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Hölder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients
 
Eric T. Sawyer McMaster University, Hamilton, ON, Canada
Richard L. Wheeden Rutgers University, New Brunswick, NJ
Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients
eBook ISBN:  978-1-4704-0451-2
Product Code:  MEMO/180/847.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $41.40
Holder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients
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Hölder Continuity of Weak Solutions to Subelliptic Equations with Rough Coefficients
Eric T. Sawyer McMaster University, Hamilton, ON, Canada
Richard L. Wheeden Rutgers University, New Brunswick, NJ
eBook ISBN:  978-1-4704-0451-2
Product Code:  MEMO/180/847.E
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $41.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1802006; 157 pp
    MSC: Primary 35;

    We study interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. In particular, we show that solutions of large classes of subelliptic equations with bounded measurable coefficients are Hölder continuous. We present two types of results dealing with such equations. The first type generalizes the celebrated Fefferman-Phong geometric characterization of subellipticity in the smooth case. We introduce a notion of \(L^q\)-subellipticity for the rough case and develop an axiomatic method which provides a near characterization of the notion of \(L^q\)-subellipticity. The second type deals with generalizing a case of Hörmanders's celebrated algebraic characterization of subellipticity for sums of squares of real analytic vector fields. In this case, we introduce a “flag condition” as a substitute for the Hörmander commutator condition which turns out to be equivalent to it in the smooth case. The question of regularity for quasilinear subelliptic equations with smooth coefficients provides motivation for our study, and we briefly indicate some applications in this direction, including degenerate Monge-Ampère equations.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Comparisons of conditions
    • 3. Proof of the general subellipticity theorem
    • 4. Reduction of the proofs of the rough diagonal extensions of Hörmander’s theorem
    • 5. Homogeneous spaces and subrepresentation inequalities
    • 6. Appendix
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1802006; 157 pp
MSC: Primary 35;

We study interior regularity of weak solutions of second order linear divergence form equations with degenerate ellipticity and rough coefficients. In particular, we show that solutions of large classes of subelliptic equations with bounded measurable coefficients are Hölder continuous. We present two types of results dealing with such equations. The first type generalizes the celebrated Fefferman-Phong geometric characterization of subellipticity in the smooth case. We introduce a notion of \(L^q\)-subellipticity for the rough case and develop an axiomatic method which provides a near characterization of the notion of \(L^q\)-subellipticity. The second type deals with generalizing a case of Hörmanders's celebrated algebraic characterization of subellipticity for sums of squares of real analytic vector fields. In this case, we introduce a “flag condition” as a substitute for the Hörmander commutator condition which turns out to be equivalent to it in the smooth case. The question of regularity for quasilinear subelliptic equations with smooth coefficients provides motivation for our study, and we briefly indicate some applications in this direction, including degenerate Monge-Ampère equations.

  • Chapters
  • 1. Introduction
  • 2. Comparisons of conditions
  • 3. Proof of the general subellipticity theorem
  • 4. Reduction of the proofs of the rough diagonal extensions of Hörmander’s theorem
  • 5. Homogeneous spaces and subrepresentation inequalities
  • 6. Appendix
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.