Partial Differential Equations of Elliptic Type

In the theory of partial differential equations, the study of elliptic equations occupies a preeminent position, both because of the importance which it assumes for various questions in mathematical physics, and because of the completeness of the results obtained up to the present time. In spite of this, even in the more classical treatises on analysis the theory of elliptic equations has been considered and illustrated only from particular points of view, while the only expositions of the whole theory, the extremely valuable ones by LICHTENSTEIN and AscoLI, have the charac­ ter of encyclopedia articles and date back to many years ago. Consequently it seemed to me that it would be of some interest to try to give an up-to-date picture of the present state of research in this area in a monograph which, without attaining the dimensions of a treatise, would nevertheless be sufficiently extensive to allow the expo­ sition, in some cases in summary form, of the various techniques used in the study of these equations.

I. Boundary value problems for linear equations.- 1. Sets of points; functions.- 2. Elliptic equations.- 3. Maximum and minimum properties of the solutions of elliptic equations.- 4. Various types of boundary value problems.- 5. Uniqueness theorems.- 6. Green’s formula.- 7. Compatibility conditions for the boundary value problems; other uniqueness theorems.- 8. Levi functions.- 9. Stokes’s formula.- 10. Fundamental solutions; Green’s functions.- II. Functions represented by integrals.- 11. Products of composition of two kernels.- 12. Functions represented by integrals.- 13. Generalized domain potentials.- 14. Generalized single layer potentials.- 15. Generalized double layer potentials.- 16. Construction of functions satisfying assigned boundary conditions.- III. Transformation of the boundary value problems into integral equations.- 17. Review of basic knowledge about integral equations.- 18. The method of potentials.- 19. Existence of fundamental solutions. Unique continuation property.- 20. Principal fundamental solutions.- 21. Transformation of the Dirichlet problem into integral equations.- 22. Transformation of Neumann’s problem into integral equations.- 23. Transformation of the oblique derivative problem into integral equations.- 24. The method of the quasi-Green’s functions.- IV Generalized solutions of the boundary value problems..- 25. Generalized elliptic operators.- 26. Equations with singular coefficients and known terms.- 27. Local properties of the solutions of elliptic equations….- 28. Generalized solutions according to Wiener of Dirichle?s problem.- 29. Generalized boundary conditions.- 30. Weak solutions of the boundary value problems.- 31. The method of Fischer-Riesz equations.- 32. The method of the minimum.- V. A priori majorization of the solutions of the boundary value problems.- 33. Orders of magnitude of the successive derivatives of a function and of their HÖLDER coefficients.- 34. Majorization in C(N,?) of the solutions of equations with constant coefficients.- 35. General majorization formulas in C(n,?).- 36. Method of continuation for the proof of the existence theorem for Dirichìe?s problem.- 37. General majorization formulas in Hk,p.- 38. Existence and regularization theorems.- 39. A priori bounds for the solutions of the second and third boundary value problem.- VI. Nonlinear equations.- 40. General properties of the solutions.- 41. Functional equations in abstract spaces.- 42. Dirichle?s problem for equations in m variables.- 43. Dirichle?s problem for equations in two variables.- 44. Equations in the analytic field.- 45. Equations in parametric form.- 46. The Neumann and oblique derivative problems.- 47. Equations of particular type.- VII. Other research on equations of second order. Equations of higher order. Systems of equations.- 48. Second order equations on a manifold.- 49. Second order equations in unbounded domains.- 50. Other problems for second order equations.- 51. Inverse problems and axiomatic theory for second order equations.- 52. Equations of higher order.- A. Elliptic operators of higher order.- B. Green’s formula. Fundamental solutions. Cauchy’s problem and the unique continuation property.- C. Local properties of the solutions.- D. Dirichle?s and general boundary value problems.- E. Equations on manifolds. Pseudo-differential operators. Non local boundary problems.- F. Equations in unbounded domains.- G. Equations in regions with degenerate boundary.- H. Mixed and transmission problems.- I. Polyharmonic functions.- J. Nonlinear equations. Questions of analyticity.- 53. Systems of equations of the first order.- 54. Canonical form of elliptic equations.- 55. Systems of higher order equations.- A. Elliptic systems.- B. Green’s formula. Fundamental matrix. Cauchy’s problem and the unique continuation property.- C. Boundary value problems.- D. Systems of second order equations.- E. Systems on manifolds. Pseudo-differential operators...- F. Systems on unbounded domains.Transmission problems.- G. Systems of equations from mathematical physics.- H. Nonlinear systems. Questions of analyticity.- 56. Degenerate elliptic equations. Questions of a small parameter.- Author Index.