caa a22 4500 605495971 CHVBK 20210128100534.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s10231-013-0364-4 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0364-4 Klimsiak Tomasz Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland aut Semilinear elliptic systems with measure data [Elektronische Daten] [Tomasz Klimsiak] We study the Dirichlet problem for systems of the form $$-\varDelta u^k=f^k(x,u)+\mu ^k,\,x\in \varOmega ,\,k=1,\ldots ,n$$ - Δ u k = f k ( x , u ) + μ k , x ∈ Ω , k = 1 , ... , n , where $$\varOmega \subset \mathbb{R }^d$$ Ω ⊂ R d is an open (possibly nonregular) bounded set, $$\mu ^1,\ldots ,\mu ^n$$ μ 1 , ... , μ n are bounded diffuse measures on $$\varOmega ,\,f=(f^1,\ldots ,f^n)$$ Ω , f = ( f 1 , ... , f n ) satisfies some mild integrability condition and the so-called angle condition. Using the methods of probabilistic Dirichlet forms theory, we show that the system has a unique solution in the generalized Sobolev space $$\dot{H}^{1}_\mathrm{loc}(\varOmega )$$ H ˙ loc 1 ( Ω ) of functions having fine gradient. We also provide a stochastic representation of the solution. The Author(s), 2013 Semilinear elliptic systems nationallicence Laplacian nationallicence Dirichlet problem nationallicence Measure data nationallicence Dirichlet form nationallicence Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/1(2015-02-01), 55-76 0373-3114 194:1<55 2015 194 10231 https://doi.org/10.1007/s10231-013-0364-4 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0364-4 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Klimsiak Tomasz Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/1(2015-02-01), 55-76 0373-3114 194:1<55 2015 194 10231