{"id":{"repo_id":"greece","oai_identifier":"oai:10442/0700"},"canonical_url":"https://search.dev.ndltd.org/etd/greece/oai:10442/0700","repository":{"repo_id":"greece","name":"Greek National Archive of PhD Theses","base_url":"https://phdtheses.ekt.gr/eadd_oai/request"},"display":{"title":"ΜΙΑ ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΠΡΟΣΕΓΓΙΣΗ ΤΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE","abstract":"THE SUBJECT WE ARE DEALING WITH CAN BE DIVIDED AS FOLLOWS: A) POSSIBILITY OF STOCHASTIC REPRESENTATION OF THE GENERALISED SOLUTION OF THE REAL MONGE-AMPERE EQUATION MU=F>0 K D, K=Φ ON ΘD, D C IRD OPEN-BOUNDED CONVEX. PARTLY IN PARALLEL TO THE WORK BY B. GAVEAN, WHO TREATED THE COMPLEX CASE, GENERAL RESULTS ARE OBTAINED. SPECIFICALLY THE SOLUTION CAN BE STOCHASTICALLY REPRESENTED IF O<F E L (D), Φ E C (ΘD), D C IRD STRICTLY CONVEX (ΘD NOT NECESSARILLY SMOOTH). PARTICULARLY IF Φ=0 A BARRIER CONDITION ON ΘD IS SUFFICIENT. IN THIS CONTEXT A SIMPLIFIEDPROOF OF KRYLER'S INEQUALITY IS OBTAINED. B) PROBABILISTIC DEMONSTRATION OF THE EXISTENCE OF THE SOLUTION OF THE MONGE- AMPERE EQUATION (IN THE GENERALISED SENSE) OPTIMAL STOCHASTIC CONTROL TECHNIQUES IN CONJUNCTION WITH A STOCHASTIC VERSION OF IDEAS DEVELOPED BY P.L. LIONS CAN ASSURE A PURELY PROBABILISTIC DEMONSTRATION OF EXISTENCE OF THE SOLUTION. THE METHOD AFTER MINOR CHANGES, CAN ALSO APPLY TO THE COMPLEX CASE. AN E-OPTIMAL CONTROL CAN BE REALISED FOR THE ASSOCIATED STOCHASTIC PROBLEM WHEN FE LP (D), P>D.","abstract_html":"THE SUBJECT WE ARE DEALING WITH CAN BE DIVIDED AS FOLLOWS: A) POSSIBILITY OF STOCHASTIC REPRESENTATION OF THE GENERALISED SOLUTION OF THE REAL MONGE-AMPERE EQUATION MU=F&gt;0 K D, K=Φ ON ΘD, D C IRD OPEN-BOUNDED CONVEX. PARTLY IN PARALLEL TO THE WORK BY B. GAVEAN, WHO TREATED THE COMPLEX CASE, GENERAL RESULTS ARE OBTAINED. SPECIFICALLY THE SOLUTION CAN BE STOCHASTICALLY REPRESENTED IF O&lt;F E L (D), Φ E C (ΘD), D C IRD STRICTLY CONVEX (ΘD NOT NECESSARILLY SMOOTH). PARTICULARLY IF Φ=0 A BARRIER CONDITION ON ΘD IS SUFFICIENT. IN THIS CONTEXT A SIMPLIFIEDPROOF OF KRYLER&#x27;S INEQUALITY IS OBTAINED. B) PROBABILISTIC DEMONSTRATION OF THE EXISTENCE OF THE SOLUTION OF THE MONGE- AMPERE EQUATION (IN THE GENERALISED SENSE) OPTIMAL STOCHASTIC CONTROL TECHNIQUES IN CONJUNCTION WITH A STOCHASTIC VERSION OF IDEAS DEVELOPED BY P.L. LIONS CAN ASSURE A PURELY PROBABILISTIC DEMONSTRATION OF EXISTENCE OF THE SOLUTION. THE METHOD AFTER MINOR CHANGES, CAN ALSO APPLY TO THE COMPLEX CASE. AN E-OPTIMAL CONTROL CAN BE REALISED FOR THE ASSOCIATED STOCHASTIC PROBLEM WHEN FE LP (D), P&gt;D.","abstract_has_math":false,"creators":["Σπηλιώτης, Ιωάννης"],"institution":"National Technical University of Athens (NTUA)","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1988,"date_issued":"1988","date_published":"1988","updated_at":"2026-07-24T02:25:24Z","subjects":["ΑΝΙΣΟΤΗΤΑ ΚΡΥΛΟΒ","ΓΕΝΙΚΕΥΜΕΝΗ ΛΥΣΗ","ΕΞΙΣΩΣΗ ΜΟΝΖ-ΑΜΠΕΡ","ΣΤΟΧΑΣΤΙΚΗ ΑΝΑΠΑΡΑΣΤΑΣΗ","ΣΤΟΧΑΣΤΙΚΟΣ ΒΕΛΤΙΣΤΟΣ ΕΛΕΓΧΟΣ","GENE LISED SOLUTION","KRYLOV'S INEQUALITY","MONGE-AMPERE EQUATION","STOCHASTIC OPTIMAL CONTROL","STOCHASTIC REPRESENTATION","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"],"languages":["gre"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/0700"],"render_values":[{"text":"10.12681/eadd/0700","href":"https://doi.org/10.12681/eadd/0700","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10442/hedi/0700","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Σπηλιώτης, Ιωάννης"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["1988"]},{"key":"dc:publisher","label":"Institution","values":["National Technical University of Athens (NTUA)","Εθνικό Μετσόβιο Πολυτεχνείο (ΕΜΠ)"]},{"key":"dc:type","label":"Dc Type","values":["PhD Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ΑΝΙΣΟΤΗΤΑ ΚΡΥΛΟΒ","ΓΕΝΙΚΕΥΜΕΝΗ ΛΥΣΗ","ΕΞΙΣΩΣΗ ΜΟΝΖ-ΑΜΠΕΡ","ΣΤΟΧΑΣΤΙΚΗ ΑΝΑΠΑΡΑΣΤΑΣΗ","ΣΤΟΧΑΣΤΙΚΟΣ ΒΕΛΤΙΣΤΟΣ ΕΛΕΓΧΟΣ","GENE LISED SOLUTION","KRYLOV'S INEQUALITY","MONGE-AMPERE EQUATION","STOCHASTIC OPTIMAL CONTROL","STOCHASTIC REPRESENTATION","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["gre"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.12681/eadd/0700","http://hdl.handle.net/10442/hedi/0700"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["THE SUBJECT WE ARE DEALING WITH CAN BE DIVIDED AS FOLLOWS: A) POSSIBILITY OF STOCHASTIC REPRESENTATION OF THE GENERALISED SOLUTION OF THE REAL MONGE-AMPERE EQUATION MU=F>0 K D, K=Φ ON ΘD, D C IRD OPEN-BOUNDED CONVEX. PARTLY IN PARALLEL TO THE WORK BY B. GAVEAN, WHO TREATED THE COMPLEX CASE, GENERAL RESULTS ARE OBTAINED. SPECIFICALLY THE SOLUTION CAN BE STOCHASTICALLY REPRESENTED IF O<F E L (D), Φ E C (ΘD), D C IRD STRICTLY CONVEX (ΘD NOT NECESSARILLY SMOOTH). PARTICULARLY IF Φ=0 A BARRIER CONDITION ON ΘD IS SUFFICIENT. IN THIS CONTEXT A SIMPLIFIEDPROOF OF KRYLER'S INEQUALITY IS OBTAINED. B) PROBABILISTIC DEMONSTRATION OF THE EXISTENCE OF THE SOLUTION OF THE MONGE- AMPERE EQUATION (IN THE GENERALISED SENSE) OPTIMAL STOCHASTIC CONTROL TECHNIQUES IN CONJUNCTION WITH A STOCHASTIC VERSION OF IDEAS DEVELOPED BY P.L. LIONS CAN ASSURE A PURELY PROBABILISTIC DEMONSTRATION OF EXISTENCE OF THE SOLUTION. THE METHOD AFTER MINOR CHANGES, CAN ALSO APPLY TO THE COMPLEX CASE. AN E-OPTIMAL CONTROL CAN BE REALISED FOR THE ASSOCIATED STOCHASTIC PROBLEM WHEN FE LP (D), P>D.","ΤΟ ΠΕΡΙΕΧΟΜΕΝΟ ΤΗΣ ΔΙΑΤΡΙΒΗΣ ΣΥΝΙΣΤΑΤΑΙ ΑΠΟ ΤΑ ΠΑΡΑΚΑΤΩ: Α) ΔΥΝΑΤΟΤΗΤΑ ΣΤΟΧΑΣΤΙΚΗΣ ΑΝΑΠΑΡΑΣΤΑΣΗΣ ΤΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ ΤΗΣ ΠΡΑΓΜΑΤΙΚΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE MU=F>0 ΣΤΟ D, Κ=Φ ΣΤΟ ΘD, DC IRD ΑΝΟΙΚΤΟ ΚΥΡΤΟ, ΦΡΑΓΜΕΝΟ. ΠΑΡΑΛΛΗΛΩΣ ΕΝ ΜΕΡΕΙ ΜΕ ΕΡΓΑΣΙΑ ΤΟΥ B . GOVEAN ΠΟΥ ΑΦΟΡΑ ΤΗΝ ΜΙΓΑΔΙΚΗ ΕΞΙΣΩΣΗ MONGE-AMPERE, ΔΙΑΤΥΠΩΝΟΝΤΑΙ ΓΕΝΙΚΑ ΑΠΟΤΕΛΕΣΜΑΤΑ ΣΧΕΤΙΚΑ ΜΕ ΤΗ ΔΥΝΑΤΟΤΗΤΑ ΣΤΟΧΑΣΤΙΚΗΣ ΑΝΑΠΑΡΑΣΤΑΣΗΣ ΤΗΣΛΥΣΗΣ. ΕΙΔΙΚΩΤΕΡΑ: Η ΛΥΣΗ Κ ΜΠΟΡΕΙ ΝΑ ΑΝΑΠΑΡΑΣΤΑΘΕΙ ΣΤΟΧΑΣΤΙΚΑ ΟΤΑΝ 0 <F E L (D), ΦΕ C(OD), DCIRD ΑΥΣΤΗΡΑ ΚΥΡΤΟ (ΘD ΟΧΙ ΚΑΤ'ΑΝΑΓΚΗ ΛΕΙΟ). ΙΔΙΑΙΤΕΡΑ ΑΝ Φ=0 ΜΙΑ BARRIER CONDITION Σ ΕΙΝΑΙ ΑΡΚΕΤΗ. Σ'ΑΥΤΑ ΤΑ ΠΛΑΙΣΙΑ ΜΙΑ ΑΠΛΟΠΟΙΗΜΕΝΗ ΑΠΟΔΕΙΞΗΤΗΣ ΑΝΙΣΟΤΗΤΑΣ KRYLOV ΜΠΟΡΕΙ ΝΑ ΕΠΙΤΕΥΧΘΕΙ. Β) ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΑΠΟΔΕΙΞΗ ΥΠΑΡΞΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ ΤΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE. ΤΕΧΝΙΚΕΣ ΤΟΥ ΒΕΛΤΙΣΤΟΥ ΣΤΟΧΑΣΤΙΚΟΥ ΕΛΕΓΧΟΥ ΕΝ ΣΥΝΔΥΑΣΜΟ ΜΕ ΤΗ ΣΤΟΧΑΣΤΙΚΗ ΕΚΔΟΧΗ ΙΔΕΩΝ ΤΟΥ P.L. LIONS, ΕΞΑΣΦΑΛΙΖΟΥΝ ΜΙΑ ΚΑΘΑΡΑ ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΑΠΟΔΕΙΞΗ ΥΠΑΡΞΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ. Η ΜΕΘΟΔΟΣ,ΜΕ ΜΙΚΡΕΣ ΑΛΛΑΓΕΣ, ΕΙΝΑΙ ΚΑΤΑΛΛΗΛΗ ΓΙΑ ΤΗΝ ΜΙΓΑΔΙΚΗ ΕΞΙΣΩΣΗ MONGE-AMPERE. ΜΕ ΒΑΣΗ ΤΗ ΔΙΑΔΙΚΑΣΙΑ ΑΥΤΗ, ΜΠΟΡΕΙ ΝΑ ΚΑΤΑΣΚΕΥΑΣΤΕΙ ΕΝΑΣ Ε-ΒΕΛΤΙΣΤΟΣ ΕΛΕΓΧΟΣ ΓΙΑ ΤΟΠΡΟΚΥΠΤΟΝ ΠΡΟΒΛΗΜΑ ΣΤΟΧΑΣΤΙΚΟΥ ΕΛΕΓΧΟΥ ΟΤΑΝ FE LP(D), P>D."]},{"key":"dc:title","label":"Title","values":["ΜΙΑ ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΠΡΟΣΕΓΓΙΣΗ ΤΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE","A PROBABILISTIC APROACH TO THE MONGE-AMPERE EQUATION"]}]}],"canonical_facts":{"dc:creator":["Σπηλιώτης, Ιωάννης"],"dc:date":["1988"],"dc:description":["THE SUBJECT WE ARE DEALING WITH CAN BE DIVIDED AS FOLLOWS: A) POSSIBILITY OF STOCHASTIC REPRESENTATION OF THE GENERALISED SOLUTION OF THE REAL MONGE-AMPERE EQUATION MU=F>0 K D, K=Φ ON ΘD, D C IRD OPEN-BOUNDED CONVEX. PARTLY IN PARALLEL TO THE WORK BY B. GAVEAN, WHO TREATED THE COMPLEX CASE, GENERAL RESULTS ARE OBTAINED. SPECIFICALLY THE SOLUTION CAN BE STOCHASTICALLY REPRESENTED IF O<F E L (D), Φ E C (ΘD), D C IRD STRICTLY CONVEX (ΘD NOT NECESSARILLY SMOOTH). PARTICULARLY IF Φ=0 A BARRIER CONDITION ON ΘD IS SUFFICIENT. IN THIS CONTEXT A SIMPLIFIEDPROOF OF KRYLER'S INEQUALITY IS OBTAINED. B) PROBABILISTIC DEMONSTRATION OF THE EXISTENCE OF THE SOLUTION OF THE MONGE- AMPERE EQUATION (IN THE GENERALISED SENSE) OPTIMAL STOCHASTIC CONTROL TECHNIQUES IN CONJUNCTION WITH A STOCHASTIC VERSION OF IDEAS DEVELOPED BY P.L. LIONS CAN ASSURE A PURELY PROBABILISTIC DEMONSTRATION OF EXISTENCE OF THE SOLUTION. THE METHOD AFTER MINOR CHANGES, CAN ALSO APPLY TO THE COMPLEX CASE. AN E-OPTIMAL CONTROL CAN BE REALISED FOR THE ASSOCIATED STOCHASTIC PROBLEM WHEN FE LP (D), P>D.","ΤΟ ΠΕΡΙΕΧΟΜΕΝΟ ΤΗΣ ΔΙΑΤΡΙΒΗΣ ΣΥΝΙΣΤΑΤΑΙ ΑΠΟ ΤΑ ΠΑΡΑΚΑΤΩ: Α) ΔΥΝΑΤΟΤΗΤΑ ΣΤΟΧΑΣΤΙΚΗΣ ΑΝΑΠΑΡΑΣΤΑΣΗΣ ΤΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ ΤΗΣ ΠΡΑΓΜΑΤΙΚΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE MU=F>0 ΣΤΟ D, Κ=Φ ΣΤΟ ΘD, DC IRD ΑΝΟΙΚΤΟ ΚΥΡΤΟ, ΦΡΑΓΜΕΝΟ. ΠΑΡΑΛΛΗΛΩΣ ΕΝ ΜΕΡΕΙ ΜΕ ΕΡΓΑΣΙΑ ΤΟΥ B . GOVEAN ΠΟΥ ΑΦΟΡΑ ΤΗΝ ΜΙΓΑΔΙΚΗ ΕΞΙΣΩΣΗ MONGE-AMPERE, ΔΙΑΤΥΠΩΝΟΝΤΑΙ ΓΕΝΙΚΑ ΑΠΟΤΕΛΕΣΜΑΤΑ ΣΧΕΤΙΚΑ ΜΕ ΤΗ ΔΥΝΑΤΟΤΗΤΑ ΣΤΟΧΑΣΤΙΚΗΣ ΑΝΑΠΑΡΑΣΤΑΣΗΣ ΤΗΣΛΥΣΗΣ. ΕΙΔΙΚΩΤΕΡΑ: Η ΛΥΣΗ Κ ΜΠΟΡΕΙ ΝΑ ΑΝΑΠΑΡΑΣΤΑΘΕΙ ΣΤΟΧΑΣΤΙΚΑ ΟΤΑΝ 0 <F E L (D), ΦΕ C(OD), DCIRD ΑΥΣΤΗΡΑ ΚΥΡΤΟ (ΘD ΟΧΙ ΚΑΤ'ΑΝΑΓΚΗ ΛΕΙΟ). ΙΔΙΑΙΤΕΡΑ ΑΝ Φ=0 ΜΙΑ BARRIER CONDITION Σ ΕΙΝΑΙ ΑΡΚΕΤΗ. Σ'ΑΥΤΑ ΤΑ ΠΛΑΙΣΙΑ ΜΙΑ ΑΠΛΟΠΟΙΗΜΕΝΗ ΑΠΟΔΕΙΞΗΤΗΣ ΑΝΙΣΟΤΗΤΑΣ KRYLOV ΜΠΟΡΕΙ ΝΑ ΕΠΙΤΕΥΧΘΕΙ. Β) ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΑΠΟΔΕΙΞΗ ΥΠΑΡΞΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ ΤΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE. ΤΕΧΝΙΚΕΣ ΤΟΥ ΒΕΛΤΙΣΤΟΥ ΣΤΟΧΑΣΤΙΚΟΥ ΕΛΕΓΧΟΥ ΕΝ ΣΥΝΔΥΑΣΜΟ ΜΕ ΤΗ ΣΤΟΧΑΣΤΙΚΗ ΕΚΔΟΧΗ ΙΔΕΩΝ ΤΟΥ P.L. LIONS, ΕΞΑΣΦΑΛΙΖΟΥΝ ΜΙΑ ΚΑΘΑΡΑ ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΑΠΟΔΕΙΞΗ ΥΠΑΡΞΗΣ ΓΕΝΙΚΕΥΜΕΝΗΣ ΛΥΣΗΣ. Η ΜΕΘΟΔΟΣ,ΜΕ ΜΙΚΡΕΣ ΑΛΛΑΓΕΣ, ΕΙΝΑΙ ΚΑΤΑΛΛΗΛΗ ΓΙΑ ΤΗΝ ΜΙΓΑΔΙΚΗ ΕΞΙΣΩΣΗ MONGE-AMPERE. ΜΕ ΒΑΣΗ ΤΗ ΔΙΑΔΙΚΑΣΙΑ ΑΥΤΗ, ΜΠΟΡΕΙ ΝΑ ΚΑΤΑΣΚΕΥΑΣΤΕΙ ΕΝΑΣ Ε-ΒΕΛΤΙΣΤΟΣ ΕΛΕΓΧΟΣ ΓΙΑ ΤΟΠΡΟΚΥΠΤΟΝ ΠΡΟΒΛΗΜΑ ΣΤΟΧΑΣΤΙΚΟΥ ΕΛΕΓΧΟΥ ΟΤΑΝ FE LP(D), P>D."],"dc:identifier":["10.12681/eadd/0700","http://hdl.handle.net/10442/hedi/0700"],"dc:language":["gre"],"dc:publisher":["National Technical University of Athens (NTUA)","Εθνικό Μετσόβιο Πολυτεχνείο (ΕΜΠ)"],"dc:subject":["ΑΝΙΣΟΤΗΤΑ ΚΡΥΛΟΒ","ΓΕΝΙΚΕΥΜΕΝΗ ΛΥΣΗ","ΕΞΙΣΩΣΗ ΜΟΝΖ-ΑΜΠΕΡ","ΣΤΟΧΑΣΤΙΚΗ ΑΝΑΠΑΡΑΣΤΑΣΗ","ΣΤΟΧΑΣΤΙΚΟΣ ΒΕΛΤΙΣΤΟΣ ΕΛΕΓΧΟΣ","GENE LISED SOLUTION","KRYLOV'S INEQUALITY","MONGE-AMPERE EQUATION","STOCHASTIC OPTIMAL CONTROL","STOCHASTIC REPRESENTATION","Φυσικές Επιστήμες","Μαθηματικά","Natural Sciences","Mathematics"],"dc:title":["ΜΙΑ ΠΙΘΑΝΟΘΕΩΡΗΤΙΚΗ ΠΡΟΣΕΓΓΙΣΗ ΤΗΣ ΕΞΙΣΩΣΗΣ MONGE-AMPERE","A PROBABILISTIC APROACH TO THE MONGE-AMPERE EQUATION"],"dc:type":["PhD Thesis"]},"updated_at":"2026-07-24T02:25:24Z"}