Steklov problem of the first class for a fractional order delay differential equation

Методом функции Грина получено решение задачи Стеклова первого класса для линейного уравнения с дробной производной Герасимова-Капуто с запаздывающим аргументом. Доказана теорема существования и единственности задачи. The solution to the Steklov problem with conditions of the first class for a linear delay differential equation with a Gerasimov-Caputo fractional derivative is obtained by Green function method. The existence and uniqueness theorem to the problem is proved.

Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

We study the Steklov problem on a subgraph with boundary $$(\Omega ,B)$$ ( Ω , B ) of a polynomial growth Cayley graph $$\Gamma$$ Γ . For $$(\Omega _l, B_l)_{l=1}^\infty$$ ( Ω l , B l ) l = 1 ∞ a sequence of subgraphs of $$\Gamma$$ Γ such that $$|\Omega _l| \longrightarrow \infty$$ | Ω l | ⟶ ∞ , we prove that for each $$k \in {\mathbb {N}}$$ k ∈ N , the kth eigenvalue tends to 0 proportionally to $$1/|B|^{\frac{1}{d-1}}$$ 1 / | B | 1 d - 1 , where d represents the growth rate of $$\Gamma$$ Γ . The method consists in associating a manifold M to $$\Gamma$$ Γ and a bounded domain $$N \subset M$$ N ⊂ M to a subgraph $$(\Omega , B)$$ ( Ω , B ) of $$\Gamma$$ Γ . We find upper bounds for the Steklov spectrum of N and transfer these bounds to $$(\Omega , B)$$ ( Ω , B ) by discretizing N and using comparison theorems.

Existence and multiplicity results for a Steklov problem involving (p(x), q(x))-Laplacian operator

In this work, we are concerned with a generalized Steklov problem with (p(x), q(x))-Laplacian operator. Under some appropriate conditions on the data involved in the elliptic problem, we prove the existence of at least three solutions using Ricceri’s three critical points theorem.

Stability estimates for an inverse Steklov problem in a class of hollow spheres

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem.

Continuity of Weighted Operators, Muckenhoupt Ap Weights, and Steklov Problem for Orthogonal Polynomials

We consider weighted operators acting on $L^p({\mathbb{R}}^d)$ and show that they depend continuously on the weight $w\in A_p({\mathbb{R}}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w({\mathbb{T}})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2({\mathbb{T}})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.