arXiv subjects
Publications and source records attributed to Frank Wikström.
Let $B$ denote Bloch's constant. We prove \[ B \ge \frac{\sqrt{3}}{4}+0.0153 = 0.448312701\ldots, \] improving the lower bounds by Chen--Gauthier ($\sqrt{3}/4+2\cdot10^{-4}$) and Xiong ($\sqrt{3}/4+3\cdot10^{-4}$). The proof refines Bonk's method through computer-assisted estimates rigorously verified using interval arithmetic.
Let $f$ be an upper semicontinuous function on a domain and consider the Perron envelope formed only with globally bounded-above subharmonic, or plurisubharmonic, minorants of $f$. Its upper semicontinuous regularization agrees with the envelope outside a polar, respectively pluripolar, set, but it is not immediate whether regularization is actually necessary. This question arises naturally when comparing quasiboundedness with quasiboundedness quasi-everywhere. We prove that in classical potential theory the restricted envelope is automatically upper semicontinuous, provided its regularization is subharmonic. The proof uses a local harmonic correction obtained from Brelot's resolutivity theorem. We then show that the corresponding pluripotential statement fails sharply by constructing a bounded B-regular complete Hartogs domain in $\mathbb{C}^2$ and a positive, continuous, unbounded pluriharmonic function $W$ whose envelope of bounded plurisubharmonic minorants is discontinuous along an analytic disc. The function $W$ nevertheless admits a positive plurisuperharmonic majorant growing faster than $W$, and it is an increasing limit of bounded plurisubharmonic functions outside a pluripolar set. Thus the exceptional pluripolar set cannot in general be removed, even under these strong growth and approximation properties and on a B-regular domain.
We introduce a moment-duality method for Korenblum's maximum principle in the Bergman space $A^2(\mathbb{D})$. Starting from an annular coefficient estimate of Wang, we show that admissibility of a constant~$c$ follows from the existence of a probability measure on $[c^2,1]$ whose ordinary and weighted moments lie on opposite sides of the Bergman moments $1/(k+1)$. This converts the norm comparison into a positive moment problem. We then give an explicit measure, consisting of eight atoms with rational data and Lebesgue measure on a terminal interval, for which the required inequalities admit a rigorous ball-arithmetic certificate. Consequently, \[ c_2\geq 0.4263, \] improving Wang's recent lower bound $c_2\geq0.3554$.
We develop an inhomogeneous form of Edwards' Jensen-measure duality for Perron envelopes constrained by Monge--Amp\`ere lower bounds. The admissible subsolution families are convex but not cones; nevertheless, the dual measures remain the homogeneous Jensen measures, and the right-hand side enters through a scalar Jensen deficit \[ B_{\mathcal{A}}(x,\mu) = \inf_{u\in\mathcal{A}} \left(\int_{\partial\Omega}u\,d\mu-u(x)\right). \] Under natural structural hypotheses we prove a boundary dual formula \[ \sup\{u(x):u\in\mathcal{A},\ u\leq\varphi\text{ on }E\} = \inf_{\mu\in J_x^\partial} \left( \int_{\partial\Omega}\varphi\,d\mu - B_{\mathcal{A}}(x,\mu) \right). \] We apply the theorem to real Alexandrov subsolutions and to complex Bedford--Taylor plurisubharmonic subsolutions with continuous density. In one real dimension the deficit is the Green-potential correction; in higher dimensions it has intrinsic stress and current interpretations. On B-regular domains, a bounded Bedford--Taylor approximation theorem identifies bounded and continuous competitors and yields a duality proof of continuity for the corresponding Dirichlet solution. Finally, for smooth strictly elliptic solutions, optimal Jensen measures are the harmonic measures of the linearized Monge--Amp\`ere operators, equivalently the boundary derivatives of the nonlinear solution map.
In this paper, we introduce a new numerical algorithm for solving the Dirichlet problem for the real Monge--Ampere equation. The idea is to represent the non-linear Monge--Ampere operator as an infimum of a class of linear elliptic operators and use Bellman's principle to construct a numeric scheme for approximating the operator attaining this infimum. Moreover, we prove convergence of the proposed algorithm (under suitable technical assumptions) and discuss its strengths and weaknesses. We also demonstrate the performance of the method on several examples with various degrees of regularity and degeneracy and compare the results to two existing methods. Our method runs considerably faster than the ones used for comparison, improving the running time by a factor of 3--10 for smooth, strictly convex examples, and by a factor of 20--100 or more for mildly degenerate examples.
We discuss two variations of Edwards' duality theorem. More precisely, we prove one version of the theorem for cones not necessarily containing all constant functions. In particular, we allow the functions in the cone to have a non-empty common zero set. In the second variation, we replace suprema of point evaluations and infima over Jensen measures by suprema of other continuous functionals and infima over a set measures defined through a natural order relation induced by the cone. As applications, we give some results on propagation of discontinuities for Perron--Bremermann envelopes in hyperconvex domains as well as a characterization of minimal elements in the order relation mentioned above.
We extend the notion of quasibounded harmonic functions to the plurisubharmonic setting. As an application, using the theory of Jensen measures, we show that certain generalized Dirichlet problems with unbounded boundary data admit unique solutions, and that these solutions are continuous outside a pluripolar set.
For an open set $V\subset\mathbb{C}^n$, denote by $\mathscr{M}_α(V)$ the family of $α$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $Ω\subset \mathbb{C}^n$, with continuous boundary (that in each variable separately allows a solution to the Dirichlet problem), a function $f \in \mathscr{M}_α(Ω\setminus f^{-1}(0))$ automatically satisfies $f\in \mathscr{M}_α(Ω)$, if it is $C^{α_j-1}$-smooth, in the $z_j$ variable, $α\in \mathbb{Z}^n_+$, up to the boundary. For a submanifold $U\subset \mathbb{C}^n$, denote by $\mathfrak{M}_α(U)$ the set of functions locally approximable by $α$-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a $C^3$-smooth hypersurface, $Ω$, a member of $\mathfrak{M}_α(Ω)$, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form.