This book presents boundary value problems for elliptic differential equations using ; _ ; methods. The boundary value problems considered are those in weak form. The methods used are illustrated by the following: Let ; be a complex Hilbert space, with norm ; ,|_ ; let ; be a locally convex linear topological space, suppose ; subset ; (the injection ; colon Vrightarrow ; is continuous) and ; is dense in ; suppose there is an involution ; rightarrowoverline ; in ; which leaves ; invariant, suppose there is a sesquilinear form ; (u,v ; (linear in ; semi-linear (or conjugate linear) in ; on ; imes ; and suppose there is an involution on ; defined by ; angleoverline f,grangle=langle f,overline grangl ; (i.e., ; verline f(g)=f(overline g ; . Let ; be the set of all ; in ; such that the map ; rightarrow a(u,v ; is continuous on ; with respect to the topology of ; Then if ; in ; there exists a unique element ; in ; such that ; angle Au,overline vrangle=a(u,v) (langle f,grangle=f(g) ; for all ; in ; If ; (u,v ; is ; elliptic, i.e., if there exists an ; lpha> ; such that ; extRe,a(u,u)geqalpha|u|_V^ ; for all ; in ; then the operator ; is an isomorphism from ; onto ; As an example of the situation just described abstractly, let ; meg ; be an open set in ; ^ ; suppose ; _0^1(Omega)subset Vsubset H'(Omega ; (Sobolev spaces), ; u,v)_V=(u,v)_ ; (u,v)_0+(u,v)_1, ; where ; u,v)_0=int_Omega uv,d ; and ; u,v)_1,0=int_Omega
abla uãoverline
abla v,d ; , ; =L^2(Omega)=Q ; suppose the involution is ; rightarrowoverline ; and ; (u,v)=(u,v)_1,0+lambda(u,v)_ ; Then ; = ; in VcolonDelta uin L^2(Omega),(-Delta u,v)_0=(u,v)_1, ; for all ; in ; nd ; =-Delta u+lambda ; and for each ; in L^2(Omega ; the unique solution ; in ; such that ; u= ; belongs to ; _2^2(Omega ; and satisfies ; artial u/partial n= ; on ; artialOmeg ; provided ; meg ; is a class ; ^ ; If ; =H_0^1(Omega ; then ; in H_2^2(Omega ; and ; = ; on ; artialOmeg ; Minor changes in the theory enable the author to handle corresponding problems for various types of higher order equations. The author proves the regularity of the solutions on the interior and up to the boundary in some cases ; ^inft ; coefficients and ; ^inft ; boundary, etc.). He also discusses the regularity of certain Green's functions. His proofs of regularity are rather complicated. This is to be excused, however, since the book was issued originally in 1957; the present book is merely a re-issue in 1967. In spite of this, the material in the book is still of considerable interest.