Mathematics Problem Archive

Find useful ways of verifying polyconvexity and quasiconvexity for stored-energy functions arising in anisotropic nonlinear elasticity....

Prove or disprove that, under reasonable growth conditions on $W$, energy minimizers satisfy the weak Euler-Lagrange equations....

Prove or disprove that, under reasonable growth conditions on $W$, an energy-minimizing deformation satisfies $\det Dy^*(x)\ge\varepsilon>0$....

Justify the Ciarlet-Nečas minimization problem, or an appropriate modification, in situations involving smooth self-contact....

Prove or disprove uniqueness of sufficiently smooth equilibrium solutions for pure-displacement problems in homogeneous bodies homeomorphic to a ball ...

Devise general methods for proving the existence of local but nonglobal minimizers and other weak equilibria in nonlinear elastostatics....

Develop local and global bifurcation theories for nonlinear elastostatics with mixed displacement-traction boundary conditions....

Clarify the status of models based on the fracture energy functional (2.31) in the source relative to classical fracture and nonlinear elastostatics....

Develop a qualitative dynamics for dynamic theories of elasticity....

Develop criteria for dynamic stability and instability of equilibria in nonlinear elasticity....

Establish the status of elasticity theory with respect to atomistic models....

For the set of energy-minimizing gradients $K(\theta)$ defined in the source, determine its quasiconvex hull $K(\theta)^{qc}$ for $\theta\le\theta_c$....

For free-energy functions $\psi(A,\theta)$ of elastic crystals, determine boundary conditions under which the minimum is attained and conditions under...

Give a rigorous derivation of models of rods, plates, and shells from three-dimensional elasticity as thickness tends to zero....

Under which additional assumptions does this principle become a rigorous theorem?...

Consider two Lagrangian fibrations $\phi: M^4 \to B$ and $\phi': {M'}^4 \to B'$. Assume that $B$ and $B'$ are affinely equivalent in the sense that th...

Do local symplectic invariants exist for diffeomorphic degenerate singularities? How many and of what kind are they? This question makes sense even in...

Describe the symplectic invariants of stable rank-one singularities described by V. V. Kalashnikov. For such singularities, one of the action variable...

Assume that we know explicit formulas for the action variables $I_1,\dots, I_n$ so that we are able to analyse their asymptotic behaviour in a neighbo...

Describe all topological types of singularities that may appear in algebraically integrable systems with a small ($\leq 3$) number of degrees of freed...

[\'{A}. Pelayo] Extend the classification of semitoric systems $F=(J,H)$ in to allow for $F$ having non-degenerate singularities with hyperbolic block...

[\'{A}. Pelayo] Consider a compact connected $2n$-dimensional symplectic manifold $M$, endowed with a Hamiltonian $(S^1)^{n-1}$-action; these are call...

[N. T. Zung] Study the topology and geometry of these singular fibers and their small neighbourhoods....

[N. T. Zung] Give a clear description of these special singular fibers....

Complete the above table: {\rm (1)} construct new examples of natural Hamiltonian systems on closed two-dimensional surfaces admitting polynomial inte...

Construct a natural Hamiltonian system with a nonconstant potential on $S^2$ that admits a nontrivial polynomial integral of degree $5$ and does not a...

Construct new examples of natural Hamiltonian systems on higher dimensional manifolds which are integrable in the class of integrals polynomial in mom...

Construct stationary axially symmetric 4-dimensional Einstein metrics admitting Killing tensors of higher order....

[Gilkey ] In the Riemannian case, is every $1$-homogeneous manifold locally homogeneous?...

In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynom...

{\it Construct a natural Hamiltonian system on the 2-sphere with a nonconstant potential which is superintegrable by integrals of degree $\ge 3$ and a...

Does there exist a non-conformally flat metric on the sphere $S^n$, $n>2$, whose geodesic flow is maximally superintegrable (in the class of integrals...

Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse....

Prove the algebraic version of Birkhoff conjecture for outer billiards in a non-Euclidean surface of constant curvature....

Prove Conjecture [source label: Descon] it the case that the foliation admits (i) a rational, (ii) an algebraic first integral....

Is it possible to choose the domain so that the dynamics of the corresponding billiard map are locally (near the 2-periodic orbit) conjugated to the d...

Are there plane billiards, other than ellipses, that possess rational caustics with two different values of the rotation numbers? Same question for ou...

Suppose that a billiard table has a sequence of convex caustics with rotation numbers converging to some number $\omega \in (0,1/2)$. Does it imply th...

{\rm (M. Bialy)} Are there multi-dimensional convex billiards, other than ellipsoids, having invariant hypersurfaces in the phase space? Same question...

{\rm (A. Glutsyuk, S. Tabachnikov )} Given two nested closed convex hypersurfaces, assume that the billiard transformations on the set of oriented lin...

{\rm (V. Retakh) Establish complete integrability of the noncommutative version of the leapfrog map. Define noncommutative versions of the pentagram m...

Consider the action of $ Aut (V,J)$ on $V$. Describe the partition of $V$ into $ Aut (V,J)$-orbits. More generally, describe the action of $ Aut (V,J)...

Find necessary and sufficient conditions for the bi-Lagrangian Grassmannian $LG(V,J)$ to be a smooth algebraic variety. Describe the partition of $LG(...

Do bi-integrable systems exist for each algebraic type?...

Compute the Jordan--Kronecker invariants for the most interesting classes of Lie algebras, and particularly for the following {\/\rm(}a{\/\rm)} semidi...

Are there any restrictions on the algebraic type of the pencils $\mathcal{A}_{x+\lambda a}$? Which algebraic types can be realised by means of an appr...

Study examples of ``quadratic $+$ linear'' Poisson pencils. Compute their algebraic types and construct complete families of polynomials in bi-involut...

Is it true that for any quadratic Poisson bracket (defined on a vector space), there exists a polynomial integrable system?...

Describe closed manifolds $M$ which admit Nijenhuis operators $L(x)$ that are algebraically regular at each point $x\in M$....

Let us fix a certain algebraic type of a linear operator, i.e., its Segre characteristic (see above). Does there exist a Nijenhuis operator $L$ in $\m...