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Solutions of Semilinear Differential Equations Related to Harmonic Functions

โœ Scribed by E.B. Dynkin

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
141 KB
Volume
170
Category
Article
ISSN
0022-1236

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โœฆ Synopsis

This is an attempt to establish a link between positive solutions of semilinear equations Lu=& (u) and Lv= (v) where L is a second order elliptic differential operator and is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of R d and let U(D), V(D) and H(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1 1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1].

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