Maz’ya V.G. Boundary Behavior of Solutions to Elliptic Equations in General Domains

Paris: European Mathematical Society, 2018. — 443 p.The present book is a detailed exposition of the author and his collaborators’ work on boundedness, continuity, and differentiability properties of solutions to elliptic equations in general domains, that is, in domains that are not a priori restricted by assumptions such as “piecewise smoothness” or being a “Lipschitz graph”. The description of the boundary behavior of such solutions is one of the most difficult problems in the theory of partial differential equations. After the famous Wiener test, the main contributions to this area were made by the author. In particular, necessary and sufficient conditions for the validity of imbedding theorems are given, which provide criteria for the unique solvability of boundary value problems of second and higher order elliptic equations. Another striking result is a test for the regularity of a boundary point for polyharmonic equations.
The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.
Keywords: Wiener test, higher order elliptic equations, elasticity systems, Zaremba problem, weighted positivity, capacityCapacitary modulus of continuity of a harmonic function
Notation and lemmas
Estimates of the solution with finite energy integral
Estimates for solutions with unbounded Dirichlet integral and the Phragmen–Lindelöf principle
Nonhomogeneous boundary condition
Nonhomogeneous equation
Refined estimate for the modulus of continuity of a harmonic function
Improvement of previous estimates for L-harmonic functions
More notations and preliminaries
L-harmonic functions vanishing on a part of the boundary
Behaviour of L-harmonic functions at infinity and near a singular point
Phragmén–Lindelöf type theorems
L-harmonic measure and non-homogeneous Dirichlet data
The Green function and solutions of the non-homogeneous equation
Continuity modulus of solutions and criterion of Hölder regularity of a point
Sufficient conditions for Hölder regularity
Comments to Chapter
Formulation of the Zaremba problem
Auxiliary assertions
Estimates for solutions of the Zaremba problem
Regularity criterion for the point at infinity
Estimates for the Green function and for the harmonic measure of the Zaremba problem
Comments to ChapterWeighted function spaces and weak solutions
Change of variables
Regularity test
The capacity cap
The capacity capK
Comments to Chapter
Preliminaries
Main result
Comments to Chapter
Construction of a special solution
Asymptotic formula for the Hölder exponent
Absence of Hölder continuity
Absence of continuity
Comments to ChapterCapacities and the L-capacitary potential
Weighted positivity of L(partial)
Further properties of the L-capacitary potential
Poincaré inequality with m-harmonic capacity
Proof of sufficiency in Theorem
Equivalence of two definitions of regularity
Regularity as a local property
Proof of necessity in Theorem
Proof of sufficiency in Theorem
Proof of necessity in Theorem
The biharmonic equation in a domain with inner cusp (n >= )
Comments to Chapter
Weighted positivity of (-Delta)m
Local estimates
Pointwise estimates for the Green function
Comments to ChapterNotations and preliminaries
Weighted positivity of (-Delta)^mu
Proof of Lemma
Non-positivity
Local estimates
Regularity of a boundary point
Comments to Chapter
Statement of results
Proof of Theorem
Proof of Theorem
Comments to ChapterIntegral identity and global estimate
Local energy and L estimates
Estimates for the Green function
The capacity Cap_P
-Regularity of a boundary point
Sufficient condition for -regularity
Necessary condition for -regularity
Examples and further properties of Cap_P and Cap
Comments to ChapterIntegral inequalities and global estimate: the case of odd dimension Part I: power weight
Preservation of positivity for solutions of ordinary differential equations
Integral inequalities and global estimate: the case of odd dimension Part II: weight g
Integral identity and global estimate: the case of even dimension Part I: power-logarithmic weight
Integral identity and global estimate: the case of even dimension Part II: weight g
Pointwise and local L estimates for solutions to the polyharmonic equation
Green function estimates
Estimates for solutions of the Dirichlet problem
Comments to ChapterRegularity of solutions to the polyharmonic equation
Higher-order regularity of a boundary point as a local property
The new notion of polyharmonic capacity
Poincaré-type inequalities
Odd dimensions
Even dimensions
Fine estimates on the quadratic forms
Scheme of the proof
Main estimates Bounds for auxiliary functions T and W related to polyharmonic potentials on the spherical shells
Conclusion of the proof
Comments to ChapterGeneral Index
Index of Mathematicians