## Dissertations - ALL

August 2017

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

#### Keywords

calculus of variations, free Lagrangian, p-harmonic energy

#### Subject Categories

Physical Sciences and Mathematics

#### 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 $\A^*$.

Open Access

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