A survey of results of St. Petersburg State university research school on nonlinear partial differential equations. I

DOI10.1134/S1063454124010035MaRDI QIDQ6638373

A. I. Nazarov, Nina Uraltseva, Darya E. Apushkinskaya, V. G. Osmolovskij, Arina Alekseevna Arkhipova

Publication date: 14 November 2024

Published in: Vestnik St. Petersburg University. Mathematics (Search for Journal in Brave)

zbMATH Keywords

variational inequalitiesboundary value problemsquasilinear equationscalculus of variations

Mathematics Subject Classification ID

History of mathematics at institutions and academies (non-university) (01A74) Boundary value problems for second-order elliptic equations (35J25) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02) History of partial differential equations (35-03) Quasilinear elliptic equations (35J62)

Cites Work

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• Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^{\infty}$ estimates for asymptotic remainders
• The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential
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• Mathematical problems of the theory of phase transitions in continuum mechanics
• Mixed boundary value problems for non-divergence type elliptic equations in unbounded domains
• On monotonicity of some functionals with variable exponent under symmetrization
• Ol'ga Aleksandrovna Ladyzhenskaya (on her 80th birthday)
• Sharp Estimates for Solutions of a Parabolic Signorini Problem
• Phase Transition in the Mechanics of Continuous Media for Big Loading
• A note on truncations in fractional Sobolev spaces
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• A borderline case of Calderón–Zygmund estimates for nonuniformly elliptic problems
• The effect of curvature in fractional Hardy–Sobolev inequality involving the spectral Dirichlet Laplacian
• A tool for symmetry breaking and multiplicity in some nonlocal problems
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• The eigenfunctions of a Sturm-Liouville problem related to generalized Lyapunov sines
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• On the ``one-dimensionality of the extremal in the Poincaré inequality in a square
• Shape optimization by the homogenization method
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• Nonstationary Venttsel problem with \(VMO_x\) leading coefficients
• On the available local asymptotic efficiency of some goodness-of-fit criteria
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• Maximum estimates for solutions to elliptic and parabolic equations on a book-type stratified set
• Fractional operators as traces of operator-valued curves
• \(L^p\)-theory of Venttsel BVPs with discontinuous data
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• On the Hölder continuity of solutions of the Venttsel' elliptic problem
• Non-critical dimensions for critical problems involving fractional Laplacians
• A counterexample to the Hopf-Oleinik lemma (elliptic case)
• On fractional Laplacians. II
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• Variational inequalities for the fractional Laplacian
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• Isoperimetric inequality and equilibrium states of a two-phase medium
• A generalization of the Pólya-Szegő inequality for one-dimensional functionals
• The nonstationary Venttsel' problem with quadratic growth with respect to the gradient
• Local gradient estimates for solutions of a simplest regularization of a class of nonuniformly elliptic equations
• Vladimir Gilelevich Maz'ya (On the occasion of his 70th anniversary)
• On exact constants in some embedding theorems of high order
• Estimates for the maxima of the moduli of the gradients for solutions of capillarity problems
• Existence of smooth solutions of problems for parabolic systems with convex constraints on the boundary of the domain
• Oblique derivative problem for non-divergence parabolic equations with time-discontinuous coefficients in a wedge
• Quasistationary phase transition problem in two-phase media. One-dimensional case. The zero surface stress coefficient
• On the symmetry of extremal in several embedding theorems
• Limit smoothness of the solutions of variational inequalities under convex constraints on the boundary of the domain
• Estimates on the boundary of a domain for the first derivatives of functions satisfying an elliptic or parabolic inequality
• The A. D. Aleksandrov maximum principle
• On the existence of an extremal function in critical Sobolev trace embedding theorem
• To the 75th birthday of Vsevolod Alekseevich Solonnikov
• A nonlinear boundary value problem with many positive solutions
• Regularity of the solutions of diagonal elliptic systems under convex constraints on the boundary of the domain
• Symmetry and related properties via the maximum principle
• Strong solutions of the generalized Signorini problem
• A nonlinear problem with an oblique derivative for parabolic equations
• Hölder estimates for bounded solutions of problems with an oblique derivative for parabolic equations of nondivergence structure
• Smoothness of solutions of problems with an obstacle
• The oblique boundary-value problem for a quasilinear parabolic equation
• Variational integrals on Orlicz-Sobolev spaces
• Linear two-phase Venttsel problems
• A survey of results on nonlinear Venttsel problems.
• Function theory and partial differential equations. Transl. from the Russian by Tamara Rozhkovkaya
• Fractional Hardy-Sobolev inequalities on half spaces
• On the sharp constant in the ``magnetic 1D embedding theorem
• Boundary value problems with free surfaces in the theory of phase transitions
• On the Pólya-Szégö inequality for functionals with variable exponent
• Thin obstacle problem: estimates of the distance to the exact solution
• On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains
• The Neumann problem for the generalized Hénon equation
• Steady states of anisotropic generalized Newtonian fluids
• Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation
• Estimate of the Hölder norm of the solutions of second-order quasilinear elliptic equations of the general form
• Tail behavior of anisotropic norms for Gaussian random fields
• Boundary estimates for the first-order derivatives of a solution to a nondivergent parabolic equation with composite right-hand side and coefficients of lower-order derivatives
• An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions
• Independence of temperatures of phase transitions of the domain occupied by a two-phase elastic medium
• On a family of extremum problems and the properties of an integral
• A note on higher order fractional Hardy-Sobolev inequalities
• On the constancy of the extremal function in the embedding theorem of fractional order
• On comparison of fractional Laplacians
• Functional a posteriori error estimates for the parabolic obstacle problem
• Behavior of solutions of one-sided variational problems on phase transitions in continuum mechanics at high temperatures
• On the attainability of the best constant in fractional Hardy-Sobolev inequalities involving the spectral Dirichlet Laplacian
• Elliptic Venttsel problems with \(VMO\) coefficients
• A generalization of the Hardy inequality
• Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions
• Sobolev inequalities for fractional Neumann Laplacians on half spaces
• Best constant in a three-parameter Poincaré inequality
• Interior gradient bounds for local minimizers of variational integrals under nonstandard growth conditions
• Abundance of entire solutions to nonlinear elliptic equations by the variational method
• On the boundary point principle for divergence-type equations
• On monotonicity of some functionals under rearrangements
• On the Sobolev and Hardy constants for the fractional Navier Laplacian
• Variational inequalities for the spectral fractional Laplacian
• On monotonicity of some functionals under monotone rearrangement with respect to one variable
• The multiplicity of positive solutions to a quasilinear equation generated by the Il'in-Caffarelli-Cohn-Nirenberg inequality
• Multiplicity of positive solutions for a critical quasilinear Neumann problem
• Multiplicity of positive solutions to the boundary-value problems for fractional Laplacians
• Dirichlet and Neumann problems to critical Emden-Fowler type equations
• Multiplicity of solutions to the quasilinear Neumann problem in the 3-dimensional case
• Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients
• Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche
• Boundary value problems of mathematical physics and related problems of function theory. Part 34. Transl. from the Russian
• Computation of phase transition temperatures for anisotropic model of a two-phase elastic medium
• On the Dirichlet problem generated by the Maz'ya-Sobolev inequality
• Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications
• A comparison theorem for nonsmooth nonlinear operators
• Regularity results for nonlocal evolution Venttsel' problems
• Multiplicity of positive solutions for the generalized Hénon equation with fractional Laplacian
• Solutions with various structures for semilinear equations in \(\mathbb{R}^n\) driven by fractional Laplacian
• Non-autonomous functionals, borderline cases and related function classes
• A centennial of the Zaremba-Hopf-Oleinik lemma
• On Fractional Laplacians
• On fractional Laplacians – 3
• Multiplicity of solutions of the Dirichlet problem for an equation with the $p$-Laplacian in a three-dimensional spherical layer
• Trace Hardy–Sobolev inequalities in cones
• The Harnack inequality and related properties for solutions of elliptic and parabolic equations with divergence-free lower-order coefficients
• The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients

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