|_{L^1(\mathbb{R}^3; \mathbb{R}^3)}. \]

This result was stunning, as Sobolev inequalities in \( L^1 \) beyond the gradient was a topic that had been almost completely ignored, save a 1972 result of M.J. Strauss on a related result for the symmetric gradient. Bourgain and Brezis's work was continued by Van Schaftingen, who in his 2013 JEMS paper (and utilizing a number of results he established leading up to this) characterized the differential operators for which such \( L^1 \) inequalities are valid. With Jean Van Schaftingen, we proved the first optimal Lorentz sharpening of this inequality

in [43], while with Felipe Hernandez and Bogdan Raita we establish such inequalities for the case of any differential operator with a first order differential co-canceling annihilator [10, 11].

Of at least equal interest to the results in [11] is the new technique introduced there and expounded upon in [5], where we show that any divergence free measure can be decomposed into Ahlfors-David regular curves such that the sum of lengths of the curves introduced in the decomposition can be bounded by a constant times the total variation of the divergence free measure. This result returns us to the Calculus of Variations, where we recall that in addition to the standard Euler-Lagrange equations, the observation of Noether was that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. In field theory, these conservation laws take the form of the continuity equation

\[ \partial_\mu j^\mu = 0 \]

for suitably defined \( j \). This framework contains the equations of conservation of energy, mass (or matter), linear momentum, angular momentum, electric charge, energy-momentum tensor, etc. and therefore under the assumption that the \( j \) in question is integrable one may be interested to apply a decomposition in the spirit of [5, 11] to such problems. The difficulty here is that the original decomposition is established in Euclidean space, while problems of mechanics often appear on Riemannian or Lorentzian manifolds. In a recently completed work, with You-Wei Chen, Jesse Goodman and Felipe Hernandez, we obtain a decomposition of the analogue of divergence free measures in the setting of metric currents. This result is immediately applicable in the setting of a Riemannian manifold, where using our decomposition we have established estimates for Hodge systems with \( L^1 \) data. More generally, one expects this decomposition to be relevant in the study of equations whose conservation laws admit \( L^1 \) integrability.

 

Publications

1. Y.-W. Chen and D. Spector, On functions of bounded \(\beta\)-dimensional mean oscillation, Adv. Calc. Var. 17 (2024), no. 3, 975-996, DOI 10.1515/acv-2022-0084. MR4767358

2. G. E. Comi, D. Spector, and G. Stefani, The fractional variation and the precise representative of \(BV^{\alpha,p}\) functions, Fract. Calc. Appl. Anal. 25 (2022), no. 2, 520-558, DOI 10.1007/s13540-022-00036-0. MR4437291

3. Y.-W. B. Chen, K. H. Ooi, and D. Spector, Capacitary maximal inequalities and applications, J. Funct. Anal. 286 (2024), no. 12, Paper No. 110396, 31, DOI 10.1016/j.jfa.2024.110396. MR4729407

4. N. Fusco and D. Spector, A remark on an integral characterization of the dual of BV, J. Math. Anal. Appl. 457 (2018), no. 2, 1370-1375, DOI 10.1016/j.jmaa.2017.01.092. MR3705358

5. J. Goodman, F. Hernandez, and D. Spector, Two approximation results for divergence free measures, Port. Math. 81 (2024), no. 3-4, 247-264, DOI 10.4171/pm/2126. MR4781637

6. R. Garg and D. Spector, On the regularity of solutions to Poisson's equation, C. R. Math. Acad. Sci. Paris 353 (2015), no. 9, 819-823, DOI 10.1016/j.crma.2015.07.001. MR3377679

7. R. Garg and D. Spector, On the role of Riesz potentials in Poisson's equation and Sobolev embeddings, Indiana Univ. Math. J. 64 (2015), no. 6, 1697-1719, DOI 10.1512/iumj-2015.64.5706. MR3436232

8. F. Gmeineder and D. Spector, On Korn-Maxwell-Sobolev inequalities, J. Math. Anal. Appl. 502 (2021), no. 1, Paper No. 125226, 14, DOI 10.1016/j.jmaa.2021.125226. MR4243713

9. J. Goodman and D. Spector, Some remarks on boundary operators of Bessel extensions, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 493-509, DOI 10.3934/dcdss.2018027. MR3732179

10. F. Hernandez, B. Raiță, and D. Spector, Endpoint L¹ estimates for Hodge systems, Math. Ann. 385 (2023), no. 3-4, 1923-1946, DOI 10.1007/s00208-022-02383-y. MR4566709

11. F. Hernandez and D. Spector, Fractional integration and optimal estimates for elliptic systems, Calc. Var. Partial Differential Equations 63 (2024), no. 5, Paper No. 117, 29, DOI 10.1007/s00526-024-02722-8. MR4739434

12. S. G. Krantz, M. M. Peloso, and D. Spector, Some remarks on L^t embeddings in the subelliptic setting, Nonlinear Anal. 202 (2021), Paper No. 112149, 11, DOI 10.1016/j.na.2020.112149. MR4156975

13. J. Lellmann, K. Papafitsoros, C. Schönlieb, and D. Spector, Analysis and application of a nonlocal Hessian, SIAM J. Imaging Sci. 8 (2015), no. 4, 2161-2202, DOI 10.1137/140993818. MR3404680

14. G. Leoni and D. Spector, Characterization of Sobolev and BV spaces, J. Funct. Anal. 261 (2011), no. 10, 2926-2958, DOI 10.1016/j.jfa.2011.07.018. MR2832587

15. G. Leoni and D. Spector, Corrigendum to "Characterization of Sobolev and BV spaces" [J. Funct. Anal. 261 (10) (2011) 2926-2958], J. Funct. Anal. 266 (2014), no. 2, 1106-1114, DOI 10.1016/j.jfa.2013.10.026. MR3132740

16. A. D. Martínez and D. Spector, An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces, Adv. Nonlinear Anal. 10 (2021), no. 1, 877-894, DOI 10.1515/anona-2020-0157. MR4191703

17. T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015), no. 1-2, 253-279, DOI 10.1007/s00526-014-0711-3. MR3299181

18. A. C. Ponce and D. Spector, On formulae decoupling the total variation of BV functions, Nonlinear Anal. 154 (2017), 241-257, DOI 10.1016/j.na.2016.08.028. MR3614653

19. A. C. Ponce and D. Spector, A note on the fractional perimeter and interpolation, C. R. Math. Acad. Sci. Paris 355 (2017), no. 9, 960-965, DOI 10.1016/j.crma.2017.09.001 (English, with English and French summaries). MR3709534

20. A. C. Ponce and D. Spector, A decomposition by non-negative functions in the Sobolev space W^k,1, Indiana Univ. Math. J. 69 (2020), no. 1, 151-169, DOI 10.1512/iumj.2020.69.8237. MR4077159

21. A. C. Ponce and D. Spector, A boxing inequality for the fractional perimeter, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (2020), no. 1, 107-141. MR4088737

22. A. C. Ponce and D. Spector, Some remarks on capacitary integrals and measure theory, Potentials and partial differential equations—the legacy of David R. Adams, Adv. Anal. Geom., vol. 8, De Gruyter, Berlin, [2023] ©2023, pp. 235-263. MR4654520

23. G. Psaradakis and D. Spector, A Leray-Trudinger inequality, J. Funct. Anal. 269 (2015), no. 1, 215-228, DOI 10.1016/j.jfa.2015.04.007. MR3345608

24. B. Raiță and D. Spector, A note on estimates for elliptic systems with L¹ data, C. R. Math. Acad. Sci. Paris 357 (2019), no. 11-12, 851-857, DOI 10.1016/j.crma.2019.11.007 (English, with English and French summaries). MR4038260

25. B. Raiță, D. Spector, and D. Stolyarov, A trace inequality for solenoidal charges, Potential Anal. 59 (2023), no. 4, 2093-2104, DOI 10.1007/s11118-022-10008-x. MR4684387

26. A. Schikorra, T.-T. Shieh, and D. Spector, L^p theory for fractional gradient PDE with VMO coefficients, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 4, 433-443, DOI 10.4171/RLM/714. MR3420498

27. A. Schikorra, T.-T. Shieh, and D. E. Spector, Regularity for a fractional p-Laplace equation, Commun. Contemp. Math. 20 (2018), no. 1, 1750003, 6, DOI 10.1142/S0219199717500031. MR3714833

28. A. Schikorra, D. Spector, and J. Van Schaftingen, An L¹-type estimate for Riesz potentials, Rev. Mat. Iberoam. 33 (2017), no. 1, 291-303, DOI 10.4171/RMI/937. MR3615452

29. I. Shafrir and D. Spector, Best constants for two families of higher order critical Sobolev embeddings, Nonlinear Anal. 177 (2018), 753-769, DOI 10.1016/j.na.2018

30. T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations, Adv. Calc. Var. 8 (2015), no. 4, 321-336, DOI 10.1515/acv-2014-0009. MR3403430

31. T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations II, Adv. Calc. Var. 11 (2018), no. 3, 289-307, DOI 10.1515/acv-2016-0056. MR3819528

32. D. Spector, Simple proofs of some results of Reshetnyak, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1681-1690, DOI 10.1090/S0002-9939-2010-10593-2. MR2763757

33. D. Spector, L^p-Taylor approximations characterize the Sobolev space W^1,p, C. R. Math. Acad. Sci. Paris 353 (2015), no. 4, 327-332, DOI 10.1016/j.crma.2015.01.010 (English, with English and French summaries). MR3319129

34. D. Spector, On a generalization of L-differentiability, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 62, 21, DOI 10.1007/s00526-016-1004-9. MR3503211

35. D. Spector, A noninequality for the fractional gradient, Port. Math. 76 (2019), no. 2, 153-168, DOI 10.4171/pm/2031. MR4065096

36. D. Spector, New directions in harmonic analysis on L¹, Nonlinear Anal. 192 (2020), 111685, 20, DOI 10.1016/j.na.2019.111685. MR4034690

37. D. Spector, An optimal Sobolev embedding for L¹, J. Funct. Anal. 279 (2020), no. 3, 108559, 26, DOI 10.1016/j.jfa.2020.108559. MR4093790

38. D. E. Spector and S. J. Spector, Uniqueness of equilibrium with sufficiently small strains in finite elasticity, Arch. Ration. Mech. Anal. 233 (2019), no. 1, 409-449, DOI 10.1007/s00205-019-01360-1. MR3974644

39. D. E. Spector and S. J. Spector, BMO and elasticity: Korn's inequality, local uniqueness in tension, J. Elasticity 143 (2021), no. 1, 85-109, DOI 10.1007/s10659-020-09805-5. MR4201698

40. D. Spector, Taylor's theorem for functionals on BMO with application to BMO local minimizers, Quart. Appl. Math. 79 (2021), no. 3, 409-417, DOI 10.1090/qam/1586. MR4288590

41. D. Spector, On Korn's first inequality in a Hardy-Sobolev space, J. Elasticity 154 (2023), no. 1-4, 187-198, DOI 10.1007/s10659-022-09976-3. MR4661780

42. D. Spector and C. B. Stockdale, On the dimensional weak-type (1,1) bound for Riesz transforms, Commun. Contemp. Math. 23 (2021), no. 7, Paper No. 2050072, 19, DOI 10.1142/S0219199720500728. MR4329315

43. D. Spector and J. Van Schaftingen, Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemma, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), no. 3, 413-436, DOI 10.4171/RLM/854. MR4002205

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