Partial Differential Equations of Elliptic Type

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.