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Soap Film Solutions to Plateau’s Problem

Plateau’s problem is to show the existence of an area-minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area-minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau’s problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincare Lemma, and good compactness properties to solve Plateau’s problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Mobius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincare Lemma, and good compactness properties of the complex to solve Plateau’s problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimension-one surfaces in ℝ n .

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Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Let \(\mathcal{X}\) be a metric space with doubling measure and L a nonnegative self-adjoint operator in \(L^{2}(\mathcal{X})\) satisfying the Davies–Gaffney estimates. Let \(\varphi:\mathcal{X}\times[0,\infty)\to[0,\infty)\) be a function such that φ(x,⋅) is an Orlicz function, \(\varphi(\cdot,t)\in\mathbb{A}_{\infty}(\mathcal{X})\) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(φ)∈(0,1], and it satisfies the uniformly reverse Hölder inequality of order 2/[2−I(φ)]. In this paper, the authors introduce a Musielak–Orlicz–Hardy space \(H_{\varphi,L}(\mathcal{X})\), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak–Orlicz BMO-type space \(\mathrm{BMO}_{\varphi,L}(\mathcal{X})\), which is further proved to be the dual space of \(H_{\varphi,L}(\mathcal{X})\) and hence whose φ-Carleson measure characterization is deduced. Characterizations of \(H_{\varphi,L}(\mathcal{X})\), including the atom, the molecule, and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize \(H_{\varphi,L}(\mathcal{X})\) in terms of the Littlewood–Paley \(g^{\ast}_{\lambda}\)-function \(g^{\ast}_{\lambda,L}\) and establish a Hörmander-type spectral multiplier theorem for L on \(H_{\varphi,L}(\mathcal{X})\). Moreover, for the Musielak–Orlicz–Hardy space H φ,L (ℝn) associated with the Schrödinger operator L:=−Δ+V, where \(0\le V\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom, and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H φ,L (ℝn) to the Musielak–Orlicz space L φ(ℝn) when i(φ)∈(0,1], and from H φ,L (ℝn) to the Musielak–Orlicz–Hardy space H φ (ℝn) when \(i(\varphi)\in(\frac{n}{n+1},1]\), where i(φ) denotes the uniformly critical lower type index of φ.

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