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A Wiener estimate for relaxed Dirichlet problems in dimension \(N\geq 2\). (English) Zbl 0815.35013

The author considers Dirichlet problems \(Lu + \mu u = \nu\) in \(\Omega\), \(L\) uniformly elliptic, \(\mu\) a nonnegative Borel measure on \(\Omega\) such that \(\mu (E) = 0\) for every Borel subset \(E\) of \(\Omega\) of capacity zero, and \(\nu\) a Kato measure, i.e. \[ \lim_{r \to o^ +} \sup_{x \in \Omega} \int_{\Omega \cap B_ y(x)} | x - y |^{2-N} d | \nu | (y) = 0 \quad \text{for} \quad N \geq 3 \] (and the usual kernel \(\log {1 \over (y - x)}\) if \(N = 2)\).
The author refines and completes, with a different technique, results of G. Dal Maso and U. Mosco [Arch. Ration. Mech. Anal. 95, 345- 387 (1986; Zbl 0634.35033)] concerning the characterization of regular Dirichlet points of \(\mu\), i.e. points \(x_ 0\) such that every local weak solution is continuous at \(x_ 0\). Furthermore, the modulus of continuity and the energy decay is estimated in terms of the Wiener modulus.
Reviewer: J.Frehse (Bonn)

MSC:

35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data

Citations:

Zbl 0634.35033