The Oblique Derivative Problem: The Poincare ProblemThe Oblique Derivative Problem is one of the classical problems in the theory of Partial Differential Equations as well as in Mathematical Physics. In a very important particular case the vector field of the problem is tangent to the boundary of a domain on a subset. This case was introduced and studied by Henri Poincaré when investigating the tides on the Earth. Apart from this, the problem arises naturally when determining the gravitational fields of the Moon, the Earth and other celestial bodies. This is the first monograph, written by one of the leading scientists in thisarea, which is completely devoted to the Oblique Derivative Problem. All the main results in this field are described with full proofs based on modern techniques. The book contains a lot of results that have been unknown to a wide audience till now. A special chapter containing extensive material from geometry, functional analysis and differential equations, which is used in the proofs, makes the book self-contained to a large extent. A short Appendix containing open problems will stimulate the reader to further research in this area. |
Contents
Introduction | 13 |
Statement of the Problem and Main Results | 79 |
Degeneracy on a Manifold | 93 |
Copyright | |
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Common terms and phrases
arbitrary function Banach space bounded c₁ Calderón operator Cauchy inequality Chapter coefficients commutator compact completes the proof converges coordinate system Corollary corresponding d₁ D₂ defined definition denote differential operator Dirichlet problem distribution domain elliptic boundary problems elliptic differential operator equality Ə₁ finite following estimate formula Fourier transform functions F gain in smoothness Green formula H₁ Hence homogeneous Hs(M integral introduce inverse operator Lemma linear operator manifold maximum principle neighborhood norm notations oblique derivative problem obtain open set operator Axe operator of order partition of unity Poincaré problem positive constant priori estimate proof of Theorem prove pseudo-differential operator quadratic form respectively restrictions right-hand side satisfies smooth function solvability space H submanifold Subsec sufficiently small supp symbol t₁ tangent Theorem 4.8 valid vector field x₁ μ₁ ди