Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown by means of Campanato spaces. These spaces make the pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. As a prerequisite device, a version of both the mean value theorem and Taylor inequality are established with the parabolic distance incorporated.
","abstract_html":"<p>Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown by means of Campanato spaces. These spaces make the pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. As a prerequisite device, a version of both the mean value theorem and Taylor inequality are established with the parabolic distance incorporated.</p>","abstract_has_math":false,"creators":["Griffin, Heather Arielle"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematics (PhD)","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Arnold, Mark E.","Harrington, Phillip S."],"advisors":["Capogna, Luca"],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-05-01T07:00:00Z","date_published":"2012-05-01T07:00:00Z","updated_at":"2026-07-24T00:59:01Z","subjects":["Pure sciences","Applied sciences","Carnot groups","Parabolic equations","Pointwise Schauder estimates","Schauder","Stratified groups","Numerical Analysis and Computation","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.uark.edu/etd/383","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Arnold, Mark E.","Harrington, Phillip S."]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Capogna, Luca"]},{"key":"dc:creator","label":"Author","values":["Griffin, Heather Arielle"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-09-29T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematics (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Pure sciences","Applied sciences","Carnot groups","Parabolic equations","Pointwise Schauder estimates","Schauder","Stratified groups","Numerical Analysis and Computation","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.uark.edu/etd/383"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown by means of Campanato spaces. These spaces make the pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. As a prerequisite device, a version of both the mean value theorem and Taylor inequality are established with the parabolic distance incorporated.
"]},{"key":"dc:title","label":"Title","values":["Pointwise Schauder Estimates of Parabolic Equations in Carnot Groups"]}]}],"canonical_facts":{"dc:contributor":["Arnold, Mark E.","Harrington, Phillip S."],"dc:contributor.advisor":["Capogna, Luca"],"dc:creator":["Griffin, Heather Arielle"],"dc:date":["2012"],"dc:date.available":["2017-09-29T07:00:00Z"],"dc:description.abstract":["Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown by means of Campanato spaces. These spaces make the pointwise nature of the estimates possible by comparing solutions to their Taylor polynomials. As a prerequisite device, a version of both the mean value theorem and Taylor inequality are established with the parabolic distance incorporated.
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