Fat sets and pointwise boundary estimates for \(p\)-harmonic functions in metric spaces. (English) Zbl 1003.31004
Let \(X\) be a linear locally convex metric space and \(\mu\) be a doubling regular Borel measure on \(X\). Assume that \((X,\mu)\) admits a \((1,q_0)\) Poincaré inequality for some \(q_0\in [1,\infty)\). The main result of the paper says that a set \(E\subset X\) which is uniformly \(p\)-fat for some \(p> q_0\) is also uniformly \(q\)-fat for some \(q<p\). Here, uniform \(p\)-fatness means that the relative (w.r.t.\(B_{2r}(x)\)) \(p\)-capacities of \(B_r(x)\cap E\) and \(B_r(x)\) are uniformly bounded away from \(0\) for all \(x\in E\) and small \(r>0\). For first-order capacities this theorem extends an earlier result of J. Lewis [Trans. Am. Math. Soc. 308, 177-196 (1988; Zbl 0668.31002)] to a metric space setting. The proof follows P. Mikkonen’s technique [Ann. Acad. Sci. Fenn., Math., Diss. 104 (1996; Zbl 0860.35041)] and uses Wolff potentials rather than representation formulae associated with the Green functions. As a corollary, a Hardy-type inequality for Newtonian functions with zero boundary values for domains with uniformly fat complements is obtained.
In the course of the proof the following interesting characterization of Radon measures in the dual of the Newtonian space of functions on \(\Omega\subset X\) with zero boundary values, \(N_0^{1,p}(\Omega)^*\), is obtained: \(\nu \in N_0^{1,p}(\Omega)^*\) if, and only if, for some \(r>0\) the Wolff potential \(W_p^\nu(x,r)\) satisfies \(\int_\Omega W_p^\nu(x,r) \nu(dx) < \infty\). Other spin-offs are estimates for the solutions of the non-linear equation \(-\text{div}(|Du|^{p-2}Du) = \nu\in N_0^{1,p}(\Omega)^*\) along with estimates for the oscillation of \(p\)-harmonic functions and \(p\)-energy minimizers near a boundary point.
In the course of the proof the following interesting characterization of Radon measures in the dual of the Newtonian space of functions on \(\Omega\subset X\) with zero boundary values, \(N_0^{1,p}(\Omega)^*\), is obtained: \(\nu \in N_0^{1,p}(\Omega)^*\) if, and only if, for some \(r>0\) the Wolff potential \(W_p^\nu(x,r)\) satisfies \(\int_\Omega W_p^\nu(x,r) \nu(dx) < \infty\). Other spin-offs are estimates for the solutions of the non-linear equation \(-\text{div}(|Du|^{p-2}Du) = \nu\in N_0^{1,p}(\Omega)^*\) along with estimates for the oscillation of \(p\)-harmonic functions and \(p\)-energy minimizers near a boundary point.
Reviewer: René L.Schilling (Brighton)
MSC:
| 31C45 | Other generalizations (nonlinear potential theory, etc.) |
| 31C15 | Potentials and capacities on other spaces |
| 35G30 | Boundary value problems for nonlinear higher-order PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
analysis on metric spaces; uniformly fat set; Poincaré inequality; Hardy inequality; \(p\)-Laplacian; \(p\)-harmonic function; Wolff potentialReferences:
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