The Oblique Derivative Problem of Potential Theory

An important part of the theory of partial differential equations is the theory of boundary problems for elliptic equations and systems of equations. Among such problems those of greatest interest are the so-called non-Fredholm boundary prob lems, whose investigation reduces, as a rule, to the study of singular integral equa tions, where the Fredholm alternative is violated for these problems. Thanks to de velopments in the theory of one-dimensional singular integral equations [28, 29], boundary problems for elliptic equations with two independent variables have been completely studied at the present time [13, 29], which cannot be said about bound ary problems for elliptic equations with many independent variables. A number of important questions in this area have not yet been solved, since one does not have sufficiently general methods for investigating them. Among the boundary problems of great interest is the oblique derivative problem for harmonic functions, which can be formulated as follows: In a domain D with sufficiently smooth boundary r find a harmonic function u(X) which, on r, satisfies the condition n n ~ au ~ . . . :;. . ai (X) ax. = f (X), . . . :;. . [ai (X)]2 = 1, i=l t i=l where aI, . . . , an,fare sufficiently smooth functions defined on r. Obviously the left side of the boundary condition is the derivative of the function u(X) in the direction of the vector P(X) with components al (X), . . . , an(X).

Inhalt

1. Foundations of Potential Theory.- 1. Harmonic Functions and Potential Theory.- 2. Green's Formula and Its Consequences.- 3. Basic Boundary Problems of Potential Theory.- 4. Investigation of Boundary Problems by the Method of Integral Equations.- 5. Harmonic Functions in Axially Symmetric Domains.- 6. General Second-Order Elliptic Equations.- 7. Functions Represented by Potential-Type Integrals.- 8. Gradient Vector Fields of Functions.- 2. Oblique Derivative Problem for Elliptic Equations.- 1. Reduction of the Oblique Derivative Problem to Fredholm Integral Equations.- 2. Reduction of the Oblique Derivative Problem for Harmonic Functions to Fredholm Equations.- 3. Simplest Properties of the Non-Fredholm Oblique Derivative Problem.- 4. Global Methods of Investigation of the Non-Fredholm Oblique Derivative Problem.- 3. Oblique Derivative Problem with Direction of Differentiation Going into the Tangent Plane.- 1. Simplest Consequences of the Maximum Principle.- 2. Generalizations of the Argument Principle.- 3. Measure of Overdeterminedness of the Oblique Derivative Problem.- 4. Oblique Derivative Problem with Polynomial Coefficients.- 5. Reduction of the Oblique Derivative Problem to a Fredholm Integrodifferential Equation.- 6. Boundary Problem for a System of Harmonic Functions.- 4. Systems of Partial Differential Equations Related to Multidimensional Generalizations of the Cauchy-Riemann System.- 1. Analog of the Riemann-Hilbert Problem.- 2. Generalization of a Holomorphic Vector.- 3. Second-Order Systems of Equations.- 4. Elliptic Systems Depending on a Parameter.- 5. Oblique Derivative Problem for Harmonic Functions of Two Variables.- 1. Boundary Properties of Conjugate Harmonic Functions.- 2. An Auxiliary Problem.- 3. Oblique Derivative Problem.- 4. ObliqueDerivative Problem with Discontinuities in the Boundary Condition.- 5. Variation of Level Lines of a Harmonic Function of a Closed Contour.- 6. Multiply Connected Domains.- References.