{"product_id":"9780821835098","title":"Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation","description":"\u003cp\u003eWe are concerned with the nonnegative solutions of $\\Delta u = u^2$ in a bounded and smooth domain in $\\mathbb{R}^d$. We prove that they are uniquely determined by their fine trace on the boundary as defined in [DK98a], thus answering a major open question of [Dy02]. In this title, a probabilistic formula for a solution in terms of its fine trace and of the Brownian snake is also provided. A major role is played by the solutions which are dominated by a harmonic function in $D$. The latters are called moderate in Dynkin's terminology. We show that every nonnegative solution of $\\Delta u = u^2$ in $D$ is the increasing limit of moderate solutions.\u003c\/p\u003e","brand":"Exclusive Books Online","offers":[{"title":"Default Title","offer_id":52684773753150,"sku":"9780821835098","price":0.0,"currency_code":"ZAR","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4357\/7662\/files\/9780821835098.jpg?v=1764858473","url":"https:\/\/exclusivebooks.co.za\/products\/9780821835098","provider":"Exclusive Books Online","version":"1.0","type":"link"}