On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data
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Nonlinear Differential Equations and Applications NoDEA
On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data Tomasz Klimsiak Abstract. We propose a new definition of renormalized solution to linear equation with self-adjoint operator generating a Markov semigroup and bounded Borel measure on the right-hand side. We give a uniqueness result and study the structure of solutions to truncated equations. Mathematics Subject Classification. Primary 35R06; Secondary 35R05, 45K05, 47G20, 35D99. Keywords. Renormalized solution, Dirichlet form, Measure data, Markov semigroup, Markov process, Green function.
1. Introduction In this paper, E is a locally compact separable metric space and m is a Radon measure on E with full support. Let (A, D(A)) be a non-positive definite selfadjoint operator on L2 (E; m) associated with some Dirichlet form (E, D(E)) on L2 (E; m). The main goal of the present paper is to give a new definition of renormalized solution to the linear equation − Au = μ
(1.1)
with general (possibly nonsmooth in the Dirichlet forms theory sense) bounded Borel measure μ on E. It is known that such a measure admits unique decomposition μ = μd + μc
(1.2)
into the absolutely continuous, with respect to the capacity Cap generated by (E, D(E)), part μd (so-called diffuse part or smooth part of μ) and the orthogonal, with respect to Cap, part μc (so-called concentrated part). The problem of right definition to (1.1) is rather subtle if we require that the solution u be unique. 0123456789().: V,-vol
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T. Klimsiak
NoDEA
In the paper, we assume that the resolvent (Rα )α>0 generated by A is Fellerian, i.e. Rα (Cb (E)) ⊂ Cb (E) for some (and hence for all) α > 0, and there exists a Green function G for A (see Sect. 2.2). Our new definition reads as follows: u ∈ B(E) is a renormalized solution to (1.1) if (i) Tk (u) := (u ∧ k) ∨ (−k) ∈ De (E), k ≥ 0, where De (E) is the extended Dirichlet space, i.e. an extension of D(E) such that De (E) with inner product E is a Hilbert space. (ii) There exists a family of bounded smooth measures (νk )k≥0 on E such that E(Tk (u), η) = μd , η + νk , η,
η ∈ De (E) ∩ Bb (E),
k ≥ 0,
(iii) νk → μc in the narrow topology, i.e. for every η ∈ Cb (E), η dνk = η dμc . lim k→∞
E
E
A similar definition of a solution to (1.1), also guaranteeing uniqueness, was introduced recently in my joint paper with Rozkosz [10]. In that paper by a solution we mean u ∈ B(E) satisfying (i) and (ii), and the following condition (iii ) limk→∞ E G(x, y) νk (dy) = E G(x, y) μc (dy) for m-a.e. x ∈ E. Condition (iii) is much simpler than (iii ) because it does not involves the notion of the Green function. One of the main results of the present paper says that (i)–(iii) still ensure uniqueness for solutions to (1.1). In the second part of the paper we show interesting properties of the family (νk )k≥0 : a structure theorem, so-called reconstruction property and the narrow convergence of variations. The above definition (i)–(iii) is a
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