| 1. |
- O'Farrell, Anthony G, et al.
(författare)
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Conjugacy of real diffeomorphisms. A survey.
- 2011
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Ingår i: St. Peterburg Mathematical Journal. - 1547-7371 .- 1061-0022. ; 22, s. 1-40
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Tidskriftsartikel (refereegranskat)abstract
- Given a group G, the conjugacy problem in G is the problem of giving an effective procedure for determining whether or not two given elements f, g of G are conjugate, i.e. whether there exists h belonging to G with fh = hg. This paper is about the conjugacy problem in the group Diffeo(I) of all diffeomorphisms of an interval I in R. There is much classical work on the subject, solving the conjugacy problem for special classes of maps. Unfortunately, it is also true that many results and arguments known to the experts are difficult to find in the literature, or simply absent. We try to repair these lacunae, by giving a systematic review, and we also include new results about the conjugacy classification in the general case.
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| 2. |
- Rozenblioum, Grigori, 1948, et al.
(författare)
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On Spectral Estimates for the Schrodinger Operators in Global Dimension 2
- 2014
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Ingår i: St Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 25:3, s. 495-505
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Tidskriftsartikel (refereegranskat)abstract
- The problem of finding eigenvalue estimates for the Schrodinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. In the paper, these results are discussed, and their counterparts are established for the operator on the combinatorial and metric graphs corresponding to the lattice Z(2).
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| 3. |
- Asekritova, Irina, et al.
(författare)
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Interpolation of Besov spaces in the nondiagonal case
- 2006
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 18:4, s. 1-9
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Tidskriftsartikel (refereegranskat)abstract
- In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points (formula presented) includes a ball of ℝm+1, we have (formula presented).
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| 4. |
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| 5. |
- Apushkinskaya, D. E., et al.
(författare)
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Lipschitz property of the free boundary in the parabolic obstacle problem
- 2004
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 15:3, s. 375-391
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Tidskriftsartikel (refereegranskat)abstract
- A parabolic obstacle problem with zero constraint is considered. It is proved, without any additional assumptions on a free boundary, that near the fixed boundary where the homogeneous Dirichlet condition is fulfilled, the boundary of the noncoincidence set is the graph of a Lipschitz function.
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| 6. |
- Baranov, A., et al.
(författare)
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Subspaces of de branges spaces with prescribed growth
- 2007
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 18:5, s. 699-716
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Tidskriftsartikel (refereegranskat)abstract
- The growth properties of de Branges spaces and their subspaces are studied. It is shown that, for each given pair of growth functions λ(r) = O(r) and λ1 = o(λ), there exist de Branges spaces of growth λ that have a de Branges subspace of growth λ1. This phenomenon cannot occur for a class of de Branges spaces that, in a certain sense, behave regularly along the real axis.
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| 7. |
- Kozlov, Vladimir, et al.
(författare)
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The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary
- 2011
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 22:6, s. 941-983
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Tidskriftsartikel (refereegranskat)abstract
- Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity γ that describes the depth O(εγ) of irregularity (ε is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.
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| 8. |
- Shimorin, Serguei
(författare)
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Branching points area theorems for univalent functions
- 2007
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 18:1, s. 141-181
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Tidskriftsartikel (refereegranskat)abstract
- Area theorems of a new type are obtained by considering branching point compositions with univalent functions. Such theorems can be formulated both in the form of integral estimates and in the form of Grunsky and Goluzin-type inequalities.
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| 9. |
- Shah gholian, Henrik
(författare)
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REGULARITY ISSUES FOR SEMILINEAR PDE-S (A NARRATIVE APPROACH)
- 2016
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Ingår i: St. Petersburg Mathematical Journal. - : American Mathematical Society (AMS). - 1061-0022 .- 1547-7371. ; 27:3, s. 577-587
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Tidskriftsartikel (refereegranskat)abstract
- Occasionally, solutions of semilinear equations have better (local) regularity properties than the linear ones if the equation is independent of space (and time) variables. The simplest example, treated by the current author, was that the solutions of Delta u = f(u), with the mere assumption that f ' >= -C, have bounded second derivatives. In this paper, some aspects of semilinear problems are discussed, with the hope to provoke a study of this type of problems from an optimal regularity point of view. It is noteworthy that the above result has so far been undisclosed for linear second order operators, with Holder coefficients. Also, the regularity of level sets of solutions as well as related quasilinear problems are discussed. Several seemingly plausible open problems that might be worthwhile are proposed.
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| 10. |
- Asekritova, Irina, et al.
(författare)
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Interpolation of Besov Spaces in the Non-Diagonal Case
- 2007
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 18:4, s. 511-516
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Tidskriftsartikel (refereegranskat)abstract
- In the nondiagonal case, interpolation spaces for a collection of Besov spaces are described. The results are consequences of the fact that, whenever the convex hull of points includes a ball of , we have whereand.
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| 11. |
- Andersson, John
(författare)
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ALMOST EVERYWHERE REGULARITY FOR THE FREE BOUNDARY OF THE p-HARMONIC OBSTACLE PROBLEM p > 2
- 2021
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Ingår i: St. Petersburg Mathematical Journal. - : American Mathematical Society (AMS). - 1061-0022 .- 1547-7371. ; 32:3, s. 415-433
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Tidskriftsartikel (refereegranskat)abstract
- Let u be a solution to the normalized p-harmonic obstacle problem with p > 2. That is, u is an element of W-1,W-p(B-1(0)), 2 < p < infinity, u >= 0 and div(vertical bar del u vertical bar(p-2) del u) = chi({u>0}) in B-1(0) where u(x) >= 0 and chi(A) is the characteristic function of the set A. The main result is that for almost every free boundary point with respect to the (n - 1)-Hausdorff measure, there is a neighborhood where the free boundary is a C-1,C-beta-graph. That is, for Hn-1- a.e. point x(0) is an element of partial derivative{u > 0}boolean AND B-1(0) there is an r > 0 such that B-r(x(0))boolean AND partial derivative{u > 0} is an element of C-1,C-beta.
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| 12. |
- Berezhnoǐ, Eugenii I., et al.
(författare)
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Represensibility of cones of monotone functions in weighted Lebesgue spaces and extrapolation of operators on these cones
- 2018
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Ingår i: St. Petersburg Mathematical Journal. - : American Mathematical Society (AMS). - 1061-0022 .- 1547-7371. ; 29:4, s. 545-574
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Tidskriftsartikel (refereegranskat)abstract
- It is shown that a sublinear operator is bounded on the cone of monotone functions if and only if a certain new operator related to the one mentioned above is bounded on a certain ideal space defined constructively. This construction is used to provide new extrapolation theorems for operators on the cone in weighted Lebesgue spaces.
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| 13. |
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| 14. |
- El Hajj, L., et al.
(författare)
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REMARKS ON THE CONVEXITY OF FREE BOUNDARIES (SCALAR AND SYSTEM CASES)
- 2021
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Ingår i: St. Petersburg Mathematical Journal. - : American Mathematical Society (AMS). - 1061-0022 .- 1547-7371. ; 32:4, s. 713-727
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Tidskriftsartikel (refereegranskat)abstract
- Convexity is discussed for several free boundary value problems in exterior domains that are generally formulated as Delta u = f(u) in Omega\D, vertical bar del u vertical bar = g on partial derivative Omega, u >= 0 in R-n, where u is assumed to be continuous in R-n, Omega = {u > 0} (is unknown), u = 1 on partial derivative D, and D is a bounded domain in R-n (n >= 2). Here g = g(x) is a given smooth function that is either strictly positive (Bernoulli-type) or identically zero (obstacle type). Properties for f will be spelled out in exact terms in the text. The interest is in the particular case where D is star-shaped or convex. The focus is on the case where f(u) lacks monotonicity, so that the recently developed tool of quasiconvex rearrangement is not applicable directly. Nevertheless, such quasiconvexity is used in a slightly different manner, and in combination with scaling and asymptotic expansion of solutions at regular points. The latter heavily relies on the regularity theory of free boundaries. Also, convexity for several systems of equations in a general framework is discussed, and some ideas along with several open problems are presented.
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| 15. |
- Frank, R. L., et al.
(författare)
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DISCRETE SCHRODINGER OPERATORS WITH DECAYINGAND OSCILLATING POTENTIALS
- 2024
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 35:1, s. 233-244
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Tidskriftsartikel (refereegranskat)abstract
- The paper is devoted to a family of discrete one-dimensional Schodingeroperators with power-like decaying potentials with rapid oscillations. In particular, the potential V(n)=lambda n(-alpha)cos(pi omega n beta)with1
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| 16. |
- Frank, Rupert L., et al.
(författare)
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Discrete Schrödinger operators with decaying and oscillating potentials
- 2024
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022 .- 1547-7371. ; 35:1, s. 233-244
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Tidskriftsartikel (refereegranskat)abstract
- The paper is devoted to a family of discrete one-dimensional Schrödinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential V (n) = λn−α cos(πωnβ) with 1 < β < 2α, it is proved that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.
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| 17. |
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| 18. |
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| 19. |
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| 20. |
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| 21. |
- Kozlov, Vladimir, et al.
(författare)
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A COMPARISON THEOREM FOR SUPER- AND SUBSOLUTIONS OF del(2)u + f(u)=0 AND ITS APPLICATION TO WATER WAVES WITH VORTICITY
- 2019
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 30:3, s. 471-483
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Tidskriftsartikel (refereegranskat)abstract
- A comparison theorem is proved for a pair of solutions that satisfy opposite nonlinear differential inequalities in a weak sense. The nonlinearity is of the form f (u) with f belonging to the class L-loc(p) and the solutions are assumed to have nonvanishing gradients in the domain, where the inequalities are considered. The comparison theorem is applied to the problem describing steady, periodic water waves with vorticity in the case of arbitrary free-surface profiles including overhanging ones. Bounds for these profiles as well as streamfunctions and admissible values of the total head are obtained.
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| 22. |
- Kozlov, Vladimir, et al.
(författare)
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FLOQUET PROBLEM AND CENTER MANIFOLD REDUCTION FOR ORDINARY DIFFERENTIAL OPERATORS WITH PERIODIC COEFFICIENTS IN HILBERT SPACES
- 2021
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 32:3, s. 531-550
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Tidskriftsartikel (refereegranskat)abstract
- A first order differential equation with a periodic operator coefficient acting in a pair of Hilbert spaces is considered. This setting models both elliptic equations with periodic coefficients in a cylinder and parabolic equations with time periodic coefficients. Our main results are a construction of a pointwise projector and a spectral splitting of the system into a finite-dimensional system of ordinary differential equations with constant coefficients and an infinite-dimensional part whose solutions have better properties in a certain sense. This complements the well-known asymptotic results for periodic hypoelliptic problems in cylinders and for elliptic problems in quasicylinders obtained by P. Kuchment and S. A. Nazarov, respectively. As an application, a center manifold reduction is presented for a class of nonlinear ordinary differential equations in Hilbert spaces with periodic coefficients. This result generalizes the known case with constant coefficients explored by A. Mielke.
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| 23. |
- Kresin, G., et al.
(författare)
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SHARP ESTIMATES FOR THE GRADIENT OF SOLUTIONS TO THE HEAT EQUATION
- 2020
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 31:3, s. 495-507
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Tidskriftsartikel (refereegranskat)abstract
- Various sharp pointwise estimates for the gradient of solutions to the heat equation are obtained. The Dirichlet and Neumann conditions are prescribed on the boundary of a half-space. All data belong to the Lebesgue space L-P. Derivation of the coefficients is based on solving certain optimization problems with respect to a vector parameter inside of an integral over the unit sphere.
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| 24. |
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| 25. |
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| 26. |
- Mazya, Vladimir
(författare)
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ELLIPTIC EQUATIONS IN CONVEX DOMAINS
- 2018
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 29:1, s. 155-164
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Tidskriftsartikel (refereegranskat)abstract
- A short survey of a series of results by the author, partly obtained in collaboration with Yu. Burago.
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| 27. |
- Mazya, Vladimir, et al.
(författare)
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ON MESO-SCALE APPROXIMATIONS FOR VIBRATIONS OF MEMBRANES WITH LOWER-DIMENSIONAL CLUSTERS OF INERTIAL INCLUSIONS
- 2021
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 32:3, s. 551-564
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Tidskriftsartikel (refereegranskat)abstract
- Formal asymptotic algorithms are considered for a class of meso-scale approximations for problems of vibration of elastic membranes that contain clusters of small inertial inclusions distributed along contours of predefined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been derived for domains containing lower-dimensional clusters of inclusions.
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| 28. |
- Mazya, Vladimir
(författare)
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SOLVABILITY CRITERIA FOR THE NEUMANN p-LAPLACIAN WITH IRREGULAR DATA
- 2019
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 30:3, s. 485-492
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Tidskriftsartikel (refereegranskat)abstract
- Necessary and sufficient conditions are found for the unique solvability of the Neumann problem for the p-Laplace operator. They characterize both the domain and measures on the right-hand sides.
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| 29. |
- Miyanishi, Y., et al.
(författare)
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EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR IN DIMENSION 3: WEYL'S LAW AND GEOMETRY
- 2020
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Ingår i: St. Petersburg Mathematical Journal. - : American Mathematical Society (AMS). - 1061-0022 .- 1547-7371. ; 31:2, s. 371-386
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Tidskriftsartikel (refereegranskat)abstract
- Asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions are treated. The region Omega subset of R-3 is bounded by a compact surface Gamma = partial derivative Omega, with certain smoothness conditions imposed. The NP operator K-Gamma, called often 'the direct value of the double layer potential', acting in L-2(Gamma), is defined by K-Gamma[psi](X) := 1/4 pi integral(Gamma) < y - x, n(y)>/vertical bar x - y vertical bar(3)psi(y) dS(y), where dS(y) is the surface element and n(y) is the outer unit normal on F. The firstnamed author proved in [27] that the singular numbers s(j) (K-Gamma) of K-Gamma and the ordered moduli of its eigenvalues lambda(j) (K-Gamma) satisfy the Weyl law s(j)(K(Gamma)) similar to vertical bar lambda(j)(K-Gamma)vertical bar similar to {3W(Gamma) - 2 pi chi(Gamma)/128 pi}(1/2) j(-1/2), under the condition that Gamma belongs to the class C-2,C-alpha with alpha > 0, where W(Gamma) and chi(Gamma) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface Gamma. Although the NP operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular numbers exists), the ordered moduli of the eigenvalues of K-Gamma satisfy the same asymptotic relation. The main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary Gamma. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of K-Gamma. A more sophisticated estimate allows us to give a natural extension of the Weyl law for the case of a smooth boundary.
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| 30. |
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| 31. |
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| 32. |
- Tkatjev, Vladimir
(författare)
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SPECTRAL PROPERTIES OF NONASSOCIATIVE ALGEBRAS AND BREAKING REGULARITY FOR NONLINEAR ELLIPTIC TYPE PDEs
- 2020
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Ingår i: St. Petersburg Mathematical Journal. - : AMER MATHEMATICAL SOC. - 1061-0022 .- 1547-7371. ; 31:2, s. 223-240
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? Our aim in the paper is twofold. First, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Second, to define an appropriate algebraic formalism, which makes the analytic part of the construction of nonclassical solutions more transparent.
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| 33. |
- Olofsson, Anders
(författare)
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Operator-valued Bergman inner functions as transfer functions
- 2008
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Ingår i: St. Petersburg Mathematical Journal. - 1061-0022. ; 19:4, s. 603-623
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Tidskriftsartikel (refereegranskat)abstract
- An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discrete time linear systems. This points to a new interaction between the fields of invariant subspace theory and mathematical systems theory.
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| 34. |
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| 35. |
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| 36. |
- Maz´ya, Vladimir, et al.
(författare)
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ON THE SOLVABILITY OF THE NEUMANN PROBLEM IN DOMAINS WITH PEAK
- 2009
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Ingår i: ST PETERSBURG MATHEMATICAL JOURNAL. - 1061-0022. ; 20:5, s. 757-790
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Tidskriftsartikel (refereegranskat)abstract
- The Neumann problem is considered for a quasilinear elliptic equation of second order in a multidimensional domain with the vertex of an isolated peak on the boundary. Under certain assumptions, the study of the solvability of this problem is reduced to a description of the dual to the Sobolev space W-p(1)(Omega) or (in the case of a homogeneous equation with nonhomogeneous boundary condition) to the boundary trace space TWp1(Omega). This description involves Sobolev classes with negative smoothness exponent on Lipschitz domains or Lipschitz surfaces, and also some weighted classes of functions on the interval (0,1) of the real line. The main results are proved on the basis of the known explicit description of the spaces TWp1(Omega) on a domain with an outward or inward cusp on the boundary.
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