arXiv subjects
Publications and source records attributed to Alexander I. Nazarov.
The first half of the 20th century in the history of Russian mathematics is striking with a combination of dramaticism, sometimes a tragedy, and outstanding achievements. The paper is devoted to St. Petersburg-Leningrad Mathematical School. It is based on a chapter in the multi-author monograph "Mathematical Petersburg. History, science, sights" (SPb: Educational projects, 2018, 336 pp. in Russian).
We provide some versions of the Zaremba-Hopf-Oleinik boundary point lemma for general elliptic and parabolic equations in divergence form under the sharp requirements on the coefficients of equations and on the boundaries of domains.
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.
We consider different fractional Neumann Laplacians of order s, 0<s<1, namely, the Restricted Neumann Laplacian, the Semirestricted Neumann Laplacian and the Spectral Neumann Laplacian. In particular, we are interested in attainability of Sobolev constants for these operators in half-spaces.
We investigate the qualitative properties of solution to the Zaremba type problem in unbounded domain for the non-divergence elliptic equation with possible degeneration at infinity. The main result is Phragmén-Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the "thickness" of its Dirichlet portion. The result is formulated in terms of so-called $s$-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain "admissibility" condition in the sequence of layers converging to infinity.
The year 2017 marked the 130th anniversary of the prominent Russian mathematician Vladimir Ivanovich Smirnov. We review some aspects of his life and his mathematical accomplishments.
Paper dedicated to qualitative study of the solution of the Zaremba type problem in Lipschitz domain with respect to the elliptic equation in non-divergent form. Main result is Landis type Growth Lemma in spherical layer for Mixed Boundary Value Problem in the class of "admissible domain". Based on the Growth Lemma Phragmén-Lindelöf theorem is proved at junction point of Dirichlet boundary and boundary over which derivative in non-tangential direction is defined.
We give a unified approach to strong maximum principles for a large class of nonlocal operators of the order $s\in(0,1)$, that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.
We study the Nemytskii operators $u\mapsto |u|$ and $u\mapsto u^{\pm}$ in fractional Sobolev spaces $H^s(\mathbb R^n)$, $s>1$.
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-Δ_Ω\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $Ω\subset\mathbb R^n$.
In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $Ω\subset\mathbb R^n$, under mild assumptions on the data.
The present paper is the natural evolution of arXiv:1308.3606. For $s>-1$ we compare two natural types of fractional Laplacians $(-Δ)^s$, namely, the "Navier" and the "Dirichlet" ones. As a main tool, we give the "dual" Caffarelli--Silvestre and Stinga--Torrea variational characterizations of these operators for $s\in(-1,0)$.
We investigate the role of the noncompact group of dilations in $\mathbb R^n$ on the difference of the quadratic forms associated to the fractional Dirichlet and Navier Laplacians. Then we apply our results to study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving the Navier-Laplacian.
In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some basic domains (rectangles, cubes, and right triangles). In the last section, we derive an a estimate of the difference between the exact solutions of two boundary value problems. Constants in Poincare type inequalities enter these estimates, which provide guaranteed a posteriori error control.
We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.
We study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving Dirichlet-Laplacian of arbitrary real order $m>0$.
We compare two natural types of fractional Laplacians $(-Δ)^s$, "Navier" and "Dirichlet" ones. We show that for $0<s<1$ their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.
We discuss the problem where the extremal function in embedding theorem of some (generally speaking, non-integer) order is the constant function. We obtain the necessary and sufficient conditions of this.