Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

Tomasz Klimsiak, and Andrzej Rozkosz. "Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms." Colloquium Mathematicae 145.1 (2016): 35-67. <http://eudml.org/doc/286342>.

@article{TomaszKlimsiak2016,
abstract = {We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where L corresponds to a lower bounded semi-Dirichlet form.},
author = {Tomasz Klimsiak, Andrzej Rozkosz},
journal = {Colloquium Mathematicae},
keywords = {semilinear elliptic equation; measure data; Dirichlet form; backward stochastic differential equation},
language = {eng},
number = {1},
pages = {35-67},
title = {Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms},
url = {http://eudml.org/doc/286342},
volume = {145},
year = {2016},
}

TY - JOUR
AU - Tomasz Klimsiak
AU - Andrzej Rozkosz
TI - Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms
JO - Colloquium Mathematicae
PY - 2016
VL - 145
IS - 1
SP - 35
EP - 67
AB - We are mainly concerned with equations of the form -Lu = f(x,u) + μ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and μ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where L corresponds to a lower bounded semi-Dirichlet form.
LA - eng
KW - semilinear elliptic equation; measure data; Dirichlet form; backward stochastic differential equation
UR - http://eudml.org/doc/286342
ER -