Related papers: Complete classification of planar p-elasticae
Let $f \in S_{\kappa}(\Gamma_0(N))$ be a Hecke eigenform at $p$ with eigenvalue $\lambda_f(p)$ for a prime $p$ not dividing $N$. Let $\alpha_p$ and $\beta_p$ be complex numbers satisfying $\alpha_p + \beta_p = \lambda_f(p)$ and $\alpha_p…
For the classical Euler's elastic problem, conjugate points are described. Inflectional elasticae admit the first conjugate point between the first and the third inflection points. All the rest elasticae do not have conjugate points.
We classify all rotational surfaces in Euclidean space whose principal curvatures $\kappa_1$ and $\kappa_2$ satisfy the linear relation $\kappa_1=a\kappa_2+b$, where $a$ and $b$ are two constants. We give a variational characterization of…
A toy top is defined as a rotationally symmetric body moving in a constant gravitational field while one point on the symmetry axis is constrained to stay in a horizontal plane. It is an integrable system similar to the Lagrange top.…
In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of…
We define a generalization of convex functions, which we call $\delta$-convex functions, and show they must satisfy interior H\"older and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear…
We study fully nonlinear elliptic equations on Hermitian manifolds through blow-up argument and partial uniform ellipticity. We apply our results to draw geometric conclusions on finding conformal Hermitian metrics with prescribed…
Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular…
Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In…
A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a…
The Planar Contraction problem is to test whether a given graph can be made planar by using at most k edge contractions. This problem is known to be NP-complete. We show that it is fixed-parameter tractable when parameterized by k.
We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor, once we know its symmetry class. In other words, we produce a rotation which brings an Elasticity tensor onto its normal form, given its…
The aim of this paper is to bring together a new type of quantum calculus, namely $p $-calculus, and variational calculus. We develop $p $-variational calculus and obtain a necessary optimality condition of Euler-Lagrange type and a…
We investigate equilibria of charged deformable materials via the minimization of an electroelastic energy. This features the coupling of elastic response and electrostatics by means of a capacitary term, which is naturally defined in…
The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order…
We introduce the Euler-Poincar\'e's characteristic with an elementary way and historically. We explain also why one should call Descartes-Poincar\'e characteristic instead of the Euler-Poincar\'e's characteristic. All the considered spaces…
Let $\mathcal{L}_\epsilon$ be a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients. We obtain the uniform $W^{1,p}$ estimate in a Lipschitz domain for solutions to the Dirichlet problem, where…
We introduce a condition on accretive matrix functions, called $p$-ellipticity, and discuss its applications to the $L^p$ theory of elliptic PDE with complex coefficients. Our examples are: (i) generalized convexity of power functions…
We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly…
The aim of this paper is to derive on the basis of the Euler's formula several analytical relations which hold for certain classes of planar graphs and which can be useful in algorithmic graph theory.