Analysis and Applications

We study the nonlinear one-dimensional p-Laplacian equation

Let X = {x ∈ ℝⁿ : a < |x| < b} be a generalized annulus and consider the Dirichlet energy functional

Let α and β be real numbers. We prove that the functional inequality

To help understand various reproducing kernels used in applied sciences, we investigate the inclusion relation of two reproducing kernel Hilbert spaces. Characterizations in terms of feature maps of the corresponding reproducing kernels are established. A full table of inclusion relations among widely-used translation invariant kernels is given. Concrete examples for Hilbert–Schmidt kernels are presented as well. We also discuss the preservation of such a relation under various operations of reproducing kernels. Finally, we briefly discuss the special inclusion with a norm equivalence.

In 1954, it was proved that if f is infinitely differentiable in the interval I and some derivatives (of order depending on x) vanish at each x, then f is a polynomial. Later, it was generalized for the multivariable case. A further extension for distributions is possible. If Ω ⊆ Rⁿ is a non-empty connected open set, and for every , there exists m(φ) ∈ N such that (D^αu)(φ) = 0 for all multi-indices α satisfying ‖α‖ = m(φ), then u is a polynomial (in distributional sense).

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.