An important topic in the calculus of variations is the study of traction-free problems, in which deformations between given domains in $\\mathbb{R}^n$ are allowed to slip on the boundary, without prescribing boundary values. For annuli $\\A = \\A(r, R)$ and $\\A^* = \\A(r_*, R_*)$, we seek the traction-free minimizer of the $p$-harmonic energy among homeomorphisms in Sobolev class $W^{1, p}(\\A, \\A^*)$. For such a mapping, the $p$-harmonic energy is defined by \\begin{equation*}
\\mathcal{E}_p[h] = \\int\\limits_{\\A} |Dh(x)|^p dx
\\end{equation*}
Classical methods fail for traction-free problems. We will use a novel approach based on the concept of free Lagrangians, described as differential forms $L(x, h(x), Dh(x))dx$ whose integral depends only on the homotopy class of $h$. We find that the solution to the $p$-harmonic variational problem depends on the relative thickness of $\\A$ and $\\A^*$.
","abstract_html":"<p>An important topic in the calculus of variations is the study of traction-free problems, in which deformations between given domains in \\mathbb{R}n are allowed to slip on the boundary, without prescribing boundary values. For annuli $\\A = \\A(r, R)$ and \\A* = \\A(r*, R*), we seek the traction-free minimizer of the $p$-harmonic energy among homeomorphisms in Sobolev class W1, p(\\A, \\A*). For such a mapping, the $p$-harmonic energy is defined by \\begin{equation*}</p> <p>\\mathcal{E}_p[h] = \\int\\limits_{\\A} |Dh(x)|^p dx</p> <p>\\end{equation*} </p> <p>Classical methods fail for traction-free problems. We will use a novel approach based on the concept of free Lagrangians, described as differential forms $L(x, h(x), Dh(x))dx$ whose integral depends only on the homotopy class of $h$. We find that the solution to the $p$-harmonic variational problem depends on the relative thickness of $\\A$ and \\A*.</p>","abstract_has_math":true,"creators":["Cuneo, Daniel"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Tadeusz Iwaniec"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-25T07:00:00Z","date_published":"2017-08-25T07:00:00Z","updated_at":"2026-07-24T04:55:19Z","subjects":["calculus of variations","free Lagrangian","p-harmonic energy","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/758","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Tadeusz Iwaniec"]},{"key":"dc:creator","label":"Author","values":["Cuneo, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["calculus of variations","free Lagrangian","p-harmonic energy","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/758"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["An important topic in the calculus of variations is the study of traction-free problems, in which deformations between given domains in $\\mathbb{R}^n$ are allowed to slip on the boundary, without prescribing boundary values. For annuli $\\A = \\A(r, R)$ and $\\A^* = \\A(r_*, R_*)$, we seek the traction-free minimizer of the $p$-harmonic energy among homeomorphisms in Sobolev class $W^{1, p}(\\A, \\A^*)$. For such a mapping, the $p$-harmonic energy is defined by \\begin{equation*}
\\mathcal{E}_p[h] = \\int\\limits_{\\A} |Dh(x)|^p dx
\\end{equation*}
Classical methods fail for traction-free problems. We will use a novel approach based on the concept of free Lagrangians, described as differential forms $L(x, h(x), Dh(x))dx$ whose integral depends only on the homotopy class of $h$. We find that the solution to the $p$-harmonic variational problem depends on the relative thickness of $\\A$ and $\\A^*$.
"]},{"key":"dc:title","label":"Title","values":["Mappings Between Annuli of Smallest p-Harmonic Energy"]}]}],"canonical_facts":{"dc:contributor":["Tadeusz Iwaniec"],"dc:creator":["Cuneo, Daniel"],"dc:description.abstract":["An important topic in the calculus of variations is the study of traction-free problems, in which deformations between given domains in $\\mathbb{R}^n$ are allowed to slip on the boundary, without prescribing boundary values. For annuli $\\A = \\A(r, R)$ and $\\A^* = \\A(r_*, R_*)$, we seek the traction-free minimizer of the $p$-harmonic energy among homeomorphisms in Sobolev class $W^{1, p}(\\A, \\A^*)$. For such a mapping, the $p$-harmonic energy is defined by \\begin{equation*}
\\mathcal{E}_p[h] = \\int\\limits_{\\A} |Dh(x)|^p dx
\\end{equation*}
Classical methods fail for traction-free problems. We will use a novel approach based on the concept of free Lagrangians, described as differential forms $L(x, h(x), Dh(x))dx$ whose integral depends only on the homotopy class of $h$. We find that the solution to the $p$-harmonic variational problem depends on the relative thickness of $\\A$ and $\\A^*$.
"],"dc:identifier":["https://surface.syr.edu/etd/758"],"dc:subject":["calculus of variations","free Lagrangian","p-harmonic energy","Physical Sciences and Mathematics"],"dc:title":["Mappings Between Annuli of Smallest p-Harmonic Energy"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:55:19Z"}