Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.1842

Foundations of the calculus of variations in generalized function algebras

Sanja Konjik·Michael Kunzinger·Michael Oberguggenberger
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of Nöther's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.

Explore related subjects

math.FAmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sanja Konjik, Michael Kunzinger, Michael Oberguggenberger. 2007-07-12. Foundations of the calculus of variations in generalized function algebras. https://arxiv.org/abs/0707.1842

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Talagrand Type

We prove that Rademacher type $s$ coincides with Talagrand type $(s,ψ_{s,s/2})$ for every $1<s<2$.

math.FA↗
arXiv2026

On partition functions of Gaussian random variables

For $β\in \mathbb{R}$, we define the partition function $$ Z_β(X) = \sum_{i=1}^N \exp(βX_i) $$ of a centered Gaussian random vector $X=(X_1, \ldots, X_N)$ with $\mathbb{E}[X_i^2]= 1$. For $q \in \mathbb{R}$, we prove a complete phase transition for the generalized $L_q$-means of $Z_β(X)$ at $q=1$. The centered Gaussian vector with covariance matrix $Δ_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q<1$, and minimizes them for $q>1$. The two regimes are governed by different principles. For $q > 1$ {and $β\ne0$}, the moment functional is globally strongly convex on the entire set of correlation matrices, with an explicit modulus of convexity and a quantitative centroid-shape stability estimate. For $q<1$, the strategy is different. We prove a universal comparison for log-concave profiles of reverse Brascamp--Lieb type. Specializing this result to the Gumbel profile yields a Laplace-transform comparison between the partition functions.

math.FA↗
arXiv2026

Morita equivalence for $L^p$-operator algebras associated with étale groupoids

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.

math.FA↗