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We obtain Euler-Lagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and…
A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…
We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider…
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…
We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale invariant even asymptotically and whose behavior can be quite far…
The notion of differentiation index for DAE systems of arbitrary order with generic second members is discussed by means of the study of the behavior of the ranks of certain Jacobian associated sub-matrices. As a by-product, we obtain upper…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups.…
We give model theoretic accounts and proofs of the existence and uniqueness of differential Galois extensions with no new constants, for logarithmic differential equations over a differential field K, when the field C of constants of K is…
We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type…
We derive and analyze high order discontinuous Galerkin methods for second-order elliptic problems on implicitely defined surfaces in $\mathbb{R}^{3}$. This is done by carefully adapting the unified discontinuous Galerkin framework of…
This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…
This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $\alpha$-order. Solutions for these equations with constant coefficients are…
We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems…
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…
We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the…
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our…
Let $C \langle t_1, \dots t_l\rangle$ be the differential field generated by $l$ differential indeterminates $\boldsymbol{t}=(t_1, \dots ,t_l)$ over an algebraically closed field $C$ of characteristic zero. We develop a lower bound…
We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…