This research is supported in part by the US National Science Foundation Grant DMS 0758500 

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RESEARCH INTERESTS:

• Partial Differential Equations

elliptic PDE's, boundary value problems, regularity, rough domains

• Harmonic Analysis

singular integral operators, function Spaces, interpolation, atomic and wavelet decompositions, maximal functions

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PUBLICATIONS:

Higher order equations

• Boundedness of the gradient of the solution and Wiener test of order one for the biharmonic equation (with Vladimir Maz'ya), Inventiones Mathematicae, 175 (2009), no. 2, 287--334.


• Boundedness of the Hessian of a biharmonic function in a convex domain (with Vladimir Maz'ya), Comm. Partial Differential Equations 33 (2008), no. 7-9, 1439--1454.


• Pointwise estimates for the polyharmonic Green function in general domains (with Vladimir Maz'ya), submitted.


• Strong localization induced by one clamped point in thin plate vibrations (with Marcel Filoche), Physical Review Letters, accepted.
  


• Spectral properties of the operator pencil and regularity of the solution for the biharmonic equation on a non-smooth domain (with Vladimir Maz'ya), in preparation.


• Green function estimates and Wiener test for higher order elliptic problems (with Vladimir Maz'ya), in preparation.




Boundary problems on Lipschitz domains

• Sharp estimates for Green potentials on non-smooth domains (with Marius Mitrea), Mathematical Research Letters, 11 (2004), 481--492.


• Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces
  (with Marius Mitrea), Annali di Matematica Pura ed Applicata (4) 185 (2006), no. 2, 155--187.


• Square-function estimates for singular integrals and applications to partial differential equations (with Marius Mitrea),
  Differential Integral Equations, 17 (2004), no. 7-8, 873--892.


• Boundary value problems for Lame system on low dimensional non-smooth domains (with Marius Mitrea), Integral methods in science and engineering, 137--160, Birkhäuser Boston, Boston, MA, 2006.


• The solution of the Chang-Krantz-Stein conjecture (with Marius Mitrea), Proceedings of the Workshop in Harmonic Analysis, Tokyo, Japan, 2007, 1--95.


• Green potential estimates and the Poisson problem on Lipschitz domains (with Marius Mitrea), in preparation.




Second order elliptic operators with rough coefficients

• Hardy and BMO spaces associated to divergence form elliptic operators (with Steve Hofmann), Mathematische Annalen, 344 (2009), no. 1, 37--116.


• Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators" (with Steve Hofmann), submitted.


• Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces (with Steve Hofmann and Alan McIntosh), in preparation.


• The connections between Dirichlet, Regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients, submitted.




Geometric measure theory

• Finite square function implies integer dimension (with Alexander Volberg), C. R. Math. Acad. Sci. Paris, 347 (2009), pp. 1271--1276.


• Boundedness of the square function and rectifiability (with Alexander Volberg), C. R. Math. Acad. Sci. Paris, 347 (2009), pp. 1051--1056.




Function spaces, interpolation

• Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and applications to problems in partial differential equations (with Nigel Kalton and Marius Mitrea), Proceedings of the Conference in honor of M.Cwikel, Contemp. Math., 2007.

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CONFERENCES, COLLOQUIA AND SEMINARS:

• Fall Southeastern Meeting of the AMS, Boca Raton, FL, October 2009.


• Conference on "Microlocal Analysis and Spectral Theory on Singular Spaces" Pennsylvania State University, State College, PA, October 2009.


• Georgia Institute of Technology, April 2009.


• Michigan State University, April 2009.


• Brown University, February 2009.


• Colloquium at Washington University, St. Louis, MO, February 2009.


• University of Missouri, Columbia, MO, February 2009.


• The conference ``Potential Theory and Analysis of Growth Processes", Laboratoire MAPMO, Université d'Orléans, France, January 2009.


• Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris VI), January 2009.



• Institute for Advanced Study, Princeton, NJ, October 2008.


• Purdue University, October 2008.


• The conference on the occasion of the 70th birthday of Vladimir Maz'ya ``Analysis, PDEs and Applications", Rome, Italy, July 2008.


• The conference ``Recent Advances in Geometric Function Theory", Syracuse University, May 2008.


• Invited one hour talk at the Second Workshop on Harmonic Analysis and Partial Differential Equations, Merida, Yucatan, Mexico, February 2008.


• University of Oregon, January 2008.


• Lehigh University, January 2008.


• Michigan State University, January 2008.


• Syracuse University, January 2008.


• Purdue University, January 2008.



• University of Minnesota, December 2007.


• University of Kentucky, December 2007.


• SIAM Conference on Analysis of PDE, Mesa, AZ, December 2007.


• Colloquium talk, Brown University,  November 2007.


• Michigan State University, October 2007.


• 2007 Fall Western Section Meeting, University of New Mexico, Albuquerque, NM, October 2007.


• 2007 Fall Central Section Meeting, DePaul University, Chicago, IL, October 2007.


• Brown University, September 2007.


• Brown University, September 2007.


• Colloquium talk, Instituto de Matematicas, Cuernavaca, Mexico, May 2007.


• The VII Joint AMS-SMM Meeting, Zacatecas, Mexico, May 2007.


• The Ohio State University, May 2007.




• University of Missouri - Columbia, November 2006.


• Syracuse University, November 2006.


• Brown University, October 2006.


• Satellite Conference to the ICM-2006 ``Harmonic and Geometrical Analysis with Applications to Partial Differential Equations", Seville, Spain, August 2006.


• The Ohio State University, May 2006.


• 2006 Spring Central Sectional Meeting, University of Notre Dame, Notre Dame, IN, April 2006.


• University of Missouri - Columbia, March 2006.




• Flinders University, Adelaide, Australia, November 2005.


• Macquarie University, Sydney, Australia, October 2005.


• Australian National University, Canberra, Australia, August 2005.


• Ohio State University, February 2005.




• The 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, Madrid (Spain), June 2004.


• The 2nd Symposium on Analysis and PDEs, Purdue University, June 2004.


• Show Me Analysis Meeting, University of Missouri-Columbia, June 2004.


• The 29th Spring Lecture Series in the Mathematical Sciences, Recent  Developments in Applied Harmonic Analysis: Multiscale Geometric Analysis, Fayetteville, Arkansas, April 2004.


• The 53rd Midwest PDE Seminar, University of Missouri, Columbia, Missouri, March 2004.


• 2004 AMS Spring Southeastern Section Meeting, Tallahassee, Florida, March 2004.


• University of Missouri - Columbia, February 2004.


• University of Tuebingen, Germany, January 2004.


• University of Ulm, Germany, January 2004.




• The 52nd Midwest Conference in PDEs, University of Minnesota, November 2003.


• University of Missouri - Columbia, October 2003.


• The 6-th TULKA Internet Seminar ``Operator matrices and delay semigroups,'' Heinrich-Fabri Institut, Blaubeuren, June 2003.


• CBMS Conference on Free Boundary Problems, Wayne State University, Detroit, Michigan, April 2003.


• The 6-th New Mexico Analysis Seminar, Albuquerque, New Mexico, March 2003.


• University of Missouri - Columbia, February 2003.


• AMS Special Session in Madison, Wisconsin, October 2002.


• 27th University of Arkansas spring lecture series, "Harmonic Analysis, Multilinear operators and Schrodinger operators," Fayetteville, Arkansas, April 2002.

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TEACHING:

• College Algebra for Calculus Bound Students, Fall 2002, University of Missouri-Columbia


• Elements of Calculus, Winter 2003, Winter 2004, Winter 2005, University of Missouri-Columbia


• Calculus II, Fall 2003, University of Missouri-Columbia


• Analytic Geometry and Calculus I, Fall 2004, University of Missouri-Columbia


• Calculus and Analytic Geometry I, Winter 2006, The Ohio State University


• Mathematical Analysis for Business II, Spring 2006, The Ohio State University


• Real Analysis I-III, Fall 2006, Winter 2007, Spring 2007, Spring 2008, The Ohio State University


• Accelerated Calculus with Analytic Geometry II, Winter 2008, The Ohio State University.

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