We extend Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity. We apply this to the interpolation problem of convex functions and convex sets introducing a new notion ...of “harmonic interpolation” that we view as a generalization of Minkowski-addition.
Let \Omega \subset \mathbb{R}^n be a convex domain, and let f:\Omega \rightarrow \mathbb{R} be a subharmonic function, \Delta f \geq 0, which satisfies f \geq 0 on the boundary \partial \Omega . Then ...<TD NOWRAP ALIGN="CENTER">\displaystyle \int _{\Omega }{f ~dx} \leq \vert\Omega \vert^{\frac {1}{n}} \int _{\partial \Omega }{f ~d\sigma }. <TD NOWRAP CLASS="eqno" WIDTH="10" ALIGN="RIGHT"> Our proof is based on a new gradient estimate for the torsion function, \Delta u = -1 with Dirichlet boundary conditions, which is of independent interest.
In the planar setting, the Radó–Kneser–Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary ...mapping is a homeomorphism. We prove the injectivity criterion of Radó–Kneser–Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injectivity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expression that is related to the Jacobian.
In this paper we calculate the norm of the bounded composition operator on the Fock space over
. This verifies a conjecture posed in Carswell B, MacCluer B, Schuster A. Composition operators on the ...Fock space. Acta Sci Math (Szeged). 2003;69:871-887.
Suppose that a holomorphic function f in a domain D in \mathbb{C}^n satisfies \vert f\vert\leq e^M on D (pointwise), where M\not \equiv -\infty is subharmonic function on D with Riesz measure \nu _M. ...Various methods are described for construction of wide classes of subharmonic test functions. These are subharmonic functions nonnegative and bounded on D\setminus S_0 for some compact set S_0\subset D that have zero limit on the boundary of D. If the integral of such a test function over D\setminus S_0 with respect to the Riesz measure \nu _M is finite and its integral with respect to the (2n-2)-dimensional Hausdorff measure over the zero set of f is infinite, f\equiv 0 on D. Thus, any new test function yields a uniqueness theorem.
In this note subharmonic and plurisubharmonic functions on a complex space are studied intrinsically. For applications subharmonicity is characterized more effectually in terms of properties that ...need be verified only locally off a thin analytic subset; these include the submean-value inequalities, the spherical (respectively, solid) monotonicity, near as well as weak subharmonicity. Several results of Gunning 9, K and L are extendable via regularity to complex spaces. In particular, plurisubharmonicity amounts (on a normal space) essentially to regularized weak plurisubharmonicity, and similarly for subharmonicity (on a general space). A generalized Hartogs’ lemma and constancy criteria for certain matrix-valued mappings are given.
We study solutions of Euler–Lagrange equations for isotropic energy functionals, generalizing a previous result on
-harmonic mappings. We classify all stored energy functions which give rise to a ...first-order differential expression whose Laplacian involves no third derivatives of the stationary solution. This classification gives rise to a new technique of finding subharmonicity results for the variational equations, and we also illustrate this technique in two examples. Firstly, we prove a subharmonicity result for the Jacobian determinant in the case of weighted Dirichlet energy. Secondly, we find optimal subharmonicity results in the case of a Neohookean-type stored energy function.
The model of free charged Bosons in an external constant magnetic field inside a cylinder, one of the few locally gauge covariant systems amenable to analytic treatment, is rigorously investigated in ...the semiclassical approximation. The model was first studied by Schafroth and is suitable for the description of quasi-bound electron pairs localized in physical space, the so-called Schafroth pairs, which occur in certain compounds. Under the assumption of existence of a solution of the semiclassical problem for which the ground state (g.s.) expectation value of the current
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