Naslov
Semilinear equations for non-local operators

Autori
Biočić, Ivan

Vrsta, podvrsta i kategorija rada
Ocjenski radovi, doktorska disertacija

Fakultet
Prirodoslovno-matematički fakultet

Mjesto
Zagreb

Datum
23.09

Godina
2022

Stranica
184

Mentor
Vondraček, Zoran

Ključne riječi
semilinear differential equations, non-local operators, killed subordinate Brownian motion, subordinate killed Brownian motion, harmonic functions

Sažetak
This doctoral thesis deals with semilinear equations for non-local operators in bounded domains in higher dimensions: For a bounded domain D ⊂ R^d, d ≥ 2, a non-local operator L, and a function f : D×R → R, the following problem is being solved Lu(x) = f(x, u(x)), x ∈ D, (1), where we also impose boundary conditions in D^c and/or on ∂D, depending on the type of the non-local operator L. The first type of non-local operators that are studied are infinitesimal generators of transient subordinate Brownian motions. Here the domain D can be any bounded domain and we impose boundary conditions both in D^c and on ∂D. A Martin representation formula for non-negative generalized harmonic functions is proved and a new type of boundary trace operator for this non-local setting is developed. A solution to problem (1) is found for a large class of functions f and conditions on f are given such that there is no solution to (1). The second type of non-local operators that are observed are infinitesimal generators of subordinate killed Brownian motions. Here a smoothness assumption on the boundary of the domain D is imposed as well as the boundary condition on ∂D in addition to (1). An integral representation formula for non-negative harmonic functions is given and also an equivalence between non-negative harmonic functions and non-negative functions that satisfy a certain mean-value property with respect to the underlying subordinate killed Brownian motion is established. A solution to the problem (1) is found for a large class of functions f and conditions on f are given such that there is no solution to (1).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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