ANNIVERSARIES

Y A R O S L A V B O R I S O V I C H L O P A T I N S K I 1 (ON H1S 7 0 T H B I R T H D A Y )

Y a . S. P o d s t r i g a c h , V. G . K o s t e n k o , a n d E. N. P a r a s y u k

November 9, 1976, m a r k e d the 70th bir thday of the eminent Soviet ma themat ic i an , a leading spec ia l i s t in the f ie ld of differential equations and m e m b e r of the Academy of Sciences of the Ukrainian SSR, P r o f e s s o r Ya ros l av Bor i sov ich Lopatinskii .

Lopat inski i ' s f i r s t scientif ic resu t t s w e r e in the f ie ld of c l a s s i ca l ana lys i s . T h e r e a f t e r , his c r e a t i v e th rus t has been comple te ly in the f ield of different ial equations~

F r o m 1946 on, Lopatinskii held the cha i r in different ial equations at Lvov Univers i ty and at the s a m e t ime s e r v e d as the sen ior sc ient i s t of the Lvov branch of the Academy of Sciences of the Ukrainian SSR.

During this pe r iod Lopatinskii was intensively in t e res t ed in p rob l ems of the genera l theory of differential equations of elliptic type and he obtained a number of fundamental resui t s .

In his pape r s of 1950-1951, he cons t ruc ted (simultaneously with and independently of F. John) local fundamental solutions of genera l elliptic s y s t e m s , and based on these he studied the analyt ic p rope r t i e s of solutions of these s y s t e m s as dependent on the smoothness of the i r coeff ic ients . This work enabled him to inves t iga te the s t r u c t u r e of the pr inc ipa l pa r t of a solution in the neighborhood of an i so la ted s ingular point.

Lopat inski i ' s r e su l t s w e r e an impor tan t contr ibut ion to the gene ra l theory of elliptic sy s t ems . They w e r e highly r e g a r d e d by the scientif ic communi ty . In 1951 he was e lec ted a cor responding m e m b e r of the Academy of Sciences of the Ukrainian SSR.

In 1953 Lopatinskii obtained a number of new resu l t s in the theory of boundary p rob t ems for elliptic s y s t e m s . In p a r t i c u l a r , he was f i r s t to fo rmula te and sotve the genera l boundary p rob lem for an elliptic s y s - tem in an a r b i t r a r y convex domain. He obtained a condition for compat ibi l i ty of s y s t e m coeff icients and bound- a ry ope ra to r s which gua ran teed reducibi l i ty of the p rob lem to a s y s t e m of r egu la r in tegra l equations of F r e d - holm type, a condition which soon b e c a m e known throughout the wor ld l i t e ra tu re as the "Lopatinskii condit ion2 This condition, as was es tab l i shed la ter , is a l so neces sa ry for normal solvabil i ty of the boundary p rob lem in a whole sequence of functional spaces (Nirenberg, Browder , Hhrmander ) .

Lopatinskii then appl ied the method which he had developed to the s t u ~ of boundary p rob t ems for second- o rde r elliptic s y s t e m s . We note that this method of fered poss ib i l i t i es for numerous genera l iza t ions s ince a c - compl i shed by many of his students.

We should a lso note that Lopatinskii was one of the f i r s t to cons ider genera l boundary p rob l ems in domains with c o r n e r points , and der ived a fo rmula for computing. the index.

With ci tat ion for his ach ievements in the development of the genera l theory of pa r t i a l different:ial equa- t ions, in 1965 Lopatinskii was e lec ted a full m e m b e r of the Academy of Sciences of the Ukrainian SSiR~

F r o m 1966 on Lopatinskii s e r v e d in Donetsk a t the Inst i tute for Applied Mathemat ics and Mechanics of the Academy of Sciences of the Ukrainian SSH. H e r e he continued his studies of genera l boundary p rob l ems in differential equations using topological methods. His work opened up b road poss ib i l i t i es for applying Morse theory to var ia t iona l elliptic p rob lems . Recently he has obtained impor tan t r esu l t s on solvabi l i ty of the Cauchy p r o b l e m for ope ra to r equations in Banach space and a l so on "a lmos t eve rywhere" solvabil i ty of genera l l inear and nonl inear boundary p rob l ems .

Lopat inski i ,s en t i re scient i f ic act ivi ty has been connected continually with his pedagogic work and with the t ra in ing of young ma themat i c i ans . He has t r a ined m o r e than 40 candidates and 10 doctors of science.

T r a n s l a t e d f rom Ukrainski i Matemat icheski i Zhurnal , VoI. 29, No. 3, pp. 364-365, May-June, 1977.

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W h e r e v e r he has worked he has been the focus Of a col tec t ive of c r e a t i v e young scho la r s .

Lopa t insk i i ' s frui tful scientif ic and pedagogic act ivi ty has been g rea t ly e s t e e m e d by the Soviet gove rn - mea t : he has been awarded the Orde r of Lenin, the Orde r of the October Revolut ion, and var ious medals .

F r o m the bot tom of our h e a r t s , we congra tu la te Ya ros l av Bor i sov ich on his 70th b i r thday, and we wish h im vigorous heal th , a long l ife, and many c r e a t i v e accompl i shmen t s .

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